Related papers: Randomized Learning of the Second-Moment Matrix of…

In this two-part work, we propose an algorithmic framework for solving non-convex problems whose objective function is the sum of a number of smooth component functions plus a convex (possibly non-smooth) or/and smooth (possibly non-convex)…

Sampling a target probability distribution with an unknown normalization constant is a fundamental challenge in computational science and engineering. Recent work shows that algorithms derived by considering gradient flows in the space of…

Given a loss function $F:\mathcal{X} \rightarrow \R^+$ that can be written as the sum of losses over a large set of inputs $a_1,\ldots, a_n$, it is often desirable to approximate $F$ by subsampling the input points. Strong theoretical…

This article studies the \emph{robust covariance matrix estimation} of a data collection $X = (x_1,\ldots,x_n)$ with $x_i = \sqrt \tau_i z_i + m$, where $z_i \in \mathbb R^p$ is a \textit{concentrated vector} (e.g., an elliptical random…

This paper presents a comprehensive analysis of a broad range of variations of the stochastic proximal point method (SPPM). Proximal point methods have attracted considerable interest owing to their numerical stability and robustness…

Smoothed analysis is a framework for analyzing the complexity of an algorithm, acting as a bridge between average and worst-case behaviour. For example, Quicksort and the Simplex algorithm are widely used in practical applications, despite…

Let $\mu$ be a radial compactly supported distribution on a harmonic $NA$ group. We prove that the right convolution operator $c_{\mu}:f \mapsto f* \mu$ maps the space of smooth $\mathfrak{v}$-radial functions onto itself if and only if the…

Training neural networks requires optimizing a loss function that may be highly irregular, and in particular neither convex nor smooth. Popular training algorithms are based on stochastic gradient descent with momentum (SGDM), for which…

We develop nonparametric regression methods for the case when the true regression function is not necessarily smooth. More specifically, our approach is using the fractional Laplacian and is designed to handle the case when the true…

We establish in this work approximation results of deep neural networks for smooth functions measured in Sobolev norms, motivated by recent development of numerical solvers for partial differential equations using deep neural networks. {Our…

We consider iterative gradient-based optimization algorithms applied to functions that are smooth and strongly convex. The fastest globally convergent algorithm for this class of functions is the Triple Momentum (TM) method. We show that if…

Randomized smoothing is a widely adopted technique for optimizing nonsmooth objective functions. However, its efficiency analysis typically relies on global Lipschitz continuity, a condition rarely met in practical applications. To address…

Stochastic Approximation (SA) is a popular approach for solving fixed-point equations where the information is corrupted by noise. In this paper, we consider an SA involving a contraction mapping with respect to an arbitrary norm, and show…

In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only…

The idea of partial smoothness in optimization blends certain smooth and nonsmooth properties of feasible regions and objective functions. As a consequence, the standard first-order conditions guarantee that diverse iterative algorithms…

Let $\pi : X\to \Lambda$ be a flat family of smooth complex projective varieties parameterized by a smooth quasi-projective variety $\Lambda$, and let $f: X\to X$ be a family of automorphisms with positive topological entropy. Suppose…

We propose nonparametric estimators for the second-order central moments of possibly anisotropic spherical random fields, within a functional data analysis context. We consider a measurement framework where each random field among an…

We review existing methods for implementing smooth functions f(A) of a sparse Hermitian matrix A on a quantum computer, and analyse a further combination of these techniques which has some advantages of simplicity and resource consumption…

Various optimal gradient-based algorithms have been developed for smooth nonconvex optimization. However, many nonconvex machine learning problems do not belong to the class of smooth functions and therefore the existing algorithms are…

The data functions that are studied in the course of functional data analysis are assembled from discrete data, and the level of smoothing that is used is generally that which is appropriate for accurate approximation of the conceptually…