# Nonsmooth rank-one matrix factorization landscape

arXiv: https://arxiv.org/abs/2211.14848 · HTML: https://mlsanity.com/paper/2211.14848/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Nov 2022.

## Built on by (most cited first, 1 of 1)

- [Can Learning Be Explained By Local Optimality In Robust Low-rank Matrix Recovery?](https://mlsanity.com/paper/2302.10963/) (2023)

## Abstract (arXiv)

We provide the first positive result on the nonsmooth optimization landscape of robust principal component analysis, to the best of our knowledge. It is the object of several conjectures and remains mostly uncharted territory. We identify a necessary and sufficient condition for the absence of spurious local minima in the rank-one case. Our proof exploits the subdifferential regularity of the objective function in order to eliminate the existence quantifier from the first-order optimality condition known as Fermat's rule.