A central challenge in machine learning is to distinguish genuine structure from chance correlations in high-dimensional data. In this work, we address this issue for the perceptron, a foundational model of neural computation. Specifically, we investigate the relationship between the pattern load $α$ and the variable selection ratio $ρ$ for which a simple perceptron can perfectly classify $P = αN$ random patterns by optimally selecting $M = ρN$ variables out of $N$ variables. While the Cover--Gardner theory establishes that a random subset of $ρN$ dimensions can separate $αN$ random patterns if and only if $α< 2ρ$, we demonstrate that optimal variable selection can surpass this bound by developing a method, based on the replica method from statistical mechanics, for enumerating the combinations of variables that enable perfect pattern classification. This not only provides a quantitative criterion for distinguishing true structure in the data from spurious regularities, but also yields the storage capacity of associative memory models with sparse asymmetric couplings.

Diffusion generative models have emerged as powerful tools for producing synthetic data from an empirically observed distribution. A common approach involves simulating the time-reversal of an Ornstein-Uhlenbeck (OU) process initialized at the true data distribution. Since the score function associated with the OU process is typically unknown, it is approximated using a trained neural network. This approximation, along with finite time simulation, time discretization and statistical approximation, introduce several sources of error whose impact on the generated samples must be carefully understood. Previous analyses have quantified the error between the generated and the true data distributions in terms of Wasserstein distance or Kullback-Leibler (KL) divergence. However, both metrics present limitations: KL divergence requires absolute continuity between distributions, while Wasserstein distance, though more general, leads to error bounds that scale poorly with dimension, rendering them impractical in high-dimensional settings. In this work, we derive an explicit, dimension-free bound on the discrepancy between the generated and the true data distributions. The bound is expressed in terms of a smooth test functional with bounded first and second derivatives. The key novelty lies in the use of this weaker, functional metric to obtain dimension-independent guarantees, at the cost of higher regularity on the test functions. As an application, we formulate and solve a variational problem to minimize the time-discretization error, leading to the derivation of an optimal time-scheduling strategy for the reverse-time diffusion. Interestingly, this scheduler has appeared previously in the literature in a different context; our analysis provides a new justification for its optimality, now grounded in minimizing the discretization bias in generative sampling.

Decision trees and random forest remain highly competitive for classification on medium-sized, standard datasets due to their robustness, minimal preprocessing requirements, and interpretability. However, a single tree suffers from high estimation variance, while large ensembles reduce this variance at the cost of substantial computational overhead and diminished interpretability. In this paper, we propose Decision Tree Embedding (DTE), a fast and effective method that leverages the leaf partitions of a trained classification tree to construct an interpretable feature representation. By using the sample means within each leaf region as anchor points, DTE maps inputs into an embedding space defined by the tree's partition structure, effectively circumventing the high variance inherent in decision-tree splitting rules. We further introduce an ensemble extension based on additional bootstrap trees, and pair the resulting embedding with linear discriminant analysis for classification. We establish several population-level theoretical properties of DTE, including its preservation of conditional density under mild conditions and a characterization of the resulting classification error. Empirical studies on synthetic and real datasets demonstrate that DTE strikes a strong balance between accuracy and computational efficiency, outperforming or matching random forest and shallow neural networks while requiring only a fraction of their training time in most cases. Overall, the proposed DTE method can be viewed either as a scalable decision tree classifier that improves upon standard split rules, or as a neural network model whose weights are learned from tree-derived anchor points, achieving an intriguing integration of both paradigms.

In this paper, we investigate a second-order stochastic algorithm for solving large-scale binary classification problems. We propose to make use of a new hybrid stochastic Newton algorithm that includes two weighted components in the Hessian matrix estimation: the first one coming from the natural Hessian estimate and the second associated with the stochastic gradient information. Our motivation comes from the fact that both parts evaluated at the true parameter of logistic regression, are equal to the Hessian matrix. This new formulation has several advantages and it enables us to prove the almost sure convergence of our stochastic algorithm to the true parameter. Moreover, we significantly improve the almost sure rate of convergence to the Hessian matrix. Furthermore, we establish the central limit theorem for our hybrid stochastic Newton algorithm. Finally, we show a surprising result on the almost sure convergence of the cumulative excess risk.

Bipartite networks are widely used to encode the ecological interactions. Being able to compare the organization of bipartite networks is a first step toward a better understanding of how environmental factors shape community structure and resilience. Yet current methods for structure detection in bipartite networks overlook shared patterns across collections of networks. We introduce the \emph{colBiSBM}, a family of probabilistic models for collections of bipartite networks that extends the classical Latent Block Model (LBM). The proposed framework assumes that networks are independent realizations of a shared mesoscale structure, encoded through common inter-block connectivity parameters. We establish identifiability conditions for the different variants of \emph{colBiSBM} and develop a variational EM algorithm for parameter estimation, coupled with an adaptation of the Integrated Classification Likelihood (ICL) criterion for model selection. We demonstrate how our approach can be used to classify networks based on their topology or organization. Simulation studies highlight the ability of \emph{colBiSBM} to recover common structures, improve clustering performance, and enhance link prediction by borrowing strength across networks. An application to plant--pollinator networks highlights how the method uncovers shared ecological roles and partitions networks into sub-collections with similar connectivity patterns. These results illustrate the methodological and practical advantages of joint modeling over separate network analyses in the study of bipartite systems.

Learning the structure of a Bayesian network from decentralized data poses two major challenges: (i) ensuring rigorous privacy guarantees for participants, and (ii) avoiding communication costs that scale poorly with dimensionality. In this work, we introduce Fed-Sparse-BNSL, a novel federated method for learning linear Gaussian Bayesian network structures that addresses both challenges. By combining differential privacy with greedy updates that target only a few relevant edges per participant, Fed-Sparse-BNSL efficiently uses the privacy budget while keeping communication costs low. Our careful algorithmic design preserves model identifiability and enables accurate structure estimation. Experiments on synthetic and real datasets demonstrate that Fed-Sparse-BNSL achieves utility close to non-private baselines while offering substantially stronger privacy and communication efficiency.

Post-training quantization (PTQ) aims to preserve model-level behavior; however, most methods focus on individual linear layers. Even recent extensions, such as QEP and LoaQ, which mitigate error propagation or target specific submodules, still rely on layer-wise formulations and fail to capture the behavior of larger submodules. We introduce Layer-Projected Coordinate Descent (LPCD), a unified framework that extends PTQ beyond layers by optimizing relaxed objectives across arbitrary submodules and projecting the solutions with layer-wise quantizers. LPCD generalizes existing methods and provides a principled approach to quantizing complex submodules while maintaining the efficiency and compatibility of layer-wise PTQ pipelines. Across diverse LLM architectures and bit-widths, LPCD-based submodule quantization consistently enhances both layer-wise PTQ methods and existing submodule approaches.

This paper introduces a comprehensive unified framework for constructing multi-view diffusion geometries through intertwined multi-view diffusion trajectories (MDTs), a class of inhomogeneous diffusion processes that iteratively combine the random walk operators of multiple data views. Each MDT defines a trajectory-dependent diffusion operator with a clear probabilistic and geometric interpretation, capturing over time the interplay between data views. Our formulation encompasses existing multi-view diffusion models, while providing new degrees of freedom for view interaction and fusion. We establish theoretical properties under mild assumptions, including ergodicity of both the point-wise operator and the process in itself. We also derive MDT-based diffusion distances, and associated embeddings via singular value decompositions. Finally, we propose various strategies for learning MDT operators within the defined operator space, guided by internal quality measures. Beyond enabling flexible model design, MDTs also offer a neutral baseline for evaluating diffusion-based approaches through comparison with randomly selected MDTs. Experiments show the practical impact of the MDT operators in a manifold learning and data clustering context.

We consider the problem of generalization of arbitrarily overparameterized two-layer ReLU Neural Networks with univariate input. Recent work showed that under square loss, flat solutions (motivated by flat / stable minima and Edge of Stability phenomenon) provably cannot overfit, but it remains unclear whether the same phenomenon holds for logistic loss. This is a puzzling open problem because existing work on logistic loss shows that gradient descent with increasing step size converges to interpolating solutions (at infinity, for the margin-separable cases). In this paper, we prove that the \emph{flatness implied generalization} is more delicate under logistic loss. On the positive side, we show that flat solutions enjoy near-optimal generalization bounds within a region between the left-most and right-most \emph{uncertain} sets determined by each candidate solution. On the negative side, we show that there exist arbitrarily flat yet overfitting solutions at infinity that are (falsely) certain everywhere, thus certifying that flatness alone is insufficient for generalization in general. We demonstrate the effects predicted by our theory in a well-controlled simulation study.

Discrete event systems are present both in observations of nature, socio economical sciences, and industrial systems. Standard analysis approaches do not usually exploit their dual event / state nature: signals are either modeled as transition event sequences, emphasizing event order alignment, or as categorical or ordinal state timeseries, usually resampled a distorting and costly operation as the observation period and number of events grow. In this work we define state transition event timeseries (STE-ts) and propose a new Selective Temporal Hamming distance (STH) leveraging both transition time and duration-in-state, avoiding costly and distorting resampling on large databases. STH generalizes both resampled Hamming and Jaccard metrics with better precision and computation time, and an ability to focus on multiple states of interest. We validate these benefits on simulated and real-world datasets.