Related papers: The Sample Complexity of Learning Lipschitz Operat…

We study the complexity of smoothed agnostic learning, recently introduced by~\cite{CKKMS24}, in which the learner competes with the best classifier in a target class under slight Gaussian perturbations of the inputs. Specifically, we focus…

We introduce a new scalable approximation for Gaussian processes with provable guarantees which hold simultaneously over its entire parameter space. Our approximation is obtained from an improved sample complexity analysis for sparse…

We describe a general technique that yields the first {\em Statistical Query lower bounds} for a range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning…

Lipschitz learning is a graph-based semi-supervised learning method where one extends labels from a labeled to an unlabeled data set by solving the infinity Laplace equation on a weighted graph. In this work we prove uniform convergence…

We propose a new random sketching approach for embedding high-dimensional Hilbert-Schmidt operators, using random input-output pairs. Such operator can then be approximated in a low-dimensional subspace of operators by solving a small…

Reconstructing an infinite-dimensional signal from a finite set of measurements is a fundamental problem in approximation theory and signal processing. While the generalized sampling (GS) framework provides a robust methodology for…

We consider the sparse polynomial approximation of a multivariate function on a tensor product domain from samples of both the function and its gradient. When only function samples are prescribed, weighted $\ell^1$ minimization has recently…

We study the complexity of smoothed agnostic learning of halfspaces on $\{\pm 1\}^n$ under uniform marginals in the model of~\cite{KM25}, where each input coordinate is independently flipped with probability $\sigma \in (0, {1}/{2})$. We…

This paper studies sample average approximation (SAA) in solving convex or strongly convex stochastic programming (SP) problems. In estimating SAA's sample efficiency, the state-of-the-art sample complexity bounds entail metric entropy…

The convergence theory for the gradient sampling algorithm is extended to directionally Lipschitz functions. Although directionally Lipschitz functions are not necessarily locally Lipschitz, they are almost everywhere differentiable and…

Since their invention, generative adversarial networks (GANs) have become a popular approach for learning to model a distribution of real (unlabeled) data. Convergence problems during training are overcome by Wasserstein GANs which minimize…

In this note we study the problem of sampling and reconstructing signals which are assumed to lie on or close to one of several subspaces of a Hilbert space. Importantly, we here consider a very general setting in which we allow infinitely…

Finding multiple solutions of non-convex optimization problems is a ubiquitous yet challenging task. Most past algorithms either apply single-solution optimization methods from multiple random initial guesses or search in the vicinity of…

We present a framework for performing efficient regression in general metric spaces. Roughly speaking, our regressor predicts the value at a new point by computing a Lipschitz extension --- the smoothest function consistent with the…

Generative neural networks (GNNs) have gained renown for efficaciously capturing intrinsic low-dimensional structure in natural images. Here, we investigate the subgaussian demixing problem for two Lipschitz signals, with GNN demixing as a…

We consider the approximation of the inverse square root of regularly accretive operators in Hilbert spaces. The approximation is of rational type and comes from the use of the Gauss-Legendre rule applied to a special integral formulation…

We initiate a program of average smoothness analysis for efficiently learning real-valued functions on metric spaces. Rather than using the Lipschitz constant as the regularizer, we define a local slope at each point and gauge the function…

Traditional measures of smoothness often fail to provide accurate $L_p$-error estimates for approximation by sampling or interpolation operators, especially for functions with low smoothness. To address this issue, we introduce a modified…

This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space,…

Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in…