The All-or-Nothing Phenomenon in Sparse Tensor PCA

TL;DR

This paper demonstrates a sharp phase transition in sparse tensor PCA, where below a critical SNR level, signals cannot be detected, and above it, signals can be almost perfectly recovered, revealing an all-or-nothing phenomenon.

Contribution

It establishes the all-or-nothing phase transition in sparse tensor PCA for all sublinear sparsity levels, extending previous results to this specific tensor setting.

Findings

Phase transition at a critical SNR level for sparse tensor PCA.

Below the threshold, no correlation with the true signal is achievable.

Above the threshold, near-perfect signal recovery is possible.

Abstract

We study the statistical problem of estimating a rank-one sparse tensor corrupted by additive Gaussian noise, a model also known as sparse tensor PCA. We show that for Bernoulli and Bernoulli-Rademacher distributed signals and \emph{for all} sparsity levels which are sublinear in the dimension of the signal, the sparse tensor PCA model exhibits a phase transition called the \emph{all-or-nothing phenomenon}. This is the property that for some signal-to-noise ratio (SNR) $\mathrm{SNR_c}$ and any fixed $\epsilon>0$, if the SNR of the model is below $\left(1-\epsilon\right)\mathrm{SNR_c}$, then it is impossible to achieve any arbitrarily small constant correlation with the hidden signal, while if the SNR is above $\left(1+\epsilon \right)\mathrm{SNR_c}$, then it is possible to achieve almost perfect correlation with the hidden signal. The all-or-nothing phenomenon was initially established in the context of sparse linear regression, and over the last year also in the context of sparse 2-tensor (matrix) PCA, Bernoulli group testing, and generalized linear models. Our results follow from a more general result showing that for any Gaussian additive model with a discrete uniform prior, the all-or-nothing phenomenon follows as a direct outcome of an appropriately defined "near-orthogonality" property of the support of the prior distribution.