Related papers: Stability of Order Preserving Transforms
In positive characteristic, in contrast to the complex analytic case, vanishing cycles are highly sensitive to test functions (the maps to the henselian traits). We study this dependence and show that on a smooth surface, this dependence is…
In this short note we prove a theorem of the Stone-Weierstrass sort for subsets of the cone of non-decreasing continuous functions on compact partially ordered sets.
The overall performance or expected excess risk of an iterative machine learning algorithm can be decomposed into training error and generalization error. While the former is controlled by its convergence analysis, the latter can be tightly…
The aim of this paper is to prove characterization theorems for higher order derivations. Among others we prove that the system defining higher order derivations is stable. Further characterization theorems in the spirit of N.~G.~de Bruijn…
This is a survey paper concerning some theorems on stochastic convex ordering and their applications to functional inequalities for convex functions. We present the recent results on those subjects
We prove that the non-commutative perspective of an operator convex function is the unique extension of the corresponding commutative perspective that preserves homogeneity and convexity.
It is well-known that every convex function admits an affine support at every interior point of a domain. Convex functions of higher order (precisely of an odd order) have a similar property: they are supported by the polynomials of degree…
The algebraic stability theorem for $\mathbb{R}$-persistence modules is a fundamental result in topological data analysis. We present a stability theorem for $n$-dimensional rectangle decomposable persistence modules up to a constant…
Given a closed, convex cone $K\subseteq \mathbb{R}^n$, a multivariate polynomial $f\in\mathbb{C}[\mathbf{z}]$ is called $K$-stable if the imaginary parts of its roots are not contained in the relative interior of $K$. If $K$ is the…
A function between two metric spaces is said to be totally bounded regular if it preserves totally bounded sets. These functions need not be continuous in general. Hence the purpose of this article is to study such functions vis-\'a-vis…
Fractional derivatives are a well-studied generalization of integer order derivatives. Naturally, for optimization, it is of interest to understand the convergence properties of gradient descent using fractional derivatives. Convergence…
We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent literature, we give a simple argument that proves spectral stability of all…
This paper is concerned with convergence of stochastic gradient algorithms with momentum terms in the nonconvex setting. A class of stochastic momentum methods, including stochastic gradient descent, heavy ball, and Nesterov's accelerated…
The main aim of this paper is the investigation of the stability problem for ordinary delay differential equations. More precisely, we would like to study the following problem. Assume that for a continuous function a given delay…
We study the exponential stability of evolutionary equations. The focus is laid on second order problems and we provide a way to rewrite them as a suitable first order evolutionary equation, for which the stability can be proved by using…
Variational analysis provides the theoretical foundations and practical tools for constructing optimization algorithms without being restricted to smooth or convex problems. We survey the central concepts in the context of a concrete but…
In the plane, we consider the problem of reconstructing a domain from the normal derivative of its Green's function (with fixed pole) relative to the Dirichlet problem for the Laplace operator. By means of the theory of conformal mappings,…
In this paper we are interested in the inverse problem of recovering a compact supported function from its truncated Fourier transform. We derive new Lipschitz stability estimates for the inversion in terms of the truncation parameter. The…
Imposing orthogonality on the layers of neural networks is known to facilitate the learning by limiting the exploding/vanishing of the gradient; decorrelate the features; improve the robustness. This paper studies the theoretical properties…
We characterize real and complex functions which, when applied entrywise to square matrices, yield a positive definite matrix if and only if the original matrix is positive definite. We refer to these transformations as sign preservers.…