On the non-detectability of spiked large random tensors

TL;DR

This paper investigates the limits of detecting low-rank large random tensors in high-dimensional noisy environments, showing that below a certain threshold, detection becomes statistically impossible.

Contribution

It extends existing results to higher-rank tensors and establishes a threshold below which detection is fundamentally impossible in high dimensions.

Findings

Detection is impossible below a specific tensor parameter threshold.

No test outperforms random guessing in the undetectable regime.

Results generalize previous rank 1 tensor detection findings.

Abstract

This paper addresses the detection of a low rank high-dimensional tensor corrupted by an additive complex Gaussian noise. In the asymptotic regime where all the dimensions of the tensor converge towards $+\infty$ at the same rate, existing results devoted to rank 1 tensors are extended. It is proved that if a certain parameter depending on the low rank tensor is below a threshold, then the null hypothesis and the presence of the low rank tensor are undistinguishable hypotheses in the sense that no test performs better than a random choice.