The all-or-nothing phenomenon in sparse tensor PCA

We study the statistical problem of estimating a rank-one sparse tensor corrupted by additive gaussian noise, a Gaussian additive model also known as sparse tensor PCA. We show that for Bernoulli and Bernoulli-Rademacher distributed signals and for all sparsity levels which are sublinear in the dimension of the signal, the sparse tensor PCA model exhibits a phase transition called the all-or-nothing phenomenon. This is the property that for some signal-to-noise ratio (SNR) SNR_c and any fixed e > 0, if the SNR of the model is below (1 - e) SNR_c, then it is impossible to achieve any arbitrarily small constant correlation with the hidden signal, while if the SNR is above (1 + e) 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.