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We give efficient algorithms for finding power-sum decomposition of an input polynomial with component s. The case of linear s 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 s and , prior work of [GHK15] yields an algorithm only when . On the other hand, the more general recent result of [GKS20] builds an algebraic approach to handle any components but only when is large enough (while yielding no bounds for or even ) 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 and quadratic s. Specifically, our algorithm succeeds in decomposing a sum of generic quadratic s for and more generally the th power-sum of generic degree- polynomials for any . 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 s have random Gaussian coefficients. 

Consider a system of m polynomial equations {pi(x)=bi}i≤m of degree D≥2 in n-dimensional variable x∈ℝn such that each coefficient of every pi and bis are chosen at random and independently from some continuous distribution. We study the basic question of determining the smallest m -- the algorithmic threshold -- for which efficient algorithms can find refutations (i.e. certificates of unsatisfiability) for such systems. This setting generalizes problems such as refuting random SAT instances, low-rank matrix sensing and certifying pseudo-randomness of Goldreich's candidate generators and generalizations. We show that for every d∈ℕ, the (n+m)O(d)-time canonical sum-of-squares (SoS) relaxation refutes such a system with high probability whenever m≥O(n)⋅(nd)D−1. We prove a lower bound in the restricted low-degree polynomial model of computation which suggests that this trade-off between SoS degree and the number of equations is nearly tight for all d. We also confirm the predictions of this lower bound in a limited setting by showing a lower bound on the canonical degree-4 sum-of-squares relaxation for refuting random quadratic polynomials. Together, our results provide evidence for an algorithmic threshold for the problem at m≳O˜(n)⋅n(1−δ)(D−1) for 2nδ-time algorithms for all δ.