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arXiv · 1610.01980

Polynomial-time Tensor Decompositions with Sum-of-Squares

Tengyu Ma·Jonathan Shi·David Steurer
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Abstract

We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete 3-tensors and learning overcomplete dictionaries with constant relative sparsity. We also give the first robust analysis for decomposing overcomplete 4-tensors in the smoothed analysis model. A key ingredient of our analysis is to establish small spectral gaps in moment matrices derived from solutions to sum-of-squares relaxations. To enable this analysis we augment sum-of-squares relaxations with spectral analogs of maximum entropy constraints.

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Tengyu Ma, Jonathan Shi, David Steurer. 2016-10-06. Polynomial-time Tensor Decompositions with Sum-of-Squares. https://arxiv.org/abs/1610.01980

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