High-Dimensional Maximum Likelihood Estimation of Multi-Spiked Tensor PCA

Abstract

We study the maximum likelihood estimation of the multi-spiked tensor PCA problem. In particular, the tensor of interest is the sum of a low-rank tensor and a tensor whose entries are independent and identically distributed standard Gaussian random variables. The low-rank tensor is a linear combination of rank-one tensors scaled by signal-to-noise ratios. The recovery of the signal vectors (which determine the rank-one tensors) is known as the multi-spiked tensor PCA problem. We prove a variational formula for the high-dimensional limit of the maximum likelihood estimation of the planted signals. This formula is achieved using conjectured results regarding the constrained ground state energy of the spherical mixed vector p-spin model from statistical physics. In this setting, we show that the high-dimensional limit is equivalent to the maximization of an infimum problem with additional penalty terms. This limit acts as a basis for the analysis of the maximum likelihood estimators performance and the investigation of the necessary conditions for their success.