Max vs Min: Tensor Decomposition and ICA with nearly Linear Sample Complexity

TL;DR

This paper introduces a new technique that significantly reduces the sample complexity of tensor decomposition and ICA algorithms, achieving nearly linear sample complexity in the dimension, which improves previous bounds.

Contribution

The paper presents a simple, general method for reducing sample complexity in tensor and matrix decomposition algorithms, with applications to ICA and Gaussian mixture models.

Findings

Polynomial-time ICA algorithm with nearly linear sample complexity

Improved bounds on sample complexity for tensor decompositions

Technique applicable to Gaussian mixture models

Abstract

We present a simple, general technique for reducing the sample complexity of matrix and tensor decomposition algorithms applied to distributions. We use the technique to give a polynomial-time algorithm for standard ICA with sample complexity nearly linear in the dimension, thereby improving substantially on previous bounds. The analysis is based on properties of random polynomials, namely the spacings of an ensemble of polynomials. Our technique also applies to other applications of tensor decompositions, including spherical Gaussian mixture models.