February 2024 Statistical-computational trade-offs in tensor PCA and related problems via communication complexity
Rishabh Dudeja, Daniel Hsu
Author Affiliations +
Ann. Statist. 52(1): 131-156 (February 2024). DOI: 10.1214/23-AOS2331


Tensor PCA is a stylized statistical inference problem introduced by Montanari and Richard to study the computational difficulty of estimating an unknown parameter from higher-order moment tensors. Unlike its matrix counterpart, Tensor PCA exhibits a statistical-computational gap, that is, a sample size regime where the problem is information-theoretically solvable but conjectured to be computationally hard. This paper derives computational lower bounds on the run-time of memory bounded algorithms for Tensor PCA using communication complexity. These lower bounds specify a trade-off among the number of passes through the data sample, the sample size and the memory required by any algorithm that successfully solves Tensor PCA. While the lower bounds do not rule out polynomial-time algorithms, they do imply that many commonly-used algorithms, such as gradient descent and power method, must have a higher iteration count when the sample size is not large enough. Similar lower bounds are obtained for non-Gaussian component analysis, a family of statistical estimation problems in which low-order moment tensors carry no information about the unknown parameter. Finally, stronger lower bounds are obtained for an asymmetric variant of Tensor PCA and related statistical estimation problems. These results explain why many estimators for these problems use a memory state that is significantly larger than the effective dimensionality of the parameter of interest.

Funding Statement

DH acknowledges support from NSF awards CCF-1740833 and IIS-1563785 and a JP Morgan Faculty Award.


This project was done when RD was a graduate student at Columbia University and a postdoctoral scholar at Harvard University.


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Rishabh Dudeja. Daniel Hsu. "Statistical-computational trade-offs in tensor PCA and related problems via communication complexity." Ann. Statist. 52 (1) 131 - 156, February 2024. https://doi.org/10.1214/23-AOS2331


Received: 1 June 2022; Revised: 1 May 2023; Published: February 2024
First available in Project Euclid: 7 March 2024

MathSciNet: MR4718410
Digital Object Identifier: 10.1214/23-AOS2331

Primary: 62F30 , 68Q17

Keywords: Statistical-computational trade-offs , tensors

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 1 • February 2024
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