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

arXiv: https://arxiv.org/abs/2609.37865 · HTML: https://mlsanity.com/paper/2609.37865/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 22 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 12 of 12)

- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Low-rank Solutions of Linear Matrix Equations via Procrustes Flow](https://mlsanity.com/paper/1507.03566/) (2015)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [On the computational and statistical complexity of over-parameterized matrix sensing](https://mlsanity.com/paper/2102.02756/) (2021)
- [Gradient descent with adaptive stepsize converges (nearly) linearly under fourth-order growth](https://mlsanity.com/paper/2409.19791/) (2024)
- [Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent](https://mlsanity.com/paper/2401.11940/) (2024)
- [Tensor Restricted Isometry Property Analysis For a Large Class of Random Measurement Ensembles](https://mlsanity.com/paper/1906.01198/) (2019)
- [Guaranteed Sampling Flexibility for Low-tubal-rank Tensor Completion](https://mlsanity.com/paper/2406.11092/) (2024)
- [Implicit Regularization for Tubal Tensor Factorizations via Gradient Descent](https://mlsanity.com/paper/2410.16247/) (2024)
- [The power of small initialization in noisy low-tubal-rank tensor recovery](https://mlsanity.com/paper/2603.02729/) (2026)
- [Efficient Low-Tubal-Rank Tensor Estimation via Alternating Preconditioned Gradient Descent](https://mlsanity.com/paper/2512.07490/) (2025)

## 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.