Daily Papers

Modern deep learning-based recommendation systems exploit hundreds to thousands of different categorical features, each with millions of different categories ranging from clicks to posts. To respect the natural diversity within the categorical data, embeddings map each category to a unique dense representation within an embedded space. Since each categorical feature could take on as many as tens of millions of different possible categories, the embedding tables form the primary memory bottleneck during both training and inference. We propose a novel approach for reducing the embedding size in an end-to-end fashion by exploiting complementary partitions of the category set to produce a unique embedding vector for each category without explicit definition. By storing multiple smaller embedding tables based on each complementary partition and combining embeddings from each table, we define a unique embedding for each category at smaller memory cost. This approach may be interpreted as using a specific fixed codebook to ensure uniqueness of each category's representation. Our experimental results demonstrate the effectiveness of our approach over the hashing trick for reducing the size of the embedding tables in terms of model loss and accuracy, while retaining a similar reduction in the number of parameters.

We address the neural network robustness problem by adding Similarity (i.e., correctly predicted depth-matches into training)-awareness and Distance-to-training-distribution-awareness to the existing output Magnitude (i.e., decision-boundary)-awareness of the softmax function. The resulting SDM activation function provides strong signals of the relative epistemic (reducible) predictive uncertainty. We use this novel behavior to further address the complementary HCI problem of mapping the output to human-interpretable summary statistics over relevant partitions of a held-out calibration set. Estimates of prediction-conditional uncertainty are obtained via a parsimonious learned transform over the class-conditional empirical CDFs of the output of a final-layer SDM activation function. For decision-making and as an intrinsic model check, estimates of class-conditional accuracy are obtained by further partitioning the high-probability regions of this calibrated output into class-conditional, region-specific CDFs. The uncertainty estimates from SDM calibration are remarkably robust to test-time distribution shifts and out-of-distribution inputs; incorporate awareness of the effective sample size; provide estimates of uncertainty from the learning and data splitting processes; and are well-suited for selective classification and conditional branching for additional test-time compute based on the predictive uncertainty, as for selective LLM generation, routing, and composition over multiple models and retrieval. Finally, we construct SDM networks, LLMs with uncertainty-aware verification and interpretability-by-exemplar as intrinsic properties. We provide open-source software implementing these results.

Embedding based product recommendations have gained popularity in recent years due to its ability to easily integrate to large-scale systems and allowing nearest neighbor searches in real-time. The bulk of studies in this area has predominantly been focused on similar item recommendations. Research on complementary item recommendations, on the other hand, still remains considerably under-explored. We define similar items as items that are interchangeable in terms of their utility and complementary items as items that serve different purposes, yet are compatible when used with one another. In this paper, we apply a novel approach to finding complementary items by leveraging dual embedding representations for products. We demonstrate that the notion of relatedness discovered in NLP for skip-gram negative sampling (SGNS) models translates effectively to the concept of complementarity when training item representations using co-purchase data. Since sparsity of purchase data is a major challenge in real-world scenarios, we further augment the model using synthetic samples to extend coverage. This allows the model to provide complementary recommendations for items that do not share co-purchase data by leveraging other abundantly available data modalities such as images, text, clicks etc. We establish the effectiveness of our approach in improving both coverage and quality of recommendations on real world data for a major online retail company. We further show the importance of task specific hyperparameter tuning in training SGNS. Our model is effective yet simple to implement, making it a great candidate for generating complementary item recommendations at any e-commerce website.

Two principles: the complementary principle and the consensus principle are widely acknowledged in the literature of multi-view learning. However, the current design of multi-head self-attention, an instance of multi-view learning, prioritizes the complementarity while ignoring the consensus. To address this problem, we propose an enhanced multi-head self-attention (EMHA). First, to satisfy the complementary principle, EMHA removes the one-to-one mapping constraint among queries and keys in multiple subspaces and allows each query to attend to multiple keys. On top of that, we develop a method to fully encourage consensus among heads by introducing two interaction models, namely inner-subspace interaction and cross-subspace interaction. Extensive experiments on a wide range of language tasks (e.g., machine translation, abstractive summarization and grammar correction, language modeling), show its superiority, with a very modest increase in model size. Our code would be available at: https://github.com/zhengkid/EIT-Enhanced-Interactive-Transformer.

The detailed images produced by Magnetic Resonance Imaging (MRI) provide life-critical information for the diagnosis and treatment of prostate cancer. To provide standardized acquisition, interpretation and usage of the complex MRI images, the PI-RADS v2 guideline was proposed. An automated segmentation following the guideline facilitates consistent and precise lesion detection, staging and treatment. The guideline recommends a division of the prostate into four zones, PZ (peripheral zone), TZ (transition zone), DPU (distal prostatic urethra) and AFS (anterior fibromuscular stroma). Not every zone shares a boundary with the others and is present in every slice. Further, the representations captured by a single model might not suffice for all zones. This motivated us to design a dual-branch convolutional neural network (CNN), where each branch captures the representations of the connected zones separately. Further, the representations from different branches act complementary to each other at the second stage of training, where they are fine-tuned through an unsupervised loss. The loss penalises the difference in predictions from the two branches for the same class. We also incorporate multi-task learning in our framework to further improve the segmentation accuracy. The proposed approach improves the segmentation accuracy of the baseline (mean absolute symmetric distance) by 7.56%, 11.00%, 58.43% and 19.67% for PZ, TZ, DPU and AFS zones respectively.

This is an essay in what might be called ``mathematical metaphysics.'' There is a fundamental duality that run through mathematics and the natural sciences. The duality starts as the logical level; it is represented by the Boolean logic of subsets and the logic of partitions since subsets and partitions are category-theoretic dual concepts. In more basic terms, it starts with the duality between the elements (Its) of subsets and the distinctions (Dits, i.e., ordered pairs of elements in different blocks) of a partition. Mathematically, the Its & Dits duality is fully developed in category theory as the reverse-the-arrows duality. The quantitative versions of subsets and partitions are developed as probability theory and information theory (based on logical entropy). Classical physics was based on a view of reality as definite all the way down. In contrast, quantum physics embodies (objective) indefiniteness. And finally, there are the two fundamental dual mechanisms at work in biology, the selectionist mechanism and the generative mechanism, two mechanisms that embody the fundamental duality.

Let T be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in T is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in R^d. We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a rm polylog(Delta)-approximation algorithm for embedding n-point ultrametrics into R^d with minimum distortion, where Delta denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in n. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.

Complementary-label learning (CLL) is a weakly-supervised learning paradigm that aims to train a multi-class classifier using only complementary labels, which indicate classes to which an instance does not belong. Despite numerous algorithmic proposals for CLL, their practical applicability remains unverified for two reasons. Firstly, these algorithms often rely on assumptions about the generation of complementary labels, and it is not clear how far the assumptions are from reality. Secondly, their evaluation has been limited to synthetically labeled datasets. To gain insights into the real-world performance of CLL algorithms, we developed a protocol to collect complementary labels from human annotators. Our efforts resulted in the creation of four datasets: CLCIFAR10, CLCIFAR20, CLMicroImageNet10, and CLMicroImageNet20, derived from well-known classification datasets CIFAR10, CIFAR100, and TinyImageNet200. These datasets represent the very first real-world CLL datasets, namely CLImage, which are publicly available at: https://github.com/ntucllab/CLImage\_Dataset. Through extensive benchmark experiments, we discovered a notable decrease in performance when transitioning from synthetically labeled datasets to real-world datasets. We investigated the key factors contributing to the decrease with a thorough dataset-level ablation study. Our analyses highlight annotation noise as the most influential factor in the real-world datasets. In addition, we discover that the biased-nature of human-annotated complementary labels and the difficulty to validate with only complementary labels are two outstanding barriers to practical CLL. These findings suggest that the community focus more research efforts on developing CLL algorithms and validation schemes that are robust to noisy and biased complementary-label distributions.

In this paper, we investigate the challenges of complementary-label learning (CLL), a specialized form of weakly-supervised learning (WSL) where models are trained with labels indicating classes to which instances do not belong, rather than standard ordinary labels. This alternative supervision is appealing because collecting complementary labels is generally cheaper and less labor-intensive. Although most existing research in CLL emphasizes the development of novel loss functions, the potential of data augmentation in this domain remains largely underexplored. In this work, we uncover that the widely-used Mixup data augmentation technique is ineffective when directly applied to CLL. Through in-depth analysis, we identify that the complementary-label noise generated by Mixup negatively impacts the performance of CLL models. We then propose an improved technique called Intra-Cluster Mixup (ICM), which only synthesizes augmented data from nearby examples, to mitigate the noise effect. ICM carries the benefits of encouraging complementary label sharing of nearby examples, and leads to substantial performance improvements across synthetic and real-world labeled datasets. In particular, our wide spectrum of experimental results on both balanced and imbalanced CLL settings justifies the potential of ICM in allying with state-of-the-art CLL algorithms, achieving significant accuracy increases of 30% and 10% on MNIST and CIFAR datasets, respectively.

For the immanent challenge of insufficiently annotated samples in the medical field, semi-supervised medical image segmentation (SSMIS) offers a promising solution. Despite achieving impressive results in delineating primary target areas, most current methodologies struggle to precisely capture the subtle details of boundaries. This deficiency often leads to significant diagnostic inaccuracies. To tackle this issue, we introduce C3S3, a novel semi-supervised segmentation model that synergistically integrates complementary competition and contrastive selection. This design significantly sharpens boundary delineation and enhances overall precision. Specifically, we develop an Outcome-Driven Contrastive Learning module dedicated to refining boundary localization. Additionally, we incorporate a Dynamic Complementary Competition module that leverages two high-performing sub-networks to generate pseudo-labels, thereby further improving segmentation quality. The proposed C3S3 undergoes rigorous validation on two publicly accessible datasets, encompassing the practices of both MRI and CT scans. The results demonstrate that our method achieves superior performance compared to previous cutting-edge competitors. Especially, on the 95HD and ASD metrics, our approach achieves a notable improvement of at least 6%, highlighting the significant advancements. The code is available at https://github.com/Y-TARL/C3S3.

Complementary-label learning (CLL) is a weakly supervised learning paradigm for multiclass classification, where only complementary labels -- indicating classes an instance does not belong to -- are provided to the learning algorithm. Despite CLL's increasing popularity, previous studies highlight two main challenges: (1) inconsistent results arising from varied assumptions on complementary label generation, and (2) high barriers to entry due to the lack of a standardized evaluation platform across datasets and algorithms. To address these challenges, we introduce libcll, an extensible Python toolkit for CLL research. libcll provides a universal interface that supports a wide range of generation assumptions, both synthetic and real-world datasets, and key CLL algorithms. The toolkit is designed to mitigate inconsistencies and streamline the research process, with easy installation, comprehensive usage guides, and quickstart tutorials that facilitate efficient adoption and implementation of CLL techniques. Extensive ablation studies conducted with libcll demonstrate its utility in generating valuable insights to advance future CLL research.

The notion of shortcut partition, introduced recently by Chang, Conroy, Le, Milenkovi\'c, Solomon, and Than [CCLMST23], is a new type of graph partition into low-diameter clusters. Roughly speaking, the shortcut partition guarantees that for every two vertices u and v in the graph, there exists a path between u and v that intersects only a few clusters. They proved that any planar graph admits a shortcut partition and gave several applications, including a construction of tree cover for arbitrary planar graphs with stretch 1+varepsilon and O(1) many trees for any fixed varepsilon in (0,1). However, the construction heavily exploits planarity in multiple steps, and is thus inherently limited to planar graphs. In this work, we breach the "planarity barrier" to construct a shortcut partition for K_r-minor-free graphs for any r. To this end, we take a completely different approach -- our key contribution is a novel deterministic variant of the cop decomposition in minor-free graphs [And86, AGG14]. Our shortcut partition for K_r-minor-free graphs yields several direct applications. Most notably, we construct the first optimal distance oracle for K_r-minor-free graphs, with 1+varepsilon stretch, linear space, and constant query time for any fixed varepsilon in (0,1). The previous best distance oracle [AG06] uses O(nlog n) space and O(log n) query time, and its construction relies on Robertson-Seymour structural theorem and other sophisticated tools. We also obtain the first tree cover of O(1) size for minor-free graphs with stretch 1+varepsilon, while the previous best (1+varepsilon)-tree cover has size O(log^2 n) [BFN19].

Clustering has been a major research topic in the field of machine learning, one to which Deep Learning has recently been applied with significant success. However, an aspect of clustering that is not addressed by existing deep clustering methods, is that of efficiently producing multiple, diverse partitionings for a given dataset. This is particularly important, as a diverse set of base clusterings are necessary for consensus clustering, which has been found to produce better and more robust results than relying on a single clustering. To address this gap, we propose DivClust, a diversity controlling loss that can be incorporated into existing deep clustering frameworks to produce multiple clusterings with the desired degree of diversity. We conduct experiments with multiple datasets and deep clustering frameworks and show that: a) our method effectively controls diversity across frameworks and datasets with very small additional computational cost, b) the sets of clusterings learned by DivClust include solutions that significantly outperform single-clustering baselines, and c) using an off-the-shelf consensus clustering algorithm, DivClust produces consensus clustering solutions that consistently outperform single-clustering baselines, effectively improving the performance of the base deep clustering framework.

We consider the problem of graph matching, or learning vertex correspondence, between two correlated stochastic block models (SBMs). The graph matching problem arises in various fields, including computer vision, natural language processing and bioinformatics, and in particular, matching graphs with inherent community structure has significance related to de-anonymization of correlated social networks. Compared to the correlated Erdos-Renyi (ER) model, where various efficient algorithms have been developed, among which a few algorithms have been proven to achieve the exact matching with constant edge correlation, no low-order polynomial algorithm has been known to achieve exact matching for the correlated SBMs with constant correlation. In this work, we propose an efficient algorithm for matching graphs with community structure, based on the comparison between partition trees rooted from each vertex, by extending the idea of Mao et al. (2021) to graphs with communities. The partition tree divides the large neighborhoods of each vertex into disjoint subsets using their edge statistics to different communities. Our algorithm is the first low-order polynomial-time algorithm achieving exact matching between two correlated SBMs with high probability in dense graphs.

Pairwise clustering, in general, partitions a set of items via a known similarity function. In our treatment, clustering is modeled as a transductive prediction problem. Thus rather than beginning with a known similarity function, the function instead is hidden and the learner only receives a random sample consisting of a subset of the pairwise similarities. An additional set of pairwise side-information may be given to the learner, which then determines the inductive bias of our algorithms. We measure performance not based on the recovery of the hidden similarity function, but instead on how well we classify each item. We give tight bounds on the number of misclassifications. We provide two algorithms. The first algorithm SACA is a simple agglomerative clustering algorithm which runs in near linear time, and which serves as a baseline for our analyses. Whereas the second algorithm, RGCA, enables the incorporation of side-information which may lead to improved bounds at the cost of a longer running time.

Finite unions of convex sets are a central object of study in discrete and computational geometry. In this paper we initiate a systematic study of complements of such unions -- i.e., sets of the form S=R^d setminus (cup_{i=1}^n K_i), where K_i are convex sets. In the first part of the paper we study isolated points in S, whose number is related to the Betti numbers of cup_{i=1}^n K_i and to its non-convexity properties. We obtain upper bounds on the number of such points, which are sharp for n=3 and significantly improve previous bounds of Lawrence and Morris (2009) for all n ll 2^d{d}. In the second part of the paper we study coverings of S by well-behaved sets. We show that S can be covered by at most g(d,n) flats of different dimensions, in such a way that each x in S is covered by a flat whose dimension equals the `local dimension' of S in the neighborhood of x. Furthermore, we determine the structure of a minimum cover that satisfies this property. Then, we study quantitative aspects of this minimum cover and obtain sharp upper bounds on its size in various settings.

Knowledge bases (KBs) and text often contain complementary knowledge: KBs store structured knowledge that can support long range reasoning, while text stores more comprehensive and timely knowledge in an unstructured way. Separately embedding the individual knowledge sources into vector spaces has demonstrated tremendous successes in encoding the respective knowledge, but how to jointly embed and reason with both knowledge sources to fully leverage the complementary information is still largely an open problem. We conduct a large-scale, systematic investigation of aligning KB and text embeddings for joint reasoning. We set up a novel evaluation framework with two evaluation tasks, few-shot link prediction and analogical reasoning, and evaluate an array of KB-text embedding alignment methods. We also demonstrate how such alignment can infuse textual information into KB embeddings for more accurate link prediction on emerging entities and events, using COVID-19 as a case study.

Model customization necessitates high-quality and diverse datasets, but acquiring such data remains time-consuming and labor-intensive. Despite the great potential of large language models (LLMs) for data synthesis, current approaches are constrained by limited seed data, model biases, and low-variation prompts, resulting in limited diversity and biased distributions with the increase of data scales. To tackle this challenge, we introduce TREESYNTH, a tree-guided subspace-based data synthesis approach inspired by decision trees. It constructs a spatial partitioning tree to recursively divide a task-specific full data space (i.e., root node) into numerous atomic subspaces (i.e., leaf nodes) with mutually exclusive and exhaustive attributes to ensure both distinctiveness and comprehensiveness before synthesizing samples within each atomic subspace. This globally dividing-and-synthesizing method finally collects subspace samples into a comprehensive dataset, effectively circumventing repetition and space collapse to ensure the diversity of large-scale data synthesis. Furthermore, the spatial partitioning tree enables sample allocation into atomic subspaces, allowing the rebalancing of existing datasets for more balanced and comprehensive distributions. Empirically, extensive experiments across diverse benchmarks consistently demonstrate the superior data diversity, model performance, and robust scalability of TREESYNTH compared to both human-crafted datasets and peer data synthesis methods, with an average performance gain reaching 10%. Besides, the consistent improvements of TREESYNTH-balanced datasets highlight its efficacious application to redistribute existing datasets for more comprehensive coverage and the induced performance enhancement. The code is available at https://github.com/cpa2001/TreeSynth.

