Tensor SVD: Statistical and Computational Limits

Anru Zhang; Dong Xia

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arXiv · 1703.02724

Tensor SVD: Statistical and Computational Limits

Anru Zhang·Dong Xia

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Abstract

In this paper, we propose a general framework for tensor singular value decomposition (tensor SVD), which focuses on the methodology and theory for extracting the hidden low-rank structure from high-dimensional tensor data. Comprehensive results are developed on both the statistical and computational limits for tensor SVD. This problem exhibits three different phases according to the signal-to-noise ratio (SNR). In particular, with strong SNR, we show that the classical higher-order orthogonal iteration achieves the minimax optimal rate of convergence in estimation; with weak SNR, the information-theoretical lower bound implies that it is impossible to have consistent estimation in general; with moderate SNR, we show that the non-convex maximum likelihood estimation provides optimal solution, but with NP-hard computational cost; moreover, under the hardness hypothesis of hypergraphic planted clique detection, there are no polynomial-time algorithms performing consistently in general.

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BibTeX RIS

Anru Zhang, Dong Xia. 2017-03-08. Tensor SVD: Statistical and Computational Limits. https://arxiv.org/abs/1703.02724

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