Selective Multiple Power Iteration: from Tensor PCA to gradient-based exploration of landscapes

TL;DR

This paper introduces SMPI, a novel tensor PCA algorithm that significantly outperforms existing methods by leveraging noise in low SNR regimes, with implications for tensor decomposition and high-dimensional optimization landscapes.

Contribution

The paper presents SMPI, a new power iteration-based algorithm for Tensor PCA that improves recovery performance and offers new theoretical insights into non-convex landscape optimization.

Findings

SMPI outperforms existing algorithms in tensor PCA tasks.

Noise plays a beneficial role in signal recovery at low SNR.

Theoretical analysis links SMPI's success to properties of high-dimensional landscapes.

Abstract

We propose Selective Multiple Power Iterations (SMPI), a new algorithm to address the important Tensor PCA problem that consists in recovering a spike $\bf{v_0}^{\otimes k}$ corrupted by a Gaussian noise tensor $\bf{Z} \in (\mathbb{R}^n)^{\otimes k}$ such that $\bf{T}=\sqrt{n} \beta \bf{v_0}^{\otimes k} + \bf{Z}$ where $\beta$ is the signal-to-noise ratio (SNR). SMPI consists in generating a polynomial number of random initializations, performing a polynomial number of symmetrized tensor power iterations on each initialization, then selecting the one that maximizes $\langle \bf{T}, \bf{v}^{\otimes k} \rangle$. Various numerical simulations for $k=3$ in the conventionally considered range $n \leq 1000$ show that the experimental performances of SMPI improve drastically upon existent algorithms and becomes comparable to the theoretical optimal recovery. We show that these unexpected performances are due to a powerful mechanism in which the noise plays a key role for the signal recovery and that takes place at low $\beta$. Furthermore, this mechanism results from five essential features of SMPI that distinguish it from previous algorithms based on power iteration. These remarkable results may have strong impact on both practical and theoretical applications of Tensor PCA. (i) We provide a variant of this algorithm to tackle low-rank CP tensor decomposition. These proposed algorithms also outperforms existent methods even on real data which shows a huge potential impact for practical applications. (ii) We present new theoretical insights on the behavior of SMPI and gradient descent methods for the optimization in high-dimensional non-convex landscapes that are present in various machine learning problems. (iii) We expect that these results may help the discussion concerning the existence of the conjectured statistical-algorithmic gap.