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RTRMC

swMATH ID:	20435
Software Authors:	Boumal, N., Absil, P.A.
Description:	RTRMC : Low-rank matrix completion via preconditioned optimization on the Grassmann manifold. We address the numerical problem of recovering large matrices of low rank when most of the entries are unknown. We exploit the geometry of the low-rank constraint to recast the problem as an unconstrained optimization problem on a single Grassmann manifold. We then apply second-order Riemannian trust-region methods (RTRMC 2) and Riemannian conjugate gradient methods (RCGMC) to solve it. A preconditioner for the Hessian is introduced that helps control the conditioning of the problem and we detail preconditioned versions of Riemannian optimization algorithms. The cost of each iteration is linear in the number of known entries. The proposed methods are competitive with state-of-the-art algorithms on a wide range of problem instances. In particular, they perform well on rectangular matrices. We further note that second-order and preconditioned methods are well suited to solve badly conditioned matrix completion tasks.
Homepage:	https://web.math.princeton.edu/~nboumal/RTRMC/index.html
Dependencies:	Matlab
Related Software:	Manopt; LMaFit; ADMiRA; PROPACK; Jellyfish; softImpute; SDPLR; PhaseLift; ROPTLIB; RASL; SDPT3; Matrix Means Toolbox; GitHub; LOBPCG; htucker; SparseMatrix; MATPOWER; LSQR; HSL-VF05; Saga
Cited in:	45 Publications

1 Publication describing the Software, including 1 Publication in zbMATH	Year
Low-rank matrix completion via preconditioned optimization on the Grassmann manifold. Zbl 1312.90092 Boumal, Nicolas; Absil, P.-A.	2015

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5	SIAM Journal on Optimization
3	Computational Statistics
3	Mathematical Programming. Series A. Series B
2	Applied Mathematics and Computation
2	Journal of Mathematical Imaging and Vision
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2	SIAM Journal on Scientific Computing
2	Journal of Inequalities and Applications
2	SIAM Journal on Imaging Sciences
1	Computers & Mathematics with Applications
1	Journal of Computational Physics
1	Automatica
1	Information Sciences
1	Journal of Computational and Applied Mathematics
1	Journal of Optimization Theory and Applications
1	Numerische Mathematik
1	SIAM Journal on Matrix Analysis and Applications
1	Journal of Scientific Computing
1	Machine Learning
1	Linear Algebra and its Applications
1	Applied and Computational Harmonic Analysis
1	Methods and Applications of Analysis
1	Advances in Computational Mathematics
1	Journal of Machine Learning Research (JMLR)
1	The Australian Journal of Mathematical Analysis and Applications
1	GAMM-Mitteilungen
1	Optimization Letters
1	Inverse Problems and Imaging
1	SIAM/ASA Journal on Uncertainty Quantification
1	SIAM Journal on Mathematics of Data Science

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35	Numerical analysis (65-XX)
25	Operations research, mathematical programming (90-XX)
15	Linear and multilinear algebra; matrix theory (15-XX)
5	Computer science (68-XX)
3	Differential geometry (53-XX)
3	Statistics (62-XX)
3	Information and communication theory, circuits (94-XX)
2	Global analysis, analysis on manifolds (58-XX)
1	Algebraic geometry (14-XX)
1	Calculus of variations and optimal control; optimization (49-XX)
1	Geometry (51-XX)
1	Probability theory and stochastic processes (60-XX)
1	Mechanics of deformable solids (74-XX)
1	Systems theory; control (93-XX)