A Random Matrix Analysis of Symmetric Tensor Unfolding

Abstract

We study a fundamental problem in machine learning: partial recovery of a planted signal $\boldsymbol{v} \in \mathbb{S}^{n-1}$ from a spiked tensor $\boldsymbol{\mathsf{X}} = \lambda \, \boldsymbol{v}^{\otimes p} + \boldsymbol{\mathsf{Z}}$, where $\boldsymbol{\mathsf{Z}}$ has symmetric Gaussian entries of variance $\asymp n^{-1}$. While previous works replace $\boldsymbol{\mathsf{Z}}$ by an i.i.d. standard Gaussian tensor to circumvent technical difficulties caused by dependencies, this paper directly analyzes the symmetric model via random matrix theory. We show that the square unfolding algorithm is subject to a sharp phase transition for partial recovery at $\lambda \asymp n^{(p-2) / 4}$, validating an open conjecture from Montanari and Richard (2014). We get precise expressions for the limiting spectral distribution, deterministic equivalent, phase transition threshold and overlap. The main message is that unfolded symmetric tensors spectrally behave like standard Wigner matrices, up to simple transformations accounting for high-order symmetries. We consider extensions to multiple orthogonal spikes, recursive unfolding and non-square unfolding via disproportionate random matrix theory, leading to an interesting conjectured statistical-computational tradeoff regarding the choice of unfolding: square versus long.