Low-tubal-rank tensor sensing shows new geometric optimization properties
Strict-Saddle Landscapes and Multi-Rank Geometry in Low-Tubal-Rank Tensor Sensing
Machine Learning
Summary
This paper studies how to find simple patterns in complex 3D data structures called tensors. The authors analyze the shape of the problem when trying to recover these patterns and prove there are no misleading dead ends (false local minima) when following their method. They also discover that the shape of the problem depends on details in the frequency domain, not just the overall complexity, revealing some directions where the search is very flat. Their experiments show these differences in practice.
What this means in practice
- •For signal processing engineers: Improve algorithms for reconstructing multidimensional signals from limited measurements by exploiting frequency-dependent rank structures.
- •For machine learning model developers: Design robust tensor completion methods aware of local geometric properties to avoid optimization traps during training.
Authors
Eugene Agyei-Kodie, Longxiu Huang, Shuang Li, Xiao Liang
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.