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We present a generative learning framework for probabilistic sampling based on an extension of the Probabilistic Learning on Manifolds (PLoM) approach, which is designed to generate statistically consistent realizations of a random vector…

What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from a sufficient…

In this paper, we propose Complex Diffusion Maps (CDM), a novel diffusion mapping framework that aims to reveal the dominant complex harmonics of high-dimensional data. Inspired by the local Gaussian kernel relevant to the heat equation and…

We develop a family of infinite-dimensional Banach manifolds of measures on an abstract measurable space, employing charts that are "balanced" between the density and log-density functions. The manifolds, $(\tilde{M}_{\lambda},\lambda\in…

We focus on semiparametric regression that has played a central role in statistics, and exploit the powerful learning ability of deep neural networks (DNNs) while enabling statistical inference on parameters of interest that offers…

Despite the success of diffusion models (DMs), we still lack a thorough understanding of their latent space. To understand the latent space $\mathbf{x}_t \in \mathcal{X}$, we analyze them from a geometrical perspective. Our approach…

Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…

One of the fundamental problems in machine learning is the estimation of a probability distribution from data. Many techniques have been proposed to study the structure of data, most often building around the assumption that observations…

Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood…

Strong generative models can accurately learn channel distributions. This could save recurring costs for physical measurements of the channel. Moreover, the resulting differentiable channel model supports training neural encoders by…

The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an…

Deep neural network classifiers partition input space into high confidence regions for each class. The geometry of these class manifolds (CMs) is widely studied and intimately related to model performance; for example, the margin depends on…

Likelihood-based, or explicit, deep generative models use neural networks to construct flexible high-dimensional densities. This formulation directly contradicts the manifold hypothesis, which states that observed data lies on a…

Recently proposed generative models for discrete data, such as Masked Diffusion Models (MDMs), exploit conditional independence approximations to reduce the computational cost of popular Auto-Regressive Models (ARMs), at the price of some…

Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the embeddings is usually determined by how well the geometry…

We present an algorithm to estimate fast and accurate depth maps from light fields via a sparse set of depth edges and gradients. Our proposed approach is based around the idea that true depth edges are more sensitive than texture edges to…

In this paper we prove two theorems. The first one is a structure result that describes the extrinsic geometry of an embedded surface with constant mean curvature (possibly zero) in a homogeneously regular Riemannian three-manifold, in any…

The existing approaches to intrinsic dimension estimation usually are not reliable when the data are nonlinearly embedded in the high dimensional space. In this work, we show that the explicit accounting to geometric properties of unknown…

The decentralized gradient descent (DGD) algorithm, and its sibling, diffusion, are workhorses in decentralized machine learning, distributed inference and estimation, and multi-agent coordination. We propose a novel, principled framework…

For a compact $(2n+1)$-dimensional smooth manifold, let $\mu_M : B Diff_\partial (D^{2n+1}) \to B Diff (M)$ be the map that is defined by extending diffeomorphisms on an embedded disc by the identity. By a classical result of Farrell and…