Efficient learning of simplices

TL;DR

This paper presents an efficient polynomial-time algorithm for learning arbitrary n-dimensional simplices from uniform samples, utilizing third-moment analysis and a novel connection to Independent Component Analysis (ICA).

Contribution

It introduces a new polynomial-time algorithm for simplex learning based on third moments and establishes a direct reduction from simplex learning to ICA.

Findings

Algorithm runs in polynomial time in n.

Uses third moments instead of fourth moments for analysis.

Establishes a reduction from learning simplices to ICA.

Abstract

We show an efficient algorithm for the following problem: Given uniformly random points from an arbitrary n-dimensional simplex, estimate the simplex. The size of the sample and the number of arithmetic operations of our algorithm are polynomial in n. This answers a question of Frieze, Jerrum and Kannan [FJK]. Our result can also be interpreted as efficiently learning the intersection of n+1 half-spaces in R^n in the model where the intersection is bounded and we are given polynomially many uniform samples from it. Our proof uses the local search technique from Independent Component Analysis (ICA), also used by [FJK]. Unlike these previous algorithms, which were based on analyzing the fourth moment, ours is based on the third moment. We also show a direct connection between the problem of learning a simplex and ICA: a simple randomized reduction to ICA from the problem of learning a simplex. The connection is based on a known representation of the uniform measure on a simplex. Similar representations lead to a reduction from the problem of learning an affine transformation of an n-dimensional l_p ball to ICA.