Integrating a notion of symmetry into point cloud neural networks is a provably effective way to improve their generalization capability. Of particular interest are E(3) equivariant point cloud networks where Euclidean transformations applied to the inputs are preserved in the outputs. Recent efforts aim to extend networks that are E(3) equivariant, to accommodate inputs made of multiple parts, each of which exhibits local E(3) symmetry. In practical settings, however, the partitioning into individually transforming regions is unknown a priori. Errors in the partition prediction would unavoidably map to errors in respecting the true input symmetry. Past works have proposed different ways to predict the partition, which may exhibit uncontrolled errors in their ability to maintain equivariance to the actual partition. To this end, we introduce APEN: a general framework for constructing approximate piecewise-E(3) equivariant point networks. Our primary insight is that functions that are equivariant with respect to a finer partition will also maintain equivariance in relation to the true partition. Leveraging this observation, we propose a design where the equivariance approximation error at each layers can be bounded solely in terms of (i) uncertainty quantification of the partition prediction, and (ii) bounds on the probability of failing to suggest a proper subpartition of the ground truth one. We demonstrate the effectiveness of APEN using two data types exemplifying part-based symmetry: (i) real-world scans of room scenes containing multiple furniture-type objects; and, (ii) human motions, characterized by articulated parts exhibiting rigid movement. Our empirical results demonstrate the advantage of integrating piecewise E(3) symmetry into network design, showing a distinct improvement in generalization compared to prior works for both classification and segmentation tasks.

In this work, we propose Reciprocal Distribution Alignment (RDA) to address semi-supervised learning (SSL), which is a hyperparameter-free framework that is independent of confidence threshold and works with both the matched (conventionally) and the mismatched class distributions. Distribution mismatch is an often overlooked but more general SSL scenario where the labeled and the unlabeled data do not fall into the identical class distribution. This may lead to the model not exploiting the labeled data reliably and drastically degrade the performance of SSL methods, which could not be rescued by the traditional distribution alignment. In RDA, we enforce a reciprocal alignment on the distributions of the predictions from two classifiers predicting pseudo-labels and complementary labels on the unlabeled data. These two distributions, carrying complementary information, could be utilized to regularize each other without any prior of class distribution. Moreover, we theoretically show that RDA maximizes the input-output mutual information. Our approach achieves promising performance in SSL under a variety of scenarios of mismatched distributions, as well as the conventional matched SSL setting. Our code is available at: https://github.com/NJUyued/RDA4RobustSSL.

Tangles were originally introduced as a concept to formalize regions of high connectivity in graphs. In recent years, they have also been discovered as a link between structural graph theory and data science: when interpreting similarity in data sets as connectivity between points, finding clusters in the data essentially amounts to finding tangles in the underlying graphs. This paper further explores the potential of tangles in data sets as a means for a formal study of clusters. Real-world data often follow a normal distribution. Accounting for this, we develop a quantitative theory of tangles in data sets drawn from Gaussian mixtures. To this end, we equip the data with a graph structure that models similarity between the points and allows us to apply tangle theory to the data. We provide explicit conditions under which tangles associated with the marginal Gaussian distributions exist asymptotically almost surely. This can be considered as a sufficient formal criterion for the separabability of clusters in the data.

A spanning tree T of graph G is a rho-approximate universal Steiner tree (UST) for root vertex r if, for any subset of vertices S containing r, the cost of the minimal subgraph of T connecting S is within a rho factor of the minimum cost tree connecting S in G. Busch et al. (FOCS 2012) showed that every graph admits 2^{O(log n)}-approximate USTs by showing that USTs are equivalent to strong sparse partition hierarchies (up to poly-logs). Further, they posed poly-logarithmic USTs and strong sparse partition hierarchies as open questions. We settle these open questions by giving polynomial-time algorithms for computing both O(log ^ 7 n)-approximate USTs and poly-logarithmic strong sparse partition hierarchies. For graphs with constant doubling dimension or constant pathwidth we improve this to O(log n)-approximate USTs and O(1) strong sparse partition hierarchies. Our doubling dimension result is tight up to second order terms. We reduce the existence of these objects to the previously studied cluster aggregation problem and what we call dangling nets.

Let F_{k,d}(n) be the maximal size of a set {A}subseteq [n] such that the equation \[a_1a_2\dots a_k=x^d, \; a_1<a_2<\ldots<a_k\] has no solution with a_1,a_2,ldots,a_kA and integer x. Erdos, S\'ark\"ozy and T. S\'os studied F_{k,2}, and gave bounds when k=2,3,4,6 and also in the general case. We study the problem for d=3, and provide bounds for k=2,3,4,6 and 9, furthermore, in the general case, as well. In particular, we refute an 18 years old conjecture of Verstra\"ete. We also introduce another function f_{k,d} closely related to F_{k,d}: While the original problem requires a_1, ldots , a_k to all be distinct, we can relax this and only require that the multiset of the a_i's cannot be partitioned into d-tuples where each d-tuple consists of d copies of the same number.

Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree tree where the weight of each node is the sum of the weights of its children. A treemap for tree is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of tree and its area is proportional to the weight of the corresponding node. An important quality criterion for treemaps is the aspect ratio of its regions. One cannot bound the aspect ratio if the regions are restricted to be rectangles. In contrast, polygonal partitions, that use convex polygons, have bounded aspect ratio. We are the first to obtain convex partitions with optimal aspect ratio O(depth(tree)). However, depth(tree) still depends on the input tree. Hence we introduce a new type of treemaps, namely orthoconvex treemaps, where regions representing leaves are rectangles, L-, and S-shapes, and regions representing internal nodes are orthoconvex polygons. We prove that any input tree, irrespective of the weights of the nodes and the depth of the tree, admits an orthoconvex treemap of constant aspect ratio. We also obtain several specialized results for single-level treemaps, that is, treemaps where the input tree has depth~1.

For any given dimension d, all reflexive d-polytopes can be found (in principle) as subpolytopes of a number of maximal polyhedra that are defined in terms of (d+1)-tuples of integers (weights), or combinations of k-tuples of weights with k<d+1. We present the results of a complete classification of sextuples of weights pertaining to the construction of all reflexive polytopes in five dimensions. We find 322 383 760 930 such weight systems. 185 269 499 015 of them give rise directly to reflexive polytopes and thereby to mirror pairs of Calabi-Yau fourfolds. These lead to 532 600 483 distinct sets of Hodge numbers.

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.

We consider the problem of constructing small coresets for k-Median in Euclidean spaces. Given a large set of data points Psubset R^d, a coreset is a much smaller set Ssubset R^d, so that the k-Median costs of any k centers w.r.t. P and S are close. Existing literature mainly focuses on the high-dimension case and there has been great success in obtaining dimension-independent bounds, whereas the case for small d is largely unexplored. Considering many applications of Euclidean clustering algorithms are in small dimensions and the lack of systematic studies in the current literature, this paper investigates coresets for k-Median in small dimensions. For small d, a natural question is whether existing near-optimal dimension-independent bounds can be significantly improved. We provide affirmative answers to this question for a range of parameters. Moreover, new lower bound results are also proved, which are the highest for small d. In particular, we completely settle the coreset size bound for 1-d k-Median (up to log factors). Interestingly, our results imply a strong separation between 1-d 1-Median and 1-d 2-Median. As far as we know, this is the first such separation between k=1 and k=2 in any dimension.

Training deep networks requires various design decisions regarding for instance their architecture, data augmentation, or optimization. In this work, we find these training variations to result in networks learning unique feature sets from the data. Using public model libraries comprising thousands of models trained on canonical datasets like ImageNet, we observe that for arbitrary pairings of pretrained models, one model extracts significant data context unavailable in the other -- independent of overall performance. Given any arbitrary pairing of pretrained models and no external rankings (such as separate test sets, e.g. due to data privacy), we investigate if it is possible to transfer such "complementary" knowledge from one model to another without performance degradation -- a task made particularly difficult as additional knowledge can be contained in stronger, equiperformant or weaker models. Yet facilitating robust transfer in scenarios agnostic to pretrained model pairings would unlock auxiliary gains and knowledge fusion from any model repository without restrictions on model and problem specifics - including from weaker, lower-performance models. This work therefore provides an initial, in-depth exploration on the viability of such general-purpose knowledge transfer. Across large-scale experiments, we first reveal the shortcomings of standard knowledge distillation techniques, and then propose a much more general extension through data partitioning for successful transfer between nearly all pretrained models, which we show can also be done unsupervised. Finally, we assess both the scalability and impact of fundamental model properties on successful model-agnostic knowledge transfer.

Numerous capability and safety techniques of Large Language Models (LLMs), including RLHF, automated red-teaming, prompt engineering, and infilling, can be cast as sampling from an unnormalized target distribution defined by a given reward or potential function over the full sequence. In this work, we leverage the rich toolkit of Sequential Monte Carlo (SMC) for these probabilistic inference problems. In particular, we use learned twist functions to estimate the expected future value of the potential at each timestep, which enables us to focus inference-time computation on promising partial sequences. We propose a novel contrastive method for learning the twist functions, and establish connections with the rich literature of soft reinforcement learning. As a complementary application of our twisted SMC framework, we present methods for evaluating the accuracy of language model inference techniques using novel bidirectional SMC bounds on the log partition function. These bounds can be used to estimate the KL divergence between the inference and target distributions in both directions. We apply our inference evaluation techniques to show that twisted SMC is effective for sampling undesirable outputs from a pretrained model (a useful component of harmlessness training and automated red-teaming), generating reviews with varied sentiment, and performing infilling tasks.

