Statistical and computational rates in high rank tensor estimation

Chanwoo Lee; Miaoyan Wang

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arXiv · 2304.04043

Statistical and computational rates in high rank tensor estimation

Chanwoo Lee·Miaoyan Wang

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Abstract

Higher-order tensor datasets arise commonly in recommendation systems, neuroimaging, and social networks. Here we develop probable methods for estimating a possibly high rank signal tensor from noisy observations. We consider a generative latent variable tensor model that incorporates both high rank and low rank models, including but not limited to, simple hypergraphon models, single index models, low-rank CP models, and low-rank Tucker models. Comprehensive results are developed on both the statistical and computational limits for the signal tensor estimation. We find that high-dimensional latent variable tensors are of log-rank; the fact explains the pervasiveness of low-rank tensors in applications. Furthermore, we propose a polynomial-time spectral algorithm that achieves the computationally optimal rate. We show that the statistical-computational gap emerges only for latent variable tensors of order 3 or higher. Numerical experiments and two real data applications are presented to demonstrate the practical merits of our methods.

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BibTeX RIS

Chanwoo Lee, Miaoyan Wang. 2023-04-08. Statistical and computational rates in high rank tensor estimation. https://arxiv.org/abs/2304.04043

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