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We analyze the Wasserstein distance ($W$-distance) between two probability distributions associated with two multidimensional jump-diffusion processes. Specifically, we analyze a temporally decoupled squared $W_2$-distance, which provides…
Comparing probability distributions is at the crux of many machine learning algorithms. Maximum Mean Discrepancies (MMD) and Wasserstein distances are two classes of distances between probability distributions that have attracted abundant…
The manifold hypothesis posits that high-dimensional data typically resides on low-dimensional sub spaces. In this paper, we assume manifold hypothesis to investigate graph-based semi-supervised learning methods. In particular, we examine…
While Bayesian neural networks (BNNs) hold the promise of being flexible, well-calibrated statistical models, inference often requires approximations whose consequences are poorly understood. We study the quality of common variational…
Modern machine learning models often employ a huge number of parameters and are typically optimized to have zero training loss; yet surprisingly, they possess near-optimal prediction performance, contradicting classical learning theory. We…
The Busemann function has recently found much interest in a variety of geometric machine learning problems, as it naturally defines projections onto geodesic rays of Riemannian manifolds and generalizes the notion of hyperplanes. As several…
The ability to model relational information using machine learning has driven advancements across various domains, from medicine to social science. While graph representation learning has become mainstream over the past decade, representing…
Statistical inference based on optimal transport offers a different perspective from that of maximum likelihood, and has increasingly gained attention in recent years. In this paper, we study univariate nonparametric shape-constrained…
This paper presents a distance-based discriminative framework for learning with probability distributions. Instead of using kernel mean embeddings or generalized radial basis kernels, we introduce embeddings based on dissimilarity of…
Understanding the space of probability measures on a metric space equipped with a Wasserstein distance is one of the fundamental questions in mathematical analysis. The Wasserstein metric has received a lot of attention in the machine…
We study shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and reduces to multilayer feedforward networks in the Euclidean case. We…
Recent works demonstrated the existence of a double-descent phenomenon for the generalization error of neural networks, where highly overparameterized models escape overfitting and achieve good test performance, at odds with the standard…
In supervised learning, it is known that overparameterized neural networks with one hidden layer provably and efficiently learn and generalize, when trained using stochastic gradient descent with a sufficiently small learning rate and…
In this paper, we present generalization bounds for the unsupervised risk in the Deep Contrastive Representation Learning framework, which employs deep neural networks as representation functions. We approach this problem from two angles.…
We consider learning in an adversarial environment, where an $\varepsilon$-fraction of samples from a distribution $P$ are arbitrarily modified (global corruptions) and the remaining perturbations have average magnitude bounded by $\rho$…
Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks as the curse of dimensionality (CoD) cannot be evaded when trying to approximate even a single ReLU neuron…
The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite…
Neural networks have proven successful at learning from complex data distributions by acting as universal function approximators. However, they are often overconfident in their predictions, which leads to inaccurate and miscalibrated…
Despite of its importance for safe machine learning, uncertainty quantification for neural networks is far from being solved. State-of-the-art approaches to estimate neural uncertainties are often hybrid, combining parametric models with…
It is now practically the norm for data to be very high dimensional in areas such as genetics, machine vision, image analysis and many others. When analyzing such data, parametric models are often too inflexible while nonparametric…