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# Statistical and Computational Limits for Tensor-on-Tensor Association Detection

*Proceedings of Thirty Sixth Conference on Learning Theory*, PMLR 195:5260-5310, 2023.

#### Abstract

In this paper, we consider the tensor-on-tensor association detection problem, where the goal is to detect whether there is an association between the tensor responses to tensor covariates linked via a low-rank tensor parameter. We first In this paper, we consider the tensor-on-tensor association detection problem, where the goal is to detect whether there is an association between the tensor responses to tensor covariates linked via a low-rank tensor parameter. We first develop tight bounds on the signal-to-noise ratio (SNR) such that the detection problem is statistically possible. We then provide testing procedures that succeed when the SNR is above the threshold. On the other hand, the statistical optimal tests often require computing the largest singular value of a given tensor, which can be NP-hard in general. To complement that, we develop efficient polynomial-time testing procedures with provable guarantees. We also develop matching lower bounds under the Statistical Query model and show that the SNRs required by the proposed polynomial-time algorithms are essential for computational efficiency. We identify a gap that appears between the SNR requirements of the optimal unconstrained-time tests and polynomial-time tests if and only if the sum of the tensor response order and the tensor covariate order is no less than three. To our best knowledge, this is the first complete characterization of the statistical and computational limits for the general tensor-on-tensor association detection problem. Our findings significantly generalize the results in the literature on signal detection in linear regression and low-rank matrix trace regression. Finally, the connection on the computational hardness of the detection problem and the corresponding estimation problem is discussed.