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

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.

Machine Learning (cs.LG); Optimization and Control (math.OC)

arXiv:2609.37865 [cs.LG]

(or arXiv:2609.37865v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.37865 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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