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

arXiv 2609.37865 ↗First posted 29 Sep 2026cs.LG math.OCcited by 0

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Abstract (arXiv)

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.

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References · 12

  1. 2017No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysiscited by 129
  2. 2015Low-rank Solutions of Linear Matrix Equations via Procrustes Flowcited by 97
  3. 2016Global Optimality of Local Search for Low Rank Matrix Recoverycited by 82
  4. 2015Exact tensor completion using t-SVDcited by 70
  5. 2021On the computational and statistical complexity of over-parameterized matrix sensingcited by 31
  6. 2024Gradient descent with adaptive stepsize converges (nearly) linearly under fourth-order growthcited by 10
  7. 2024Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descentcited by 7
  8. 2019Tensor Restricted Isometry Property Analysis For a Large Class of Random Measurement Ensemblescited by 3
  9. 2024Guaranteed Sampling Flexibility for Low-tubal-rank Tensor Completioncited by 3
  10. 2024Implicit Regularization for Tubal Tensor Factorizations via Gradient Descentcited by 2
  11. 2026The power of small initialization in noisy low-tubal-rank tensor recoverycited by 2
  12. 2025Efficient Low-Tubal-Rank Tensor Estimation via Alternating Preconditioned Gradient Descentcited by 1

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