# Nonlinear matrix recovery using optimization on the Grassmann manifold

arXiv: https://arxiv.org/abs/2109.06095 · HTML: https://mlsanity.com/paper/2109.06095/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 41 references that are in the map.
- First posted on arXiv 13 Sep 2021.

## Builds on (most cited first, 8 of 8)

- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Polynomial Matrix Completion for Missing Data Imputation and Transductive Learning](https://mlsanity.com/paper/1912.06989/) (2019)
- [Algebraic Variety Models for High-Rank Matrix Completion](https://mlsanity.com/paper/1703.09631/) (2017)
- [Tensor Methods for Nonlinear Matrix Completion](https://mlsanity.com/paper/1804.10266/) (2018)
- [Robust Non-Linear Matrix Factorization for Dictionary Learning, Denoising, and Clustering](https://mlsanity.com/paper/2005.01317/) (2020)
- [Online high rank matrix completion](https://mlsanity.com/paper/2002.08934/) (2020)

## Built on by (most cited first, 2 of 2)

- [An extrapolated and provably convergent algorithm for nonlinear matrix decomposition with the ReLU function](https://mlsanity.com/paper/2503.23832/) (2025)
- [Registration of algebraic varieties using Riemannian optimization](https://mlsanity.com/paper/2401.08562/) (2024)

## Abstract (arXiv)

We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by a constrained non-convex optimization problem involving the Grassmann manifold. We propose two sets of algorithms, one arising from Riemannian optimization and the other as an alternating minimization scheme, both of which include first- and second-order variants. Both sets of algorithms have theoretical guarantees. In particular, for the alternating minimization, we establish global convergence and worst-case complexity bounds. Additionally, using the Kurdyka-Lojasiewicz property, we show that the alternating minimization converges to a unique limit point. We provide extensive numerical results for the recovery of union of subspaces and clustering under entry sampling and dense Gaussian sampling. Our methods are competitive with existing approaches and, in particular, high accuracy is achieved in the recovery using Riemannian second-order methods.