A Scalable Factorization Approach for High-Order Structured Tensor Recovery

TL;DR

This paper introduces a scalable Riemannian gradient descent framework for high-order tensor decomposition, providing convergence guarantees with polynomial scaling in tensor order, improving efficiency over existing methods.

Contribution

The authors develop a unified Riemannian optimization approach for tensor factorization, with convergence guarantees that scale polynomially with tensor order, enhancing scalability and theoretical understanding.

Findings

RGD converges linearly to the ground-truth tensor.

Convergence rate scales polynomially with tensor order N.

The framework applies to various tensor formats like Tucker and tensor-train.

Abstract

Tensor decompositions, which represent an $N$-order tensor using approximately $N$ factors of much smaller dimensions, can significantly reduce the number of parameters. This is particularly beneficial for high-order tensors, as the number of entries in a tensor grows exponentially with the order. Consequently, they are widely used in signal recovery and data analysis across domains such as signal processing, machine learning, and quantum physics. A computationally and memory-efficient approach to these problems is to optimize directly over the factors using local search algorithms such as gradient descent, a strategy known as the factorization approach in matrix and tensor optimization. However, the resulting optimization problems are highly nonconvex due to the multiplicative interactions between factors, posing significant challenges for convergence analysis and recovery guarantees. In this paper, we present a unified framework for the factorization approach to solving various tensor decomposition problems. Specifically, by leveraging the canonical form of tensor decompositions--where most factors are constrained to be orthonormal to mitigate scaling ambiguity--we apply Riemannian gradient descent (RGD) to optimize these orthonormal factors on the Stiefel manifold. Under a mild condition on the loss function, we establish a Riemannian regularity condition for the factorized objective and prove that RGD converges to the ground-truth tensor at a linear rate when properly initialized. Notably, both the initialization requirement and the convergence rate scale polynomially rather than exponentially with $N$, improving upon existing results for Tucker and tensor-train format tensors.