We show how the Schur-Weyl duality that exists between the partition algebra and the symmetric group results in a stronger theoretical foundation for characterising all of the possible permutation equivariant neural networks whose layers are some tensor power of the permutation representation M_n of the symmetric group S_n. In doing so, we unify two separate bodies of literature, and we correct some of the major results that are now widely quoted by the machine learning community. In particular, we find a basis of matrices for the learnable, linear, permutation equivariant layer functions between such tensor power spaces in the standard basis of M_n by using an elegant graphical representation of a basis of set partitions for the partition algebra and its related vector spaces. Also, we show how we can calculate the number of weights that must appear in these layer functions by looking at certain paths through the McKay quiver for M_n. Finally, we describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.

We present GSPMD, an automatic, compiler-based parallelization system for common machine learning computations. It allows users to write programs in the same way as for a single device, then give hints through a few annotations on how to distribute tensors, based on which GSPMD will parallelize the computation. Its representation of partitioning is simple yet general, allowing it to express different or mixed paradigms of parallelism on a wide variety of models. GSPMD infers the partitioning for every operator based on limited user annotations, making it convenient to scale existing single-device programs. It solves several technical challenges for production usage, allowing GSPMD to achieve 50% to 62% compute utilization on up to 2048 Cloud TPUv3 cores for models with up to one trillion parameters.

Combining several (sample approximations of) distributions, which we term sub-posteriors, into a single distribution proportional to their product, is a common challenge. Occurring, for instance, in distributed 'big data' problems, or when working under multi-party privacy constraints. Many existing approaches resort to approximating the individual sub-posteriors for practical necessity, then find either an analytical approximation or sample approximation of the resulting (product-pooled) posterior. The quality of the posterior approximation for these approaches is poor when the sub-posteriors fall out-with a narrow range of distributional form, such as being approximately Gaussian. Recently, a Fusion approach has been proposed which finds an exact Monte Carlo approximation of the posterior, circumventing the drawbacks of approximate approaches. Unfortunately, existing Fusion approaches have a number of computational limitations, particularly when unifying a large number of sub-posteriors. In this paper, we generalise the theory underpinning existing Fusion approaches, and embed the resulting methodology within a recursive divide-and-conquer sequential Monte Carlo paradigm. This ultimately leads to a competitive Fusion approach, which is robust to increasing numbers of sub-posteriors.

We study the H-chromatic symmetric functions X_G^H (introduced in (arXiv:2011.06063) as a generalization of the chromatic symmetric function (CSF) X_G), which track homomorphisms from the graph G to the graph H. We focus first on the case of self-chromatic symmetric functions (self-CSFs) X_G^G, making some progress toward a conjecture from (arXiv:2011.06063) that the self-CSF, like the normal CSF, is always different for different trees. In particular, we show that the self-CSF distinguishes trees from non-trees with just one exception, we check using Sage that it distinguishes all trees on up to 12 vertices, and we show that it determines the number of legs of a spider and the degree sequence of a caterpillar given its spine length. We also show that the self-CSF detects the number of connected components of a forest, again with just one exception. Then we prove some results about the power sum expansions for H-CSFs when H is a complete bipartite graph, in particular proving that the conjecture from (arXiv:2011.06063) about p-monotonicity of ω(X_G^H) for H a star holds as long as H is sufficiently large compared to G. We also show that the self-CSFs of complete multipartite graphs form a basis for the ring Λ of symmetric functions, and we give some construction of bases for the vector space Λ^n of degree n symmetric functions using H-CSFs X_G^H where H is a fixed graph that is not a complete graph, answering a question from (arXiv:2011.06063) about whether such bases exist. However, we show that there generally do not exist such bases with G fixed, even with loops, answering another question from (arXiv:2011.06063). We also define the H-chromatic polynomial as an analogue of the chromatic polynomial, and ask when it is the same for different graphs.

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock's synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.

We study the problem of detecting the correlation between two Gaussian databases XinR^{ntimes d} and Y^{ntimes d}, each composed of n users with d features. This problem is relevant in the analysis of social media, computational biology, etc. We formulate this as a hypothesis testing problem: under the null hypothesis, these two databases are statistically independent. Under the alternative, however, there exists an unknown permutation sigma over the set of n users (or, row permutation), such that X is rho-correlated with Y^sigma, a permuted version of Y. We determine sharp thresholds at which optimal testing exhibits a phase transition, depending on the asymptotic regime of n and d. Specifically, we prove that if rho^2dto0, as dtoinfty, then weak detection (performing slightly better than random guessing) is statistically impossible, irrespectively of the value of n. This compliments the performance of a simple test that thresholds the sum all entries of X^TY. Furthermore, when d is fixed, we prove that strong detection (vanishing error probability) is impossible for any rho<rho^star, where rho^star is an explicit function of d, while weak detection is again impossible as long as rho^2dto0. These results close significant gaps in current recent related studies.

Recent progress in 3D object generation has greatly improved both the quality and efficiency. However, most existing methods generate a single mesh with all parts fused together, which limits the ability to edit or manipulate individual parts. A key challenge is that different objects may have a varying number of parts. To address this, we propose a new end-to-end framework for part-level 3D object generation. Given a single input image, our method generates high-quality 3D objects with an arbitrary number of complete and semantically meaningful parts. We introduce a dual volume packing strategy that organizes all parts into two complementary volumes, allowing for the creation of complete and interleaved parts that assemble into the final object. Experiments show that our model achieves better quality, diversity, and generalization than previous image-based part-level generation methods.

k-Clustering in R^d (e.g., k-median and k-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality n, it remains open to find sublinear-time quantum algorithms. We give quantum algorithms that find coresets for k-clustering in R^d with O(nkd^{3/2}) query complexity. Our coreset reduces the input size from n to poly(kepsilon^{-1}d), so that existing alpha-approximation algorithms for clustering can run on top of it and yield (1 + epsilon)alpha-approximation. This eventually yields a quadratic speedup for various k-clustering approximation algorithms. We complement our algorithm with a nearly matching lower bound, that any quantum algorithm must make Omega(nk) queries in order to achieve even O(1)-approximation for k-clustering.

We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case. We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.

We propose a graph clustering formulation based on multicut (a.k.a. weighted correlation clustering) on the complete graph. Our formulation does not need specification of the graph topology as in the original sparse formulation of multicut, making our approach simpler and potentially better performing. In contrast to unweighted correlation clustering we allow for a more expressive weighted cost structure. In dense multicut, the clustering objective is given in a factorized form as inner products of node feature vectors. This allows for an efficient formulation and inference in contrast to multicut/weighted correlation clustering, which has at least quadratic representation and computation complexity when working on the complete graph. We show how to rewrite classical greedy algorithms for multicut in our dense setting and how to modify them for greater efficiency and solution quality. In particular, our algorithms scale to graphs with tens of thousands of nodes. Empirical evidence on instance segmentation on Cityscapes and clustering of ImageNet datasets shows the merits of our approach.

We introduce fast algorithms for correlation clustering with respect to the Min Max objective that provide constant factor approximations on complete graphs. Our algorithms are the first purely combinatorial approximation algorithms for this problem. We construct a novel semi-metric on the set of vertices, which we call the correlation metric, that indicates to our clustering algorithms whether pairs of nodes should be in the same cluster. The paper demonstrates empirically that, compared to prior work, our algorithms sacrifice little in the objective quality to obtain significantly better run-time. Moreover, our algorithms scale to larger networks that are effectively intractable for known algorithms.

Contrastive learning (CL) methods effectively learn data representations in a self-supervision manner, where the encoder contrasts each positive sample over multiple negative samples via a one-vs-many softmax cross-entropy loss. By leveraging large amounts of unlabeled image data, recent CL methods have achieved promising results when pretrained on large-scale datasets, such as ImageNet. However, most of them consider the augmented views from the same instance are positive pairs, while views from other instances are negative ones. Such binary partition insufficiently considers the relation between samples and tends to yield worse performance when generalized on images in the wild. In this paper, to further improve the performance of CL and enhance its robustness on various datasets, {we propose a doubly CL strategy that separately compares positive and negative samples within their own groups, and then proceeds with a contrast between positive and negative groups}. We realize this strategy with contrastive attraction and contrastive repulsion (CACR), which makes the query not only exert a greater force to attract more distant positive samples but also do so to repel closer negative samples. Theoretical analysis reveals that CACR generalizes CL's behavior by positive attraction and negative repulsion, and it further considers the intra-contrastive relation within the positive and negative pairs to narrow the gap between the sampled and true distribution, which is important when datasets are less curated. With our extensive experiments, CACR not only demonstrates good performance on CL benchmarks, but also shows better robustness when generalized on imbalanced image datasets. Code and pre-trained checkpoints are available at https://github.com/JegZheng/CACR-SSL.

