arXiv subjects

Vladimir Nikishkin

Publications and source records attributed to Vladimir Nikishkin.

4 recordsLinked to original sources

arXiv2017

Near-Optimal Closeness Testing of Discrete Histogram Distributions

We investigate the problem of testing the equivalence between two discrete histograms. A {\em $k$-histogram} over $[n]$ is a probability distribution that is piecewise constant over some set of $k$ intervals over $[n]$. Histograms have been extensively studied in computer science and statistics. Given a set of samples from two $k$-histogram distributions $p, q$ over $[n]$, we want to distinguish (with high probability) between the cases that $p = q$ and $\|p-q\|_1 \geq ε$. The main contribution of this paper is a new algorithm for this testing problem and a nearly matching information-theoretic lower bound. Specifically, the sample complexity of our algorithm matches our lower bound up to a logarithmic factor, improving on previous work by polynomial factors in the relevant parameters. Our algorithmic approach applies in a more general setting and yields improved sample upper bounds for testing closeness of other structured distributions as well.

cs.DS↗

arXiv2015

Optimal Algorithms and Lower Bounds for Testing Closeness of Structured Distributions

We give a general unified method that can be used for $L_1$ {\em closeness testing} of a wide range of univariate structured distribution families. More specifically, we design a sample optimal and computationally efficient algorithm for testing the equivalence of two unknown (potentially arbitrary) univariate distributions under the $\mathcal{A}_k$-distance metric: Given sample access to distributions with density functions $p, q: I \to \mathbb{R}$, we want to distinguish between the cases that $p=q$ and $\|p-q\|_{\mathcal{A}_k} \ge ε$ with probability at least $2/3$. We show that for any $k \ge 2, ε>0$, the {\em optimal} sample complexity of the $\mathcal{A}_k$-closeness testing problem is $Θ(\max\{ k^{4/5}/ε^{6/5}, k^{1/2}/ε^2 \})$. This is the first $o(k)$ sample algorithm for this problem, and yields new, simple $L_1$ closeness testers, in most cases with optimal sample complexity, for broad classes of structured distributions.

cs.DS↗

arXiv2014

Testing Identity of Structured Distributions

We study the question of identity testing for structured distributions. More precisely, given samples from a {\em structured} distribution $q$ over $[n]$ and an explicit distribution $p$ over $[n]$, we wish to distinguish whether $q=p$ versus $q$ is at least $ε$-far from $p$, in $L_1$ distance. In this work, we present a unified approach that yields new, simple testers, with sample complexity that is information-theoretically optimal, for broad classes of structured distributions, including $t$-flat distributions, $t$-modal distributions, log-concave distributions, monotone hazard rate (MHR) distributions, and mixtures thereof.

cs.DS↗

arXiv2012

Amortized communication complexity of an equality predicate

We study the communication complexity of a direct sum of independent copies of the equality predicate. We prove that the probabilistic communication complexity of this problem is equal to O(N); computational complexity of the proposed protocol is polynomial in size of inputs. Our protocol improves the result achieved in 1995(Feder, Kushilevitz, Naor, Nisan). Our construction is based on two techniques: Nisan's pseudorandom generator (1992) and Smith's string synchronization algorithm (2007).

cs.CC↗