# Polynomial-Time Power-Sum Decomposition of Polynomials

We give efficient algorithms for finding power-sum decomposition of an input polynomial P(x)= ∑_i≤ m p_i(x)^d with component p_is. The case of linear p_is is equivalent to the well-studied tensor decomposition problem while the quadratic case occurs naturally in studying identifiability of non-spherical Gaussian mixtures from low-order moments. Unlike tensor decomposition, both the unique identifiability and algorithms for this problem are not well-understood. For the simplest setting of quadratic p_is and d=3, prior work of Ge, Huang and Kakade yields an algorithm only when m ≤Õ(√(n)). On the other hand, the more general recent result of Garg, Kayal and Saha builds an algebraic approach to handle any m=n^O(1) components but only when d is large enough (while yielding no bounds for d=3 or even d=100) and only handles an inverse exponential noise. Our results obtain a substantial quantitative improvement on both the prior works above even in the base case of d=3 and quadratic p_is. Specifically, our algorithm succeeds in decomposing a sum of m ∼Õ(n) generic quadratic p_is for d=3 and more generally the dth power-sum of m ∼ n^2d/15 generic degree-K polynomials for any K ≥ 2. Our algorithm relies only on basic numerical linear algebraic primitives, is exact (i.e., obtain arbitrarily tiny error up to numerical precision), and handles an inverse polynomial noise when the p_is have random Gaussian coefficients. Our main tool is a new method for extracting the linear span of p_is by studying the linear subspace of low-order partial derivatives of the input P. For establishing polynomial stability of our algorithm in average-case, we prove inverse polynomial bounds on the smallest singular value of certain correlated random matrices with low-degree polynomial entries that arise in our analyses.

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