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We study a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint.
Riemannian geometry
Manfredo Perdigao Do Carmo and J Flaherty Francis · 1992
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Perturbation analysis of the orthogonal Procrustes problem
Inge Söderkvist · 1993
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The geometry of algorithms with orthogonality constraints
Alan Edelman, Tomás A Arias, and Steven T Smith · 1998
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Matrix algorithms: volume 1: basic decompositions
Gilbert W Stewart · 1998
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Uniqueness of matrix square roots and an application
Charles R Johnson, Kazuyoshi Okubo, and Robert Reams · 2001
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Local minima and convergence in low-rank semidefinite programming
Samuel Burer and Renato DC Monteiro · 2005
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First order error propagation of the Procrustes method for 3d attitude estimation
Leo Dorst · 2005
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A differential geometric approach to the geometric mean of symmetric positive-definite matrices
Maher Moakher · 2005
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Geometric means in a novel vector space structure on symmetric positive-definite matrices
Vincent Arsigny, Pierre Fillard, Xavier Pennec, and Nicholas Ayache · 2007
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Optimization algorithms on matrix manifolds
P-A Absil, Robert Mahony, and Rodolphe Sepulchre · 2009
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Matrix completion from a few entries
Raghunandan H Keshavan, Sewoong Oh, and Andrea Montanari · 2009
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Variational analysis
R Tyrrell Rockafellar and Roger J-B Wets · 2009
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Embedded geometry of the set of symmetric positive semidefinite matrices of fixed rank
Bart Vandereycken, P-A Absil, and Stefan Vandewalle · 2009
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Low-rank optimization on the cone of positive semidefinite matrices
Michel Journée, Francis Bach, P-A Absil, and Rodolphe Sepulchre · 2010
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Rtrmc: A Riemannian trust-region method for low-rank matrix completion
Nicolas Boumal and P-A Absil · 2011
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Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements
Emmanuel J Candes and Yaniv Plan · 2011
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Linear regression under fixed-rank constraints: a Riemannian approach
Gilles Meyer, Silvere Bonnabel, and Rodolphe Sepulchre · 2011
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Smooth manifolds
John M Lee · 2013
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Low-rank matrix completion by Riemannian optimization
Bart Vandereycken · 2013
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Fixed-rank matrix factorizations and Riemannian low-rank optimization
Bamdev Mishra, Gilles Meyer, Silvère Bonnabel, and Rodolphe Sepulchre · 2014
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Phase retrieval via Wirtinger flow: Theory and algorithms
Emmanuel J Candès, Xiaodong Li, and Mahdi Soltanolkotabi · 2015
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Yudong Chen and Martin J Wainwright · 2015
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Escaping from saddle points-online stochastic gradient for tensor decomposition
Rong Ge, Furong Huang, Chi Jin, and Yang Yuan · 2015
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Guaranteed matrix completion via nonconvex factorization
Ruoyu Sun and Zhi-Quan Luo · 2015
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When are nonconvex problems not scary?
Ju Sun, Qing Qu, and John Wright · 2015
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A convergent gradient descent algorithm for rank minimization and semidefinite programming from random linear measurements
Qinqing Zheng and John Lafferty · 2015
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A nonconvex optimization framework for low rank matrix estimation
Tuo Zhao, Zhaoran Wang, and Han Liu · 2015
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Dropping convexity for faster semi-definite optimization
Srinadh Bhojanapalli, Anastasios Kyrillidis, and Sujay Sanghavi · 2016
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Global optimality of local search for low rank matrix recovery
Srinadh Bhojanapalli, Behnam Neyshabur, and Nati Srebro · 2016
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Complete dictionary recovery over the sphere i: Overview and the geometric picture
Ju Sun, Qing Qu, and John Wright · 2016
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Low-rank solutions of linear matrix equations via Procrustes flow
Stephen Tu, Ross Boczar, Max Simchowitz, Mahdi Soltanolkotabi, and Benjamin Recht · 2016
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Guarantees of Riemannian optimization for low rank matrix recovery
Ke Wei, Jian-Feng Cai, Tony F Chan, and Shingyu Leung · 2016
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Gradient descent can take exponential time to escape saddle points
Simon S Du, Chi Jin, Jason D Lee, Michael I Jordan, Aarti Singh, and Barnabas Poczos · 2017
A well-tempered landscape for non-convex robust subspace recovery
Tyler Maunu, Teng Zhang, and Gilad Lerman · 2019
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Escaping from saddle points on Riemannian manifolds
Yue Sun, Nicolas Flammarion, and Maryam Fazel · 2019
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Sharp restricted isometry bounds for the inexistence of spurious local minima in nonconvex matrix recovery
Richard Y Zhang, Somayeh Sojoudi, and Javad Lavaei · 2019
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An introduction to optimization on smooth manifolds
Nicolas Boumal · 2020
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Deterministic guarantees for Burer-Monteiro factorizations of smooth semidefinite programs
Nicolas Boumal, Vladislav Voroninski, and Afonso S Bandeira · 2020
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Leave-one-out approach for matrix completion: Primal and dual analysis
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No spurious local minima in nonconvex low rank problems: A unified geometric analysis
Rong Ge, Chi Jin, and Yi Zheng · 2017
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How to escape saddle points efficiently
Chi Jin, Rong Ge, Praneeth Netrapalli, Sham M Kakade, and Michael I Jordan · 2017
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Solving sdps for synchronization and maxcut problems via the grothendieck inequality
Song Mei, Theodor Misiakiewicz, Andrea Montanari, and Roberto Imbuzeiro Oliveira · 2017
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Non-square matrix sensing without spurious local minima via the Burer-Monteiro approach
Dohyung Park, Anastasios Kyrillidis, Constantine Carmanis, and Sujay Sanghavi · 2017
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A unified computational and statistical framework for nonconvex low-rank matrix estimation
Lingxiao Wang, Xiao Zhang, and Quanquan Gu · 2017
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Blind deconvolution by a steepest descent algorithm on a quotient manifold
Wen Huang and Paul Hand · 2018
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Lijun Ding and Yudong Chen · 2020
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Yuetian Luo, Wen Huang, Xudong Li, and Anru R Zhang · 2020
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Quotient geometry with simple geodesics for the manifold of fixed-rank positive-semidefinite matrices
Estelle Massart and P-A Absil · 2020
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Accelerating ill-conditioned low-rank matrix estimation via scaled gradient descent
Tian Tong, Cong Ma, and Yuejie Chi · 2020
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On critical points of quadratic low-rank matrix optimization problems
André Uschