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
