Related papers: Robustness of statistical models

We prove that for every smooth compact manifold $M$ and any $r \ge 1$, whenever there is an open domain in $\mathrm{Diff}^r(M)$ exhibiting a persistent homoclinic tangency related to a basic set with a sectionally dissipative periodic…

The Gaussian-filtered Navier-Stokes equations are examined theoretically and a generalized theory of their numerical stability is proposed. Using the exact expansion series of subfilter-scale stresses or integration by parts, the terms…

We construct an injective map from the set of holomorphic equivalence classes of neighborhoods $M$ of a compact complex manifold $C$ into ${\mathbb C}^m$ for some $m<\infty$ when $(TM)|_C$ is fixed and the normal bundle of $C$ in $M$ is…

We obtain a generic regularity result for stationary integral $n$-varifolds with only strongly isolated singularities inside $N$-dimensional Riemannian manifolds, in absence of any restriction on the dimension ($n\geq 2$) and codimension.…

We prove Llarull-type rigidity for $S^{n-m}\times\mathbb{T}^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times\mathbb{T}^m$ whose spherical projection is area non-increasing, and…

The Hammersley-Clifford theorem states that if the support of a Markov random field has a safe symbol then it is a Gibbs state with some nearest neighbour interaction. In this paper we generalise the theorem with an added condition that the…

Randomized smoothing has shown promising certified robustness against adversaries in classification tasks. Despite such success with only zeroth-order access to base models, randomized smoothing has not been extended to a general form of…

We prove a version of Whitney's strong embedding theorem for isometric embeddings within the general setting of the Nash-Kuiper h-principle. More precisely, we show that any $n$-dimensional smooth compact manifold admits infinitely many…

We provide a theoretical framework for a wide class of generalized posteriors that can be viewed as the natural Bayesian posterior counterpart of the class of M-estimators in the frequentist world. We call the members of this class…

In this article we study two "strong" topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modeled on a locally convex space. Namely, we construct Whitney type topologies…

In many structured prediction problems, complex relationships between variables are compactly defined using graphical structures. The most prevalent graphical prediction methods---probabilistic graphical models and large margin…

We prove that for a group $\mathrm{SO}_n(\mathrm{R}) \subset G \subset \mathrm{GL}_n (\mathrm{R})$, any $G$-structure on a smooth manifold can be endowed with a torsion free connection which is locally the Levi-Civita connection of a…

Feature attributions are a popular tool for explaining the behavior of Deep Neural Networks (DNNs), but have recently been shown to be vulnerable to attacks that produce divergent explanations for nearby inputs. This lack of robustness is…

Let $T\subset{\mathbb R}^n$ be a semialgebraic set and let $\mu\ge0$ be a non-negative integer. We say that $T$ is a {\em Nash $\mu$-approximation target space} (or a $({\mathcal N},\mu)$-${\tt ats}$ for short) if it has the following…

We investigate a special case of hereditary property in graphs, referred to as {\em robustness}. A property (or structure) is called robust in a graph $G$ if it is inherited by all the connected spanning subgraphs of $G$. We motivate this…

A $\mathrm{G}_2$-structure on a $7$-manifold $M$ is called a $\mathrm{G}_2T$-structure if $M$ admits a $\mathrm{G}_2$-connection $\nabla^T$ with totally skew-symmetric torsion $T_\varphi$. If furthermore, $T_\varphi$ is closed then it is…

It follows from the Marcus-Spielman-Srivastava proof of the Kadison-Singer conjecture that if $G=(V,E)$ is a $\Delta$-regular dense expander then there is an edge-induced subgraph $H=(V,E_H)$ of $G$ of constant maximum degree which is also…

Stochastic dynamical systems consisting of non-invertible continuous maps on an interval are studied. It is proved that if they satisfy the recently introduced so-called $\mu$-injectivity and some mild assumptions, then proximality,…

We establish a correspondence between information geometry and gauge theory. First, we define an important class of statistical manifolds, that is normalized and satisfies a conservation field equation. Second, we prove that for a…

In this paper, we study the entropy of a hard random geometric graph (RGG), a commonly used model for spatial networks, where the connectivity is governed by the distances between the nodes. Formally, given a connection range $r$, a hard…