Bias mitigation in image classification has been widely researched, and existing methods have yielded notable results. However, most of these methods implicitly assume that a given image contains only one type of known or unknown bias, failing to consider the complexities of real-world biases. We introduce a more challenging scenario, agnostic biases mitigation, aiming at bias removal regardless of whether the type of bias or the number of types is unknown in the datasets. To address this difficult task, we present the Partition-and-Debias (PnD) method that uses a mixture of biases-specific experts to implicitly divide the bias space into multiple subspaces and a gating module to find a consensus among experts to achieve debiased classification. Experiments on both public and constructed benchmarks demonstrated the efficacy of the PnD. Code is available at: https://github.com/Jiaxuan-Li/PnD.

We study the approximability of an existing framework for clustering edge-colored hypergraphs, which is closely related to chromatic correlation clustering and is motivated by machine learning and data mining applications where the goal is to cluster a set of objects based on multiway interactions of different categories or types. We present improved approximation guarantees based on linear programming, and show they are tight by proving a matching integrality gap. Our results also include new approximation hardness results, a combinatorial 2-approximation whose runtime is linear in the hypergraph size, and several new connections to well-studied objectives such as vertex cover and hypergraph multiway cut.

Clustering is a NP-hard problem. Thus, no optimal algorithm exists, heuristics are applied to cluster the data. Heuristics can be very resource-intensive, if not applied properly. For substantially large data sets computational efficiencies can be achieved by reducing the input space if a minimal loss of information can be achieved. Clustering algorithms, in general, face two common problems: 1) these converge to different settings with different initial conditions and; 2) the number of clusters has to be arbitrarily decided beforehand. This problem has become critical in the realm of big data. Recently, clustering algorithms have emerged which can speedup computations using parallel processing over the grid but face the aforementioned problems. Goals: Our goals are to find methods to cluster data which: 1) guarantee convergence to the same settings irrespective of the initial conditions; 2) eliminate the need to establish the number of clusters beforehand, and 3) can be applied to cluster large datasets. Methods: We introduce a method that combines probabilistic and combinatorial clustering methods to produce repeatable and compact clusters that are not sensitive to initial conditions. This method harnesses the power of k-means (a combinatorial clustering method) to cluster/partition very large dimensional datasets and uses the Gaussian Mixture Model (a probabilistic clustering method) to validate the k-means partitions. Results: We show that this method produces very compact clusters that are not sensitive to initial conditions. This method can be used to identify the most 'separable' set in a dataset which increases the 'clusterability' of a dataset. This method also eliminates the need to specify the number of clusters in advance.

A new information theoretic condition is presented for reconstructing a discrete random variable X based on the knowledge of a set of discrete functions of X. The reconstruction condition is derived from Shannon's 1953 lattice theory with two entropic metrics of Shannon and Rajski. Because such a theoretical material is relatively unknown and appears quite dispersed in different references, we first provide a synthetic description (with complete proofs) of its concepts, such as total, common and complementary informations. Definitions and properties of the two entropic metrics are also fully detailed and shown compatible with the lattice structure. A new geometric interpretation of such a lattice structure is then investigated that leads to a necessary (and sometimes sufficient) condition for reconstructing the discrete random variable X given a set { X_1,ldots,X_{n} } of elements in the lattice generated by X. Finally, this condition is illustrated in five specific examples of perfect reconstruction problems: reconstruction of a symmetric random variable from the knowledge of its sign and absolute value, reconstruction of a word from a set of linear combinations, reconstruction of an integer from its prime signature (fundamental theorem of arithmetic) and from its remainders modulo a set of coprime integers (Chinese remainder theorem), and reconstruction of the sorting permutation of a list from a minimal set of pairwise comparisons.

Strategies for the generation of periodic discrete structures with identical two-point correlation are developed. Starting from a pair of root structures, which are not related by translation, phase inversion or axis reflections, child structures of arbitrary resolution (i.e., pixel or voxel numbers) and number of phases (i.e., material phases/species) can be generated by means of trivial embedding based phase extension, application of kernels and/or phase coalescence, such that the generated structures inherit the two-point-correlation equivalence. Proofs of the inheritance property are provided by means of the Discrete Fourier Transform theory. A Python 3 implementation of the results is offered by the authors through the Github repository https://github.com/DataAnalyticsEngineering/EQ2PC in order to make the provided results reproducible and useful for all interested readers. Examples for the generation of structures are demonstrated, together with applications in the homogenization theory of periodic media.

Large Language Models (LLMs) have demonstrated impressive performance across various downstream tasks. When training these models, there is a growing inclination to process more tokens on larger training scales but with relatively smaller model sizes. Zero Redundancy Optimizer (ZeRO), although effective in conventional training environments, grapples with scaling challenges when confronted with this emerging paradigm. To this end, we propose a novel LLM training framework AMSP, which undertakes a granular partitioning of model states, encompassing parameters (P), gradient (G), and optimizer states (OS). Specifically, AMSP(1) builds a unified partitioning space, enabling independent partitioning strategies for P, G, and OS; (2) incorporates a scale-aware partitioner to autonomously search for optimal partitioning strategies: (3) designs a dedicated communication optimizer to ensure proficient management of data placement discrepancies arising from diverse partitioning strategies. Our evaluations show that AMSP achieves up to 90.3% scaling efficiency across 1024 GPUs.

Instructing the model to generate a sequence of intermediate steps, a.k.a., a chain of thought (CoT), is a highly effective method to improve the accuracy of large language models (LLMs) on arithmetics and symbolic reasoning tasks. However, the mechanism behind CoT remains unclear. This work provides a theoretical understanding of the power of CoT for decoder-only transformers through the lens of expressiveness. Conceptually, CoT empowers the model with the ability to perform inherently serial computation, which is otherwise lacking in transformers, especially when depth is low. Given input length n, previous works have shown that constant-depth transformers with finite precision poly(n) embedding size can only solve problems in TC^0 without CoT. We first show an even tighter expressiveness upper bound for constant-depth transformers with constant-bit precision, which can only solve problems in AC^0, a proper subset of TC^0. However, with T steps of CoT, constant-depth transformers using constant-bit precision and O(log n) embedding size can solve any problem solvable by boolean circuits of size T. Empirically, enabling CoT dramatically improves the accuracy for tasks that are hard for parallel computation, including the composition of permutation groups, iterated squaring, and circuit value problems, especially for low-depth transformers.

We propose a cluster-based frame selection strategy to mitigate information leakage in video-derived frames datasets. By grouping visually similar frames before splitting into training, validation, and test sets, the method produces more representative, balanced, and reliable dataset partitions.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data. Most SSL approaches rely on strong, well-established, handcrafted data augmentations to generate diverse views for representation learning. However, designing such augmentations requires domain-specific knowledge and implicitly imposes representational invariances on the model, which can limit generalization. In this work, we propose an unsupervised representation learning method that replaces augmentations by generating views using orthonormal bases and overcomplete frames. We show that embeddings learned from orthonormal and overcomplete spaces reside on distinct manifolds, shaped by the geometric biases introduced by representing samples in different spaces. By jointly leveraging the complementary geometry of these distinct manifolds, our approach achieves superior performance without artificially increasing data diversity through strong augmentations. We demonstrate the effectiveness of our method on nine datasets across five temporal sequence tasks, where signal-specific characteristics make data augmentations particularly challenging. Without relying on augmentation-induced diversity, our method achieves performance gains of up to 15--20\% over existing self-supervised approaches. Source code: https://github.com/eth-siplab/Learning-with-FrameProjections

In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.

Dense Retrieval (DR) has achieved state-of-the-art first-stage ranking effectiveness. However, the efficiency of most existing DR models is limited by the large memory cost of storing dense vectors and the time-consuming nearest neighbor search (NNS) in vector space. Therefore, we present RepCONC, a novel retrieval model that learns discrete Representations via CONstrained Clustering. RepCONC jointly trains dual-encoders and the Product Quantization (PQ) method to learn discrete document representations and enables fast approximate NNS with compact indexes. It models quantization as a constrained clustering process, which requires the document embeddings to be uniformly clustered around the quantization centroids and supports end-to-end optimization of the quantization method and dual-encoders. We theoretically demonstrate the importance of the uniform clustering constraint in RepCONC and derive an efficient approximate solution for constrained clustering by reducing it to an instance of the optimal transport problem. Besides constrained clustering, RepCONC further adopts a vector-based inverted file system (IVF) to support highly efficient vector search on CPUs. Extensive experiments on two popular ad-hoc retrieval benchmarks show that RepCONC achieves better ranking effectiveness than competitive vector quantization baselines under different compression ratio settings. It also substantially outperforms a wide range of existing retrieval models in terms of retrieval effectiveness, memory efficiency, and time efficiency.

