The Wasserstein barycenter is defined as the mean of a set of probability measures under the optimal transport metric, and has numerous applications spanning machine learning, statistics, and computer graphics. In practice these input measures are empirical distributions built from sensitive datasets, motivating a differentially private (DP) treatment. We present, to our knowledge, the first algorithms for computing Wasserstein barycenters under differential privacy. Empirically, on synthetic data, MNIST, and large-scale U.S. population datasets, our methods produce high-quality private barycenters with strong accuracy-privacy tradeoffs.

We propose new Markov chain Monte Carlo algorithms to sample a uniform distribution on a convex body $K$. Our algorithms are based on the Alternating Sampling Framework/proximal sampler, which uses Gibbs sampling on an augmented distribution and assumes access to the so-called restricted Gaussian oracle (RGO). The key contribution of this work is the efficient implementation of RGO for uniform sampling on $K$ via rejection sampling and access to either a projection oracle or a separation oracle on $K$. In both oracle cases, we establish non-asymptotic complexities to obtain unbiased samples where the accuracy is measured in Rényi divergence or $χ^2$-divergence.

Understanding how urban residents travel between neighborhoods and destinations is critical for transportation planning, mobility management, and public health. By mining historical origin-to-destination flow patterns with spatial, temporal, and functional relations among urban places, we estimate probabilities of visits from neighborhoods to specific destinations. These probabilities capture neighborhood-level contributions to citywide vehicular and foot traffic, supporting demand estimation, accessibility assessment, and multimodal planning. Particularly, we introduce VisitHGNN, a heterogeneous, relation-specific graph neural network designed to predict visit probabilities at individual Points of interest (POIs). POIs are characterized using numerical, JSON-derived, and textual attributes, augmented with fixed summaries of POI--POI spatial proximity, temporal co-activity, and brand affinity, while census block groups (CBGs) are described with 72 socio-demographic variables. CBGs are connected via spatial adjacency, and POIs and CBGs are linked through distance-annotated cross-type edges. Inference is constrained to a distance-based candidate set of plausible origin CBGs, and training minimizes a masked Kullback-Leibler (KL) divergence to yield probability distribution across the candidate set. Using weekly mobility data from Fulton County, Georgia, USA, VisitHGNN achieves strong predictive performance with mean KL divergence of 0.287, MAE of 0.008, Top-1 accuracy of 0.853, and R-square of 0.892, substantially outperforming pairwise MLP and distance-only baselines, and aligning closely with empirical visitation patterns (NDCG@50 = 0.966); Recall@5 = 0.611). The resulting distributions closely mirror observed travel behavior with high fidelity, highlighting the model's potential for decision support in urban planning, transportation policy, mobility system design, and public health.

What is in the model? Abstract For many scientific questions, understanding the underlying mechanism is the goal. To help investigators better understand the underlying mechanism, variable selection is a crucial step that permits the identification of the most associated regression variables of interest. A variable selection method consists of model evaluation using an information criterion and a search of the model space. Here, we provide a comprehensive comparison of variable selection methods using performance measures of correct identification rate (CIR), recall, and false discovery rate (FDR). We consider the BIC and AIC for evaluating models, and exhaustive, greedy, LASSO path, and stochastic search approaches for searching the model space; we also consider LASSO using cross validation. We perform simulation studies for linear and generalized linear models that parametrically explore a wide range of realistic sample sizes, effect sizes, and correlations among regression variables. We consider model spaces with a small and larger number of potential regressors. The results show that the exhaustive search BIC and stochastic search BIC outperform the other methods when considering the performance measures on small and large model spaces, respectively. These approaches result in the highest CIR and lowest FDR, which collectively may support long-term efforts towards increasing replicability in research.

Deep neural networks excel in high-dimensional problems, outperforming models such as kernel methods, which suffer from the curse of dimensionality. However, the theoretical foundations of this success remain poorly understood. We follow the idea that the compositional structure of the learning task is the key factor determining when deep networks outperform other approaches. Taking a step towards formalizing this idea, we consider a simple compositional model, namely the multi-index model (MIM). In this context, we introduce and study hyper-kernel ridge regression (HKRR), an approach blending neural networks and kernel methods. Our main contribution is a sample complexity result demonstrating that HKRR can adaptively learn MIM, overcoming the curse of dimensionality. Further, we exploit the kernel nature of the estimator to develop ad hoc optimization approaches. Indeed, we contrast alternating minimization and alternating gradient methods both theoretically and numerically. These numerical results complement and reinforce our theoretical findings.

We introduce the Information-Estimation Metric (IEM), a novel form of distance function derived from an underlying continuous probability density over a domain of signals. The IEM is rooted in a fundamental relationship between information theory and estimation theory, which links the log-probability of a signal with the errors of an optimal denoiser, applied to noisy observations of the signal. In particular, the IEM between a pair of signals is obtained by comparing their denoising error vectors over a range of noise amplitudes. Geometrically, this amounts to comparing the score vector fields of the blurred density around the signals over a range of blur levels. We prove that the IEM is a valid global metric and derive a closed-form expression for its local second-order approximation, which yields a Riemannian metric. For Gaussian-distributed signals, the IEM coincides with the Mahalanobis distance. But for more complex distributions, it adapts, both locally and globally, to the geometry of the distribution. In practice, the IEM can be computed using a learned denoiser (analogous to generative diffusion models) and solving a one-dimensional integral. To demonstrate the value of our framework, we learn an IEM on the ImageNet database. Experiments show that this IEM is competitive with or outperforms state-of-the-art supervised image quality metrics in predicting human perceptual judgments.

Adaptive randomized pivoting (ARP) is a recently proposed and highly effective algorithm for column subset selection. This paper reinterprets the ARP algorithm by drawing connections to the volume sampling distribution and active learning algorithms for linear regression. As consequences, this paper presents new analysis for the ARP algorithm and faster implementations using rejection sampling.

We examine gradient descent in matrix factorization and show that under large step sizes the parameter space develops a fractal structure. We derive the exact critical step size for convergence in scalar-vector factorization and show that near criticality the selected minimizer depends sensitively on the initialization. Moreover, we show that adding regularization amplifies this sensitivity, generating a fractal boundary between initializations that converge and those that diverge. The analysis extends to general matrix factorization with orthogonal initialization. Our findings reveal that near-critical step sizes induce a chaotic regime of gradient descent where the long-term dynamics are unpredictable and there are no simple implicit biases, such as towards balancedness, minimum norm, or flatness.

Interpreting the internal behavior of large language models trained on code remains a critical challenge, particularly for applications demanding trust, transparency, and semantic robustness. We propose Code Concept Analysis (CoCoA): a global post-hoc interpretability framework that uncovers emergent lexical, syntactic, and semantic structures in a code language model's representation space by clustering contextualized token embeddings into human-interpretable concept groups. We propose a hybrid annotation pipeline that combines static analysis tool-based syntactic alignment with prompt-engineered large language models (LLMs), enabling scalable labeling of latent concepts across abstraction levels. We analyse the distribution of concepts across layers and across three finetuning tasks. Emergent concept clusters can help identify unexpected latent interactions and be used to identify trends and biases within the model's learned representations. We further integrate LCA with local attribution methods to produce concept-grounded explanations, improving the coherence and interpretability of token-level saliency. Empirical evaluations across multiple models and tasks show that LCA discovers concepts that remain stable under semantic-preserving perturbations (average Cluster Sensitivity Index, CSI = 0.288) and evolve predictably with fine-tuning. In a user study on the programming-language classification task, concept-augmented explanations disambiguated token roles and improved human-centric explainability by 37 percentage points compared with token-level attributions using Integrated Gradients.

Composite optimization problems involve minimizing the composition of a smooth map with a convex function. Such objectives arise in numerous data science and signal processing applications, including phase retrieval, blind deconvolution, and collaborative filtering. The subgradient method achieves local linear convergence when the composite loss is well-conditioned. However, if the smooth map is, in a certain sense, ill-conditioned or overparameterized, the subgradient method exhibits much slower sublinear convergence even when the convex function is well-conditioned. To overcome this limitation, we introduce a Levenberg-Morrison-Marquardt subgradient method that converges linearly under mild regularity conditions at a rate determined solely by the convex function. Further, we demonstrate that these regularity conditions hold for several problems of practical interest, including square-variable formulations, matrix sensing, and tensor factorization. Numerical experiments illustrate the benefits of our method.