Related papers: On Divergence-based Distance Functions for Multipl…
Path planning is an important problem in robotics. One way to plan a path between two points $x,y$ within a (not necessarily simply-connected) planar domain $\Omega$, is to define a non-negative distance function $d(x,y)$ on…
Distance functions between points in a domain are sometimes used to automatically plan a gradient-descent path towards a given target point in the domain, avoiding obstacles that may be present. A key requirement from such distance…
By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation ($BV$) in terms of suitable vector fields on a complete and separable metric measure space $(\mathbb{X},d,\mu)$…
We consider the problem of computing the distance between two piecewise-linear bivariate functions $f$ and $g$ defined over a common domain $M$. We focus on the distance induced by the $L_2$-norm, that is $\|f-g\|_2=\sqrt{\iint_M (f-g)^2}$.…
It is a generally shared opinion that significant information about the topology of a bounded domain $\Omega $ of a riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial\Omega}$, %, $d:\Omega\rightarrow…
Let the space $\mathbb{R}^n$ be endowed with a Minkowski structure $M$ (that is $M\colon \mathbb{R}^n \to [0,+\infty)$ is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class $C^2$),…
Density-based directed distances -- particularly known as divergences -- between probability distributions are widely used in statistics as well as in the adjacent research fields of information theory, artificial intelligence and machine…
This paper relates parameter distance to gradient breakdown for a broad class of nonlinear compositional functions. The analysis leads to a new distance function called deep relative trust and a descent lemma for neural networks. Since the…
This work establishes computable bounds between f-divergences for probability measures within a generalized quasi-$\varepsilon_{(M,m)}$-neighborhood framework. We make the following key contributions. (1) a unified characterization of local…
Let $\Omega$ be a domain in a smooth complete Finsler manifold, and let $G$ be the largest open subset of $\Omega$ such that for every $x$ in $G$ there is a unique closest point from $\partial \Omega$ to $x$ (measured in the Finsler…
There are many applications that benefit from computing the exact divergence between 2 discrete probability measures, including machine learning. Unfortunately, in the absence of any assumptions on the structure or independencies within…
Dempster-Shafer theory is widely applied in uncertainty modelling and knowledge reasoning due to its ability of expressing uncertain information. A distance between two basic probability assignments(BPAs) presents a measure of performance…
Divergence functions are measures of distance or dissimilarity between probability distributions that serve various purposes in statistics and applications. We propose decompositions of Wasserstein and Cram\'er distances$-$which compare two…
In this paper, we propose the Fourier Discrepancy Function, a new discrepancy to compare discrete probability measures. We show that this discrepancy takes into account the geometry of the underlying space. We prove that the Fourier…
This paper defines a distance function that measures the dissimilarity between planar geometric figures formed with straight lines. This function can in turn be used in partial matching of different geometric figures. For a given pair of…
We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^2$, with respect to some asymmetric norms. We allow the boundary of the domain to have corners. We obtain an explicit formula for the second…
The energy distance is used to test distributional equality, and as a loss function in machine learning. While $D^2(X, Y)=0$ only when $X\sim Y$, the sensitivity to different moments is of practical importance. This work considers $D^2(X,…
We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius of $h$. For a simply connected domain $D$ in the plane, let $\omega_h(0,\cdot;D)$ be the discrete harmonic measure at $0\in D$…
Computing the similarity between two probability distributions is a recurring theme across control. We introduce a unified family of distances between the probability distributions of two random variables that is based on the discrepancy…
In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information…