Post-hoc out-of-distribution (OOD) detection has garnered intensive attention in reliable machine learning. Many efforts have been dedicated to deriving score functions based on logits, distances, or rigorous data distribution assumptions to identify low-scoring OOD samples. Nevertheless, these estimate scores may fail to accurately reflect the true data density or impose impractical constraints. To provide a unified perspective on density-based score design, we propose a novel theoretical framework grounded in Bregman divergence, which extends distribution considerations to encompass an exponential family of distributions. Leveraging the conjugation constraint revealed in our theorem, we introduce a ConjNorm method, reframing density function design as a search for the optimal norm coefficient p against the given dataset. In light of the computational challenges of normalization, we devise an unbiased and analytically tractable estimator of the partition function using the Monte Carlo-based importance sampling technique. Extensive experiments across OOD detection benchmarks empirically demonstrate that our proposed ConjNorm has established a new state-of-the-art in a variety of OOD detection setups, outperforming the current best method by up to 13.25% and 28.19% (FPR95) on CIFAR-100 and ImageNet-1K, respectively.

Kolmogorov-Arnold Networks (KANs) replace scalar weights with per-edge vectors of basis coefficients, thereby boosting expressivity and accuracy but at the same time resulting in a multiplicative increase in parameters and memory. We propose MetaCluster, a framework that makes KANs highly compressible without sacrificing accuracy. Specifically, a lightweight meta-learner, trained jointly with the KAN, is used to map low-dimensional embedding to coefficient vectors, shaping them to lie on a low-dimensional manifold that is amenable to clustering. We then run K-means in coefficient space and replace per-edge vectors with shared centroids. Afterwards, the meta-learner can be discarded, and a brief fine-tuning of the centroid codebook recovers any residual accuracy loss. The resulting model stores only a small codebook and per-edge indices, exploiting the vector nature of KAN parameters to amortize storage across multiple coefficients. On MNIST, CIFAR-10, and CIFAR-100, across standard KANs and ConvKANs using multiple basis functions, MetaCluster achieves a reduction of up to 80times in parameter storage, with no loss in accuracy. Code will be released upon publication.

We resolve an open problem of Hanneke on the subject of universally consistent online learning with non-i.i.d. processes and unbounded losses. The notion of an optimistically universal learning rule was defined by Hanneke in an effort to study learning theory under minimal assumptions. A given learning rule is said to be optimistically universal if it achieves a low long-run average loss whenever the data generating process makes this goal achievable by some learning rule. Hanneke posed as an open problem whether, for every unbounded loss, the family of processes admitting universal learning are precisely those having a finite number of distinct values almost surely. In this paper, we completely resolve this problem, showing that this is indeed the case. As a consequence, this also offers a dramatically simpler formulation of an optimistically universal learning rule for any unbounded loss: namely, the simple memorization rule already suffices. Our proof relies on constructing random measurable partitions of the instance space and could be of independent interest for solving other open questions. We extend the results to the non-realizable setting thereby providing an optimistically universal Bayes consistent learning rule.

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

In this paper, we consider the minimum submodular cost allocation (MSCA) problem. The input of MSCA is k non-negative submodular functions f_1,ldots,f_k on the ground set N given by evaluation oracles, and the goal is to partition N into k (possibly empty) sets X_1,ldots,X_k so that sum_{i=1}^k f_i(X_i) is minimized. In this paper, we focus on the case when f_1,ldots,f_k are monotone (denoted by Mono-MSCA). We provide a natural LP-relaxation for Mono-MSCA, which is equivalent to the convex program relaxation introduced by Chekuri and Ene. We show that the integrality gap of the LP-relaxation is at most k/2, which yields a k/2-approximation algorithm for Mono-MSCA. We also show that the integrality gap of the LP-relaxation is at least k/2-epsilon for any constant epsilon>0 when k is fixed.

Finite symmetric groups S_n are essential in fields such as combinatorics, physics, and chemistry. However, learning a probability distribution over S_n poses significant challenges due to its intractable size and discrete nature. In this paper, we introduce SymmetricDiffusers, a novel discrete diffusion model that simplifies the task of learning a complicated distribution over S_n by decomposing it into learning simpler transitions of the reverse diffusion using deep neural networks. We identify the riffle shuffle as an effective forward transition and provide empirical guidelines for selecting the diffusion length based on the theory of random walks on finite groups. Additionally, we propose a generalized Plackett-Luce (PL) distribution for the reverse transition, which is provably more expressive than the PL distribution. We further introduce a theoretically grounded "denoising schedule" to improve sampling and learning efficiency. Extensive experiments show that our model achieves state-of-the-art or comparable performances on solving tasks including sorting 4-digit MNIST images, jigsaw puzzles, and traveling salesman problems. Our code is released at https://github.com/DSL-Lab/SymmetricDiffusers.

Markov categories are a novel framework to describe and treat problems in probability and information theory. In this work we combine the categorical formalism with the traditional quantitative notions of entropy, mutual information, and data processing inequalities. We show that several quantitative aspects of information theory can be captured by an enriched version of Markov categories, where the spaces of morphisms are equipped with a divergence or even a metric. As it is customary in information theory, mutual information can be defined as a measure of how far a joint source is from displaying independence of its components. More strikingly, Markov categories give a notion of determinism for sources and channels, and we can define entropy exactly by measuring how far a source or channel is from being deterministic. This recovers Shannon and R\'enyi entropies, as well as the Gini-Simpson index used in ecology to quantify diversity, and it can be used to give a conceptual definition of generalized entropy.

Quantifying the similarity between two mathematical structures or datasets constitutes a particularly interesting and useful operation in several theoretical and applied problems. Aimed at this specific objective, the Jaccard index has been extensively used in the most diverse types of problems, also motivating some respective generalizations. The present work addresses further generalizations of this index, including its modification into a coincidence index capable of accounting also for the level of relative interiority between the two compared entities, as well as respective extensions for sets in continuous vector spaces, the generalization to multiset addition, densities and generic scalar fields, as well as a means to quantify the joint interdependence between two random variables. The also interesting possibility to take into account more than two sets has also been addressed, including the description of an index capable of quantifying the level of chaining between three structures. Several of the described and suggested eneralizations have been illustrated with respect to numeric case examples. It is also posited that these indices can play an important role while analyzing and integrating datasets in modeling approaches and pattern recognition activities, including as a measurement of clusters similarity or separation and as a resource for representing and analyzing complex networks.

We develop a framework, in the style of Adler, for interpreting the notion of "witnessing" that has appeared (usually as a variant of Kim's Lemma) in different areas of neostability theory as a binary relation between abstract independence relations. This involves extending the relativisations of Kim-independence and Conant-independence due to Mutchnik to arbitrary independence relations. After developing this framework, we show that several results from simplicity, NTP_2, NSOP_1, and beyond follow as instances of general theorems for abstract independence relations. In particular, we prove the equivalence between witnessing and symmetry and the implications from this notion to chain local character and the weak independence theorem, and recover some partial converses. Finally, we use this framework to prove a dichotomy between NSOP_1 and Kruckman and Ramsey's BTP that applies to most known NSOP_4 examples in the literature.

In this paper, we address the problem of generalized category discovery (GCD), \ie, given a set of images where part of them are labelled and the rest are not, the task is to automatically cluster the images in the unlabelled data, leveraging the information from the labelled data, while the unlabelled data contain images from the labelled classes and also new ones. GCD is similar to semi-supervised learning (SSL) but is more realistic and challenging, as SSL assumes all the unlabelled images are from the same classes as the labelled ones. We also do not assume the class number in the unlabelled data is known a-priori, making the GCD problem even harder. To tackle the problem of GCD without knowing the class number, we propose an EM-like framework that alternates between representation learning and class number estimation. We propose a semi-supervised variant of the Gaussian Mixture Model (GMM) with a stochastic splitting and merging mechanism to dynamically determine the prototypes by examining the cluster compactness and separability. With these prototypes, we leverage prototypical contrastive learning for representation learning on the partially labelled data subject to the constraints imposed by the labelled data. Our framework alternates between these two steps until convergence. The cluster assignment for an unlabelled instance can then be retrieved by identifying its nearest prototype. We comprehensively evaluate our framework on both generic image classification datasets and challenging fine-grained object recognition datasets, achieving state-of-the-art performance.

The top-down and bottom-up methods are two mainstreams of referring segmentation, while both methods have their own intrinsic weaknesses. Top-down methods are chiefly disturbed by Polar Negative (PN) errors owing to the lack of fine-grained cross-modal alignment. Bottom-up methods are mainly perturbed by Inferior Positive (IP) errors due to the lack of prior object information. Nevertheless, we discover that two types of methods are highly complementary for restraining respective weaknesses but the direct average combination leads to harmful interference. In this context, we build Win-win Cooperation (WiCo) to exploit complementary nature of two types of methods on both interaction and integration aspects for achieving a win-win improvement. For the interaction aspect, Complementary Feature Interaction (CFI) provides fine-grained information to top-down branch and introduces prior object information to bottom-up branch for complementary feature enhancement. For the integration aspect, Gaussian Scoring Integration (GSI) models the gaussian performance distributions of two branches and weightedly integrates results by sampling confident scores from the distributions. With our WiCo, several prominent top-down and bottom-up combinations achieve remarkable improvements on three common datasets with reasonable extra costs, which justifies effectiveness and generality of our method.

