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We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time…
We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe…
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…
The Integral Image algorithm is often applied in tasks that require efficient integration over images, such as object detection. In this paper we discuss theoretical aspects of the algorithm's continuous version. We suggest to define the…
We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the…
A discretisation scheme that preserves topological features of a physical problem is extended so that differential geometric structures can be approximated in a consistent way thus giving access to the study of physical systems which are…
We introduce a continuous domain for function spaces over topological spaces which are not core-compact. Notable examples of such topological spaces include the real line with the upper limit topology, which is used in solution of initial…
Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond…
Given a finite number of samples of a continuous set-valued function F, mapping an interval to non-empty compact subsets of $\mathbb{R}^d$, $F: [a,b] \to K(\mathbb{R}^d)$, we discuss the problem of computing good approximations of F. We…
Minimax optimization has been central in addressing various applications in machine learning, game theory, and control theory. Prior literature has thus far mainly focused on studying such problems in the continuous domain, e.g.,…
We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments. This problem has important applications in several areas of numerical…
In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and…
We introduce a rotation-invariant representation of planar shapes. In particular, this representation encodes shapes as vectors such that the Euclidean distance between them serves as a valid shape distance. For standardized, star-shaped…
Any symmetric affinity function $w: V\times V \to \mathbb{R}_+$ defined on a discrete set $V$ induces Euclidean space structure on $V$. In particular, an undirected graph specified by an affinity (or adjacency) matrix can be considered as a…
We consider approximations of a continuous function on a countable normed Fr\'{e}chet space by analytic and $*$-analytic. Also we found a criterium of the existence of an extension of a continuous function from a dense subspace of a…
This paper focuses on minimizing a smooth function combined with a nonsmooth regularization term on a compact Riemannian submanifold embedded in the Euclidean space under a decentralized setting. Typically, there are two types of approaches…
We extend the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to…
We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold $M$. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the…
Holomorphic functions are amazing because their values in an ever so small disk in the complex plane completely determine the function values at arbitrary points in their maximum possible domain. The process of extending such a function…
In this paper we investigate the approximation of continuous functions on the Wasserstein space by smooth functions, with smoothness meant in the sense of Lions differentiability. In particular, in the case of a Lipschitz function we are…