mlsanityData to 7 Oct 2026

Riemannian Perspective on Matrix Factorization

arXiv 2102.00937 ↗First posted 1 Feb 2021math.OC cs.LG math.DGcited by 3

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Abstract (arXiv)

We study the non-convex matrix factorization approach to matrix completion via Riemannian geometry. Based on an optimization formulation over a Grassmannian manifold, we characterize the landscape based on the notion of principal angles between subspaces. For the fully observed case, our results show that there is a region in which the cost is geodesically convex, and outside of which all critical points are strictly saddle. We empirically study the partially observed case based on our findings.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 14 of its 46 references that are in the map. First posted on arXiv 1 Feb 2021.

Builds on

14 of 46 references.

References · 14

  1. 2017No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysiscited by 127
  2. 2016Matrix Completion has No Spurious Local Minimumcited by 124
  3. 2014Guaranteed Matrix Completion via Non-convex Factorizationcited by 81
  4. 2016Global Optimality of Local Search for Low Rank Matrix Recoverycited by 81
  5. 2015Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guaranteescited by 68
  6. 2015Complete Dictionary Recovery over the Sphere I: Overview and the Geometric Picturecited by 49
  7. 2016Non-square matrix sensing without spurious local minima via the Burer-Monteiro approachcited by 38
  8. 2013Understanding Alternating Minimization for Matrix Completioncited by 32
  9. 2014Global Convergence of Stochastic Gradient Descent for Some Non-convex Matrix Problemscited by 25
  10. 2017A Well-Tempered Landscape for Non-convex Robust Subspace Recoverycited by 18
  11. 2014Fast matrix completion without the condition numbercited by 17
  12. 2018Subgradient Descent Learns Orthogonal Dictionariescited by 17
2 more
  1. 2020Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initializationcited by 8
  2. 2020Maximizing robustness of point-set registration by leveraging non-convexitycited by 3

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Cited by year

Citing papers per year: 2021: 2; 2022: 0; 2023: 0; 2024: 1.

3 papers.

  1. 2021Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimizationcited by 3
  2. 2021On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimizationcited by 3
  3. 2024Riemannian Bilevel Optimizationcited by 0