Strict-Saddle Landscapes and Multi-Rank Geometry in Low-Tubal-Rank Tensor Sensing

2026-09-29

AI Summary

This study addresses the lack of theoretical characterization regarding the optimization landscape and local geometry in low tubal-rank tensor sensing under arbitrary multi-rank profiles. By leveraging balanced tensor decomposition and the tubal restricted isometry property (tRIP), combined with multi-rank analysis in the Fourier domain, this work provides the first rigorous quantification of the strict saddle landscape for this problem. Theoretically, it establishes that the objective function is free of spurious local minima, thereby guaranteeing global convergence. Furthermore, it reveals fundamentally distinct local geometric structures between uniform and non-uniform rank settings. Notably, the analysis demonstrates that non-uniform ranks induce quartic flat directions due to over-parameterization in hidden frequencies. These findings offer new insights into the intrinsic complexity of tensor recovery landscapes.

Abstract

We study the optimization landscape of low-tubal-rank tensor sensing through a balanced factorization. Under a tubal restricted isometry condition, we establish a quantitative strict-saddle landscape with no spurious local minima for arbitrary Fourier multi-rank profiles. We further show that the local geometry depends on the Fourier-slice ranks rather than the tubal rank alone. Uniform ranks yield quadratic growth transverse to the solution orbit, whereas nonuniform ranks produce quartically flat directions through hidden frequency-wise overparameterization, even when the factor width equals the exact tubal rank. Numerical experiments illustrate the global optimization behavior and the contrasting local geometries.

Problem

• Low-tubal-rank tensor sensing
• Optimization landscape
• Strict-saddle
• Fourier multi-rank
• Spurious local minima

Innovation

• Low-Tubal-Rank Tensor Sensing
• Strict-Saddle Landscape
• Fourier Multi-Rank Geometry
• Balanced Factorization
• Overparameterization