We introduce Psi-Turing Machines (Psi-TM): classical Turing machines equipped with a constant-depth introspection interface iota and an explicit per-step information budget B(d,n)=c,dlog_2 n . With the interface frozen, we develop an information-theoretic lower-bound toolkit: Budget counting, Psi -Fooling, and Psi -Fano, with worked examples L_k and L_k^{phase} . We prove an oracle-relative separation P^{Psi} neq NP^{Psi} and a strict depth hierarchy, reinforced by an Anti-Simulation Hook that rules out polynomial emulation of iota_k using many calls to iota_{k-1} under the budget regime. We also present two independent platforms (Psi-decision trees and interface-constrained circuits IC-AC^{0}/IC-NC^{1}) and bridges that transfer bounds among machine, tree, and circuit with explicit poly/log losses. The model preserves classical computational power outside iota yet enables precise oracle-aware statements about barriers (relativization; partial/conditional progress on natural proofs and proof complexity). The aim is a standardized minimal introspection interface with clearly accounted information budgets.

The analysis of scientific data and complex multivariate systems requires information quantities that capture relationships among multiple random variables. Recently, new information-theoretic measures have been developed to overcome the shortcomings of classical ones, such as mutual information, that are restricted to considering pairwise interactions. Among them, the concept of information synergy and redundancy is crucial for understanding the high-order dependencies between variables. One of the most prominent and versatile measures based on this concept is O-information, which provides a clear and scalable way to quantify the synergy-redundancy balance in multivariate systems. However, its practical application is limited to simplified cases. In this work, we introduce SOmegaI, which allows for the first time to compute O-information without restrictive assumptions about the system. Our experiments validate our approach on synthetic data, and demonstrate the effectiveness of SOmegaI in the context of a real-world use case.

Cell instance segmentation in cytology images has significant importance for biology analysis and cancer screening, while remains challenging due to 1) the extensive overlapping translucent cell clusters that cause the ambiguous boundaries, and 2) the confusion of mimics and debris as nuclei. In this work, we proposed a De-overlapping Network (DoNet) in a decompose-and-recombined strategy. A Dual-path Region Segmentation Module (DRM) explicitly decomposes the cell clusters into intersection and complement regions, followed by a Semantic Consistency-guided Recombination Module (CRM) for integration. To further introduce the containment relationship of the nucleus in the cytoplasm, we design a Mask-guided Region Proposal Strategy (MRP) that integrates the cell attention maps for inner-cell instance prediction. We validate the proposed approach on ISBI2014 and CPS datasets. Experiments show that our proposed DoNet significantly outperforms other state-of-the-art (SOTA) cell instance segmentation methods. The code is available at https://github.com/DeepDoNet/DoNet.

We study differentially private (DP) algorithms for recovering clusters in well-clustered graphs, which are graphs whose vertex set can be partitioned into a small number of sets, each inducing a subgraph of high inner conductance and small outer conductance. Such graphs have widespread application as a benchmark in the theoretical analysis of spectral clustering. We provide an efficient (epsilon,delta)-DP algorithm tailored specifically for such graphs. Our algorithm draws inspiration from the recent work of Chen et al., who developed DP algorithms for recovery of stochastic block models in cases where the graph comprises exactly two nearly-balanced clusters. Our algorithm works for well-clustered graphs with k nearly-balanced clusters, and the misclassification ratio almost matches the one of the best-known non-private algorithms. We conduct experimental evaluations on datasets with known ground truth clusters to substantiate the prowess of our algorithm. We also show that any (pure) epsilon-DP algorithm would result in substantial error.

Organizing unstructured image collections into semantic clusters is a long-standing challenge. Traditional deep clustering techniques address this by producing a single data partition, whereas multiple clustering methods uncover diverse alternative partitions-but only when users predefine the clustering criteria. Yet expecting users to specify such criteria a priori for large, unfamiliar datasets is unrealistic. In this work, we introduce the task of Open-ended Semantic Multiple Clustering (OpenSMC), which aims to automatically discover clustering criteria from large, unstructured image collections, revealing interpretable substructures without human input. Our framework, X-Cluster: eXploratory Clustering, treats text as a reasoning proxy: it concurrently scans the entire image collection, proposes candidate criteria in natural language, and groups images into meaningful clusters per criterion. To evaluate progress, we release COCO-4c and Food-4c benchmarks, each annotated with four grouping criteria. Experiments show that X-Cluster effectively reveals meaningful partitions and enables downstream applications such as bias discovery and social media image popularity analysis. We will open-source code and data to encourage reproducibility and further research.

Circuits based on sum-product structure have become a ubiquitous representation to compactly encode knowledge, from Boolean functions to probability distributions. By imposing constraints on the structure of such circuits, certain inference queries become tractable, such as model counting and most probable configuration. Recent works have explored analyzing probabilistic and causal inference queries as compositions of basic operators to derive tractability conditions. In this paper, we take an algebraic perspective for compositional inference, and show that a large class of queries - including marginal MAP, probabilistic answer set programming inference, and causal backdoor adjustment - correspond to a combination of basic operators over semirings: aggregation, product, and elementwise mapping. Using this framework, we uncover simple and general sufficient conditions for tractable composition of these operators, in terms of circuit properties (e.g., marginal determinism, compatibility) and conditions on the elementwise mappings. Applying our analysis, we derive novel tractability conditions for many such compositional queries. Our results unify tractability conditions for existing problems on circuits, while providing a blueprint for analysing novel compositional inference queries.

This paper develops a hierarchical clustering algorithm for weighted projective spaces P_{q}, utilizing a Finsler metric d_F([z], [w]) and its rational analogue d_{F,Q}([z], [w]) to define distances that preserve the non-Euclidean geometry of these quotient manifolds. Defined via geodesic integrals of a scaling invariant Finsler norm weighted by the grades q = (q_0, q_1, dots, q_n), these metrics satisfy true metric properties including the triangle inequality, overcoming the limitations of the non-metric dissimilarity measure from prior work.

Recent works have demonstrated reasonable success of representation learning in hypercomplex space. Specifically, "fully-connected layers with Quaternions" (4D hypercomplex numbers), which replace real-valued matrix multiplications in fully-connected layers with Hamilton products of Quaternions, both enjoy parameter savings with only 1/4 learnable parameters and achieve comparable performance in various applications. However, one key caveat is that hypercomplex space only exists at very few predefined dimensions (4D, 8D, and 16D). This restricts the flexibility of models that leverage hypercomplex multiplications. To this end, we propose parameterizing hypercomplex multiplications, allowing models to learn multiplication rules from data regardless of whether such rules are predefined. As a result, our method not only subsumes the Hamilton product, but also learns to operate on any arbitrary nD hypercomplex space, providing more architectural flexibility using arbitrarily 1/n learnable parameters compared with the fully-connected layer counterpart. Experiments of applications to the LSTM and Transformer models on natural language inference, machine translation, text style transfer, and subject verb agreement demonstrate architectural flexibility and effectiveness of the proposed approach.

We give explicit low-rank bilinear non-commutative schemes for multiplying structured n times n matrices with 2 leq n leq 5, which serve as building blocks for recursive algorithms with improved multiplicative factors in asymptotic complexity. Our schemes are discovered over F_2 or F_3 and lifted to Z or Q. Using a flip graph search over tensor decompositions, we derive schemes for general, upper-triangular, lower-triangular, symmetric, and skew-symmetric inputs, as well as products of a structured matrix with its transpose. In particular, we obtain 4 times 4 rank-34 schemes: (i) multiplying a general matrix by its transpose using 10 recursive calls, improving the factor from 26/41 (0.634) to 8/13 (0.615); and (ii) multiplying an upper-triangular matrix by a general matrix using 12 recursive calls, improving the factor from 8/13 (0.615) to 22/37 (0.595). Additionally, using F_3 flip graphs, we discover schemes over Q that fundamentally require the inverse of 2, including a 2 times 2 symmetric-symmetric multiplication of rank 5 and a 3 times 3 skew-symmetric-general multiplication of rank 14 (improving upon AlphaTensor's 15).

The higher-order correlation clustering problem is an expressive model, and recently, local search heuristics have been proposed for several applications. Certifying optimality, however, is NP-hard and practically hampered already by the complexity of the problem statement. Here, we focus on establishing partial optimality conditions for the special case of complete graphs and cubic objective functions. In addition, we define and implement algorithms for testing these conditions and examine their effect numerically, on two datasets.

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.

In this paper, we propose a unified approach to harness quantum conformal methods for multi-output distributions, with a particular emphasis on two experimental paradigms: (i) a standard 2-qubit circuit scenario producing a four-dimensional outcome distribution, and (ii) a multi-basis measurement setting that concatenates measurement probabilities in different bases (Z, X, Y) into a twelve-dimensional output space. By combining a multioutput regression model (e.g., random forests) with distributional conformal prediction, we validate coverage and interval-set sizes on both simulated quantum data and multi-basis measurement data. Our results confirm that classical conformal prediction can effectively provide coverage guarantees even when the target probabilities derive from inherently quantum processes. Such synergy opens the door to next-generation quantum-classical hybrid frameworks, providing both improved interpretability and rigorous coverage for quantum machine learning tasks. All codes and full reproducible Colab notebooks are made available at https://github.com/detasar/QECMMOU.

In this work, we focus on the problem of retrieving relevant arguments for a query claim covering diverse aspects. State-of-the-art methods rely on explicit mappings between claims and premises, and thus are unable to utilize large available collections of premises without laborious and costly manual annotation. Their diversity approach relies on removing duplicates via clustering which does not directly ensure that the selected premises cover all aspects. This work introduces a new multi-step approach for the argument retrieval problem. Rather than relying on ground-truth assignments, our approach employs a machine learning model to capture semantic relationships between arguments. Beyond that, it aims to cover diverse facets of the query, instead of trying to identify duplicates explicitly. Our empirical evaluation demonstrates that our approach leads to a significant improvement in the argument retrieval task even though it requires less data.

We define a class of finite groups based on the properties of the closed twins of their power graphs and study the structure of those groups. As a byproduct, we obtain results about finite groups admitting a partition by cyclic subgroups.

Learning unnormalized statistical models (e.g., energy-based models) is computationally challenging due to the complexity of handling the partition function. To eschew this complexity, noise-contrastive estimation~(NCE) has been proposed by formulating the objective as the logistic loss of the real data and the artificial noise. However, as found in previous works, NCE may perform poorly in many tasks due to its flat loss landscape and slow convergence. In this paper, we study it a direct approach for optimizing the negative log-likelihood of unnormalized models from the perspective of compositional optimization. To tackle the partition function, a noise distribution is introduced such that the log partition function can be written as a compositional function whose inner function can be estimated with stochastic samples. Hence, the objective can be optimized by stochastic compositional optimization algorithms. Despite being a simple method, we demonstrate that it is more favorable than NCE by (1) establishing a fast convergence rate and quantifying its dependence on the noise distribution through the variance of stochastic estimators; (2) developing better results for one-dimensional Gaussian mean estimation by showing our objective has a much favorable loss landscape and hence our method enjoys faster convergence; (3) demonstrating better performance on multiple applications, including density estimation, out-of-distribution detection, and real image generation.

Unsupervised image representations have significantly reduced the gap with supervised pretraining, notably with the recent achievements of contrastive learning methods. These contrastive methods typically work online and rely on a large number of explicit pairwise feature comparisons, which is computationally challenging. In this paper, we propose an online algorithm, SwAV, that takes advantage of contrastive methods without requiring to compute pairwise comparisons. Specifically, our method simultaneously clusters the data while enforcing consistency between cluster assignments produced for different augmentations (or views) of the same image, instead of comparing features directly as in contrastive learning. Simply put, we use a swapped prediction mechanism where we predict the cluster assignment of a view from the representation of another view. Our method can be trained with large and small batches and can scale to unlimited amounts of data. Compared to previous contrastive methods, our method is more memory efficient since it does not require a large memory bank or a special momentum network. In addition, we also propose a new data augmentation strategy, multi-crop, that uses a mix of views with different resolutions in place of two full-resolution views, without increasing the memory or compute requirements much. We validate our findings by achieving 75.3% top-1 accuracy on ImageNet with ResNet-50, as well as surpassing supervised pretraining on all the considered transfer tasks.

Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.

We present Submatrix-wise Vector Embedding Learner (Swivel), a method for generating low-dimensional feature embeddings from a feature co-occurrence matrix. Swivel performs approximate factorization of the point-wise mutual information matrix via stochastic gradient descent. It uses a piecewise loss with special handling for unobserved co-occurrences, and thus makes use of all the information in the matrix. While this requires computation proportional to the size of the entire matrix, we make use of vectorized multiplication to process thousands of rows and columns at once to compute millions of predicted values. Furthermore, we partition the matrix into shards in order to parallelize the computation across many nodes. This approach results in more accurate embeddings than can be achieved with methods that consider only observed co-occurrences, and can scale to much larger corpora than can be handled with sampling methods.

One of the major difficulties in medical image segmentation is the high variability of these images, which is caused by their origin (multi-centre), the acquisition protocols (multi-parametric), as well as the variability of human anatomy, the severity of the illness, the effect of age and gender, among others. The problem addressed in this work is the automatic semantic segmentation of lumbar spine Magnetic Resonance images using convolutional neural networks. The purpose is to assign a class label to each pixel of an image. Classes were defined by radiologists and correspond to different structural elements like vertebrae, intervertebral discs, nerves, blood vessels, and other tissues. The proposed network topologies are variants of the U-Net architecture. Several complementary blocks were used to define the variants: Three types of convolutional blocks, spatial attention models, deep supervision and multilevel feature extractor. This document describes the topologies and analyses the results of the neural network designs that obtained the most accurate segmentations. Several of the proposed designs outperform the standard U-Net used as baseline, especially when used in ensembles where the output of multiple neural networks is combined according to different strategies.

Instance segmentation on point clouds is crucially important for 3D scene understanding. Most SOTAs adopt distance clustering, which is typically effective but does not perform well in segmenting adjacent objects with the same semantic label (especially when they share neighboring points). Due to the uneven distribution of offset points, these existing methods can hardly cluster all instance points. To this end, we design a novel divide-and-conquer strategy named PBNet that binarizes each point and clusters them separately to segment instances. Our binary clustering divides offset instance points into two categories: high and low density points (HPs vs. LPs). Adjacent objects can be clearly separated by removing LPs, and then be completed and refined by assigning LPs via a neighbor voting method. To suppress potential over-segmentation, we propose to construct local scenes with the weight mask for each instance. As a plug-in, the proposed binary clustering can replace the traditional distance clustering and lead to consistent performance gains on many mainstream baselines. A series of experiments on ScanNetV2 and S3DIS datasets indicate the superiority of our model. In particular, PBNet ranks first on the ScanNetV2 official benchmark challenge, achieving the highest mAP.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

By classifying infinite-width neural networks and identifying the *optimal* limit, Tensor Programs IV and V demonstrated a universal way, called muP, for *widthwise hyperparameter transfer*, i.e., predicting optimal hyperparameters of wide neural networks from narrow ones. Here we investigate the analogous classification for *depthwise parametrizations* of deep residual networks (resnets). We classify depthwise parametrizations of block multiplier and learning rate by their infinite-width-then-depth limits. In resnets where each block has only one layer, we identify a unique optimal parametrization, called Depth-muP that extends muP and show empirically it admits depthwise hyperparameter transfer. We identify *feature diversity* as a crucial factor in deep networks, and Depth-muP can be characterized as maximizing both feature learning and feature diversity. Exploiting this, we find that absolute value, among all homogeneous nonlinearities, maximizes feature diversity and indeed empirically leads to significantly better performance. However, if each block is deeper (such as modern transformers), then we find fundamental limitations in all possible infinite-depth limits of such parametrizations, which we illustrate both theoretically and empirically on simple networks as well as Megatron transformer trained on Common Crawl.

This paper studies the fair range clustering problem in which the data points are from different demographic groups and the goal is to pick k centers with the minimum clustering cost such that each group is at least minimally represented in the centers set and no group dominates the centers set. More precisely, given a set of n points in a metric space (P,d) where each point belongs to one of the ell different demographics (i.e., P = P_1 uplus P_2 uplus cdots uplus P_ell) and a set of ell intervals [alpha_1, beta_1], cdots, [alpha_ell, beta_ell] on desired number of centers from each group, the goal is to pick a set of k centers C with minimum ell_p-clustering cost (i.e., (sum_{vin P} d(v,C)^p)^{1/p}) such that for each group iin ell, |Ccap P_i| in [alpha_i, beta_i]. In particular, the fair range ell_p-clustering captures fair range k-center, k-median and k-means as its special cases. In this work, we provide efficient constant factor approximation algorithms for fair range ell_p-clustering for all values of pin [1,infty).

We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are assumed to be linear subspaces, this reduces to the classical problem of subspace clustering, which has been studied extensively over the past two decades. Unfortunately, many real-world datasets such as natural images can not be well approximated by linear subspaces. On the other hand, numerous works have attempted to learn an appropriate transformation of the data, such that data is mapped from a union of general non-linear manifolds to a union of linear subspaces (with points from the same manifold being mapped to the same subspace). However, many existing works have limitations such as assuming knowledge of the membership of samples to clusters, requiring high sampling density, or being shown theoretically to learn trivial representations. In this paper, we propose to optimize the Maximal Coding Rate Reduction metric with respect to both the data representation and a novel doubly stochastic cluster membership, inspired by state-of-the-art subspace clustering results. We give a parameterization of such a representation and membership, allowing efficient mini-batching and one-shot initialization. Experiments on CIFAR-10, -20, -100, and TinyImageNet-200 datasets show that the proposed method is much more accurate and scalable than state-of-the-art deep clustering methods, and further learns a latent linear representation of the data.

In this note we establish a 1-to-1 correspondence between the class of generalized quadrangles with ovoids and the class of balanced incomplete block designs that posses a non-triangular local resolution system and have the appropriate parameters. We present a non-triangular local resolution system for a difference family BIBD construction of Sprott.