Optimizing Orthogonalized Tensor Deflation via Random Tensor Theory

TL;DR

This paper develops an optimized tensor deflation method for recovering low-rank signals from noisy tensors, especially addressing non-orthogonal components using random tensor theory, with proven optimality in a specific model.

Contribution

It introduces a novel parameterized deflation algorithm for non-orthogonal tensors and proves its optimality within the studied model, extending tensor recovery techniques.

Findings

Derived asymptotic analysis of deflation on non-orthogonal tensors

Proposed an optimized tensor deflation algorithm

Proven optimality of the algorithm in the specific tensor model

Abstract

This paper tackles the problem of recovering a low-rank signal tensor with possibly correlated components from a random noisy tensor, or so-called spiked tensor model. When the underlying components are orthogonal, they can be recovered efficiently using tensor deflation which consists of successive rank-one approximations, while non-orthogonal components may alter the tensor deflation mechanism, thereby preventing efficient recovery. Relying on recently developed random tensor tools, this paper deals precisely with the non-orthogonal case by deriving an asymptotic analysis of a parameterized deflation procedure performed on an order-three and rank-two spiked tensor. Based on this analysis, an efficient tensor deflation algorithm is proposed by optimizing the parameter introduced in the deflation mechanism, which in turn is proven to be optimal by construction for the studied tensor model. The same ideas could be extended to more general low-rank tensor models, e.g., higher ranks and orders, leading to more efficient tensor methods with a broader impact on machine learning and beyond.