IEEE Transactions on Information Theory· 2018Q1
Tensor SVD: Statistical and Computational Limits
- 181citations
- Q1SCImago
- 2018year
Short summary
A new framework for tensor SVD reveals three distinct phases based on signal-to-noise ratio (SNR), showing classical methods are optimal at high SNR but fail at low SNR, while optimal estimation at moderate SNR is NP-hard.
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Key points
- Tensor SVD performance exhibits three phases based on SNR: strong, moderate, and weak.
- Classical higher-order orthogonal iteration achieves minimax optimal estimation rates in the strong SNR phase.
- Consistent estimation is information-theoretically impossible in the weak SNR phase.
- Optimal estimation in the moderate SNR phase is NP-hard, with no known polynomial-time consistent algorithms.
AI-generated from the title and abstract; the full text is not read.
Abstract
In this paper, we propose a general framework for tensor singular value decomposition (tensor singular value decomposition (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.
The authors' abstract, as published at the source. IEEE Transactions on Information Theory, 2018 · DOI ↗
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Field: Computational Mathematics
Computational MathematicsMathematics