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Diffusion models have demonstrated remarkable empirical success in the recent years and are considered one of the state-of-the-art generative models in modern AI. These models consist of a forward process, which gradually diffuses the data…

We propose Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) for learning distributions on submanifolds of Euclidean space that are level sets of functions, including most of the manifolds relevant to applications. Existing…

We consider the additive superimposition of an extensive number of independent Euclidean Random Matrices in the high-density regime. The resolvent is computed with techniques from free probability theory, as well as with the replica method…

Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply…

Generative models, such as large language models or text-to-image diffusion models, can generate relevant responses to user-given queries. Response-based vector embeddings of generative models facilitate statistical analysis and inference…

We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and…

Many machine learning applications require operating on a spatially distributed dataset. Despite technological advances, privacy considerations and communication constraints may prevent gathering the entire dataset in a central unit. In…

Recently, diffusion probabilistic models (DPMs) have achieved promising results in diverse generative tasks. A typical DPM framework includes a forward process that gradually diffuses the data distribution and a reverse process that…

Many stochastic optimization problems include chance constraints that enforce constraint satisfaction with a specific probability; however, solving an optimization problem with chance constraints assumes that the solver has access to the…

We study the spectrum of a random matrix, whose elements depend on the Euclidean distance between points randomly distributed in space. This problem is widely studied in the context of the Instantaneous Normal Modes of fluids and is…

Data which lie in the space $\mathcal{P}_{m\,}$, of $m \times m$ symmetric positive definite matrices, (sometimes called tensor data), play a fundamental role in applications including medical imaging, computer vision, and radar signal…

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible…

The problem of overdispersion in multivariate count data is a challenging issue. Nowadays, it covers a central role mainly due to the relevance of modern technologies data, such as Next Generation Sequencing and textual data from the web or…

We present a data-driven approach for distributionally robust chance constrained optimization problems (DRCCPs). We consider the case where the decision maker has access to a finite number of samples or realizations of the uncertainty. The…

This paper begins with a description of methods for estimating image probability density functions that reflects the observation that such data is usually constrained to lie in restricted regions of the high-dimensional image space-not…

This paper explores methods for estimating or approximating the total variation distance and the chi-squared divergence of probability measures within topological sample spaces, using independent and identically distributed samples. Our…

Data analysis in high-dimensional spaces aims at obtaining a synthetic description of a data set, revealing its main structure and its salient features. We here introduce an approach providing this description in the form of a topography of…

Addressing real-world optimization problems becomes particularly challenging when analytic objective functions or constraints are unavailable. While numerous studies have addressed the issue of unknown objectives, limited research has…

Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to…

Diffusion generative models (DMs) have achieved promising results in image and graph generation. However, real-world graphs, such as social networks, molecular graphs, and traffic graphs, generally share non-Euclidean topologies and hidden…