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Title: Polynomial-Time Power-Sum Decomposition of Polynomials
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.  more » « less
Award ID(s):
2047933
PAR ID:
10407348
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)
Page Range / eLocation ID:
956 to 967
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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