# Convergence results for projected line-search methods on varieties of low-rank matrices via Lojasiewicz inequality

arXiv: https://arxiv.org/abs/1402.5284 · HTML: https://mlsanity.com/paper/1402.5284/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2014.

## Built on by (most cited first, 10 of 10)

- [Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay](https://mlsanity.com/paper/2206.08756/) (2022)
- [On the Local Linear Rate of Consensus on the Stiefel Manifold](https://mlsanity.com/paper/2101.09346/) (2021)
- [Manifold constrained steepest descent for smooth and closed-set optimization](https://mlsanity.com/paper/2601.21487/) (2026)
- [Tensor train completion: local recovery guarantees via Riemannian optimization](https://mlsanity.com/paper/2110.03975/) (2021)
- [New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition](https://mlsanity.com/paper/2101.11108/) (2021)
- [A space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints](https://mlsanity.com/paper/2501.13830/) (2025)
- [O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds](https://mlsanity.com/paper/2309.10775/) (2023)
- [Error Bound Analysis for the Regularized Loss of Deep Linear Neural Networks](https://mlsanity.com/paper/2502.11152/) (2025)
- [Asymptotic Escape of Spurious Critical Points on the Low-rank Matrix Manifold](https://mlsanity.com/paper/2107.09207/) (2021)
- [Heaviside Low-Rank Support Matrix Machine](https://mlsanity.com/paper/2603.00491/) (2026)

## Abstract (arXiv)

The aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety M≤ k of real m × n matrices of rank at most k. Such methods extend Riemannian optimization methods, which are successfully used on the smooth manifold Mk of rank-k matrices, to its closure by taking steps along gradient-related directions in the tangent cone, and afterwards projecting back to M≤ k. Considering such a method circumvents the difficulties which arise from the nonclosedness and the unbounded curvature of Mk. The pointwise convergence is obtained for real-analytic functions on the basis of a Lojasiewicz inequality for the projection of the antigradient to the tangent cone. If the derived limit point lies on the smooth part of M≤ k, i.e. in Mk, this boils down to more or less known results, but with the benefit that asymptotic convergence rate estimates (for specific step-sizes) can be obtained without an a priori curvature bound, simply from the fact that the limit lies on a smooth manifold. At the same time, one can give a convincing justification for assuming critical points to lie in Mk: if X is a critical point of f on M≤ k, then either X has rank k, or nabla f(X) = 0.