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In this note we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity…
The sampling of functions of bounded variation (BV) is a long-standing problem in op- timization. The ability to sample such functions has relevance in the field of variational inverse problems, where the standard theory fails to guarantee…
We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…
In this report, we consider extended real-valued functions on some real vector space. Gerstewitz functionals are used to construct all translative functions. We derive formulas for translative functions which are lower semicontinuous,…
In this work, we introduce a new class of non-convex functions, called implicit concave functions, which are compositions of a concave function with a continuously differentiable mapping. We analyze the properties of their minimization by…
Consider a convex function that is invariant under an group of transformations. If it has a minimizer, does it also have an invariant minimizer? Variants of this problem appear in nonparametric statistics and in a number of adjacent fields.…
This paper provides an unique dual representation of set-valued lower semi-continuous quasiconvex and convex functions. The results are based on a duality result for increasing set valued functions.
Recent years have witnessed a hot wave of deep neural networks in various domains; however, it is not yet well understood theoretically. A theoretical characterization of deep neural networks should point out their approximation ability and…
This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or…
Submodularity is one of the most well-studied properties of problem classes in combinatorial optimization and many applications of machine learning and data mining, with strong implications for guaranteed optimization. In this thesis, we…
This work is concerned with the convex analysis of functions defined on (not necessarily finite-dimensional) Hilbert spaces whose values depend solely on a certain ``spectrum'' of the arguments, a class we term ``spectral functions.'' We…
We study maximal regularity with respect to continuous functions for strongly continuous semigroups on locally convex spaces as well as its relation to the notion of admissible operators. This extends several results for classical strongly…
We investigate the computation of the gradient of the value function in parametric convex optimization problems. We derive general expression for the gradient of the value function in terms of the cost function, constraints and Lagrange…
A scale of the Frechet spaces of exponential type entire functions of one complex variable is considered. Certain special properties of subsets of these spaces consisting of Laguerre entire functions, which are obtained as uniform limits on…
In this paper, we show that some fundamental results for smooth mappings (e.g., the Brouwer degree formula, the implicit function and inverse function theorems, the mean value theorem, Sard's theorem, Hadamard's global invertibility…
We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the…
We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth…
Based on a refinement of the notion of internal sets in Colombeau's theory, so-called strongly internal sets, we introduce the space of generalized smooth functions, a maximal extension of Colombeau generalized functions. Generalized smooth…
In this paper, we unify and improve existing results on characterizing strict and almost stricty convex functions via subdifferential mapping, Moreau envelope, and proximal mappings. In particular, it is shown that if a convex function is…
This note presents an approach to studying the iterates of a mapping whose restriction to the complement of a finite set is continuous and open. The main examples to which the approach can be applied are piecewise monotone mappings defined…