Tensor SVD: Statistical and Computational Limits

Tensors, or high-order arrays, attract much attention in recent research. In this paper, we propose a general framework for tensor principal component analysis (tensor PCA), which focuses on the methodology and theory for extracting the hidden low-rank structure from the high-dimensional tensor data. A unified solution is provided to tensor PCA in both statistical limits and computational costs. The problem exhibits three different phases according to the signal-noise-ratio (SNR). In particular, with strong SNR, we propose a fast spectral power iteration method that achieves the minimax optimal rate of convergence in estimation; with weak SNR, the information-theoretical lower bound shows 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 the hypergraphic planted clique detection, there are no polynomial-time algorithms performing consistently in general. Simulation studies show that the proposed spectral power iteration method has good performs under a variety of settings.