Related papers: Estimating the distance from testable affine-invar…
Let us denote by $\mathcal K_n$ the hyperspace of all convex bodies of $\mathbb R^n$ equipped with the Hausdorff distance topology. An affine invariant point $p$ is a continuous and Aff(n)-equivariant map $p:\mathcal K_n\to \mathbb R^n$,…
Let $V$ be an inner product space, and $x, y \in V$; the conjecture is made that, for any $p \in [1, \infty]$, the function $d_p(x, y):=\|x-y\|/(\|x\|^p+ \|y\|^p)^{1/p}$ is a distance on $V$.
Given an unknown $\mathbb{R}^n$-valued function $f$ on a metric space $X$, can we approximate the persistent homology of $f$ from a finite sampling of $X$ with known pairwise distances and function values? This question has been answered in…
This article proposes a method to consistently estimate functionals $\frac1p\sum_{i=1}^pf(\lambda_i(C_1C_2))$ of the eigenvalues of the product of two covariance matrices $C_1,C_2\in\mathbb{R}^{p\times p}$ based on the empirical estimates…
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…
Given a piecewise linear (PL) function $p$ defined on an open subset of $\R^n$, one may construct by elementary means a unique polyhedron with multiplicities $\D(p)$ in the cotangent bundle $\R^n\times \R^{n*}$ representing the graph of the…
In this paper, using altering distance functions we obtain a generalization of the results due to B.K. Das and S. Gupta as well as the results given by B. Samet and H. Yazid. Moreover, we study the so-called property P for the contraction…
We show that for any constant $\epsilon > 0$ and $p \ge 1$, it is possible to distinguish functions $f : \{0,1\}^n \to [0,1]$ that are submodular from those that are $\epsilon$-far from every submodular function in $\ell_p$ distance with a…
Applications in machine learning and data mining require computing pairwise Lp distances in a data matrix A. For massive high-dimensional data, computing all pairwise distances of A can be infeasible. In fact, even storing A or all pairwise…
Given two distributions $\mathcal{P}$ and $\mathcal{Q}$ over a high-dimensional domain $\{0,1\}^n$, and a parameter $\varepsilon$, the goal of distance estimation is to determine the statistical distance between $\mathcal{P}$ and…
The goal of property testing is to quickly distinguish between objects which satisfy a property and objects that are $\epsilon$-far from satisfying the property. There are now several general results in this area which show that natural…
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…
Property testers are fast, randomized "election polling"-type algorithms that determine if an input (e.g., graph or hypergraph) has a certain property or is $\varepsilon$-far from the property. In the dense graph model of property testing,…
This paper studies the approximation of generalized ridge functions, namely of functions which are constant along some submanifolds of $\mathbb{R}^N$. We introduce the notion of linear-sleeve functions, whose function values only depend on…
This paper presents a general study of one-dimensional differentiability for functionals defined on convex domains that are not necessarily open. The local approximation is carried out using affine functionals, as opposed to linear…
One of the prevailing ideas in geometric and topological data analysis is to provide descriptors that encode useful information about hidden objects from observed data. The Reeb graph is one such descriptor for a given scalar function. The…
This paper addresses the problem of finding a B-term wavelet representation of a given discrete function $f \in \real^n$ whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance…
Fix a prime $p$ and a positive integer $R$. We study the property testing of functions $\mathbb F_p^n\to[R]$. We say that a property is testable if there exists an oblivious tester for this property with one-sided error and constant query…
Reasoning about distance is indispensable for establishing or avoiding contact in manipulation tasks. To this end, we present an online approach for learning implicit representations of signed distance using piecewise polynomial basis…
We study the maximum Hamming distance (or rather, the complementary notion of "minimum approximability") of a general function on a finite group $G$ to either of the sets $\operatorname{End}(G)$ and $\operatorname{Aff}(G)$, of group…