Related papers: Max-convolution through numerics and tropical geom…
Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation…
In this work, we examine the process of Tropical Polynomial Division, a geometric method which seeks to emulate the division of regular polynomials, when applied to those of the max-plus semiring. This is done via the approximation of the…
Metric magnitude is a measure of the "size" of point clouds with many desirable geometric properties. It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization…
We present a systematic study of the reconstruction of a non-negative function via maximum entropy approach utilizing the information contained in a finite number of moments of the function. For testing the efficacy of the approach, we…
The purpose of the paper is to provide a characterization of the error of the best polynomial approximation of composite functions in weighted spaces. Such a characterization is essential for the convergence analysis of numerical methods…
We give near-optimal algorithms for computing an ellipsoidal rounding of a convex polytope whose vertices are given in a stream. The approximation factor is linear in the dimension (as in John's theorem) and only loses an excess logarithmic…
A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian…
Optimization algorithms and Monte Carlo sampling algorithms have provided the computational foundations for the rapid growth in applications of statistical machine learning in recent years. There is, however, limited theoretical…
We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give estimates…
For 24 years, it has been an open problem to obtain improved bounds, for the maximal function over a sparse sequence of discrete spherical averages, going beyond the range for the full discrete spherical maximal function. I formulate a…
In the simultaneous Max-Cut problem, we are given $k$ weighted graphs on the same set of $n$ vertices, and the goal is to find a cut of the vertex set so that the minimum, over the $k$ graphs, of the cut value is as large as possible.…
We consider the maximum vertex-weighted matching problem (MVM) for non-bipartite graphs. In earlier work we have described a 2/3-approximation algorithm for the MVM on bipartite graphs (Dobrian, Halappanavar, Pothen and Al-Herz, SIAM J.…
We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly…
Motivated by the geometric reduction of Cauchy--Szeg\H{o} projections on quadratic surfaces of higher codimension (Nagel--Ricci--Stein, 2001) and recent developments on the real-variable theory adapted to twisted multiparameter structures…
We present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions. The main motivator behind this numerical method is estimating solutions to Adverse Selection…
We study convex and quasiconvex functions on a metric graph. Given a set of points in the metric graph, we consider the largest convex function below the prescribed datum. We characterize this largest convex function as the unique largest…
We consider multidimensional optimization problems that are formulated in the framework of tropical mathematics to minimize functions defined on vectors over a tropical semifield (a semiring with idempotent addition and invertible…
Tensor networks represent the state-of-the-art in computational methods across many disciplines, including the classical simulation of quantum many-body systems and quantum circuits. Several applications of current interest give rise to…
Many problems in machine learning can be formulated as optimizing a convex functional over a vector space of measures. This paper studies the convergence of the mirror descent algorithm in this infinite-dimensional setting. Defining Bregman…
In this paper we analyze several new methods for solving nonconvex optimization problems with the objective function formed as a sum of two terms: one is nonconvex and smooth, and another is convex but simple and its structure is known.…