Scalable symmetric Tucker tensor decomposition

TL;DR

This paper introduces scalable algorithms for symmetric Tucker tensor decomposition, particularly for large moment tensors, using projected gradient descent and eigenvalue methods, with proven convergence and practical efficiency.

Contribution

It develops scalable adaptations of PGD and HOEVD for large symmetric tensors, enabling efficient decomposition of high-dimensional moment tensors.

Findings

Algorithms are efficient and scalable for large datasets.

Moment tensor decompositions are applicable to real-world data.

Convergence of PGD on the Grassmannian is theoretically established.

Abstract

We study the best low-rank Tucker decomposition of symmetric tensors. The motivating application is decomposing higher-order multivariate moments. Moment tensors have special structure and are important to various data science problems. We advocate for projected gradient descent (PGD) method and higher-order eigenvalue decomposition (HOEVD) approximation as computation schemes. Most importantly, we develop scalable adaptations of the basic PGD and HOEVD methods to decompose sample moment tensors. With the help of implicit and streaming techniques, we evade the overhead cost of building and storing the moment tensor. Such reductions make computing the Tucker decomposition realizable for large data instances in high dimensions. Numerical experiments demonstrate the efficiency of the algorithms and the applicability of moment tensor decompositions to real-world datasets. Finally we study the convergence on the Grassmannian manifold, and prove that the update sequence derived by the PGD solver achieves first- and second-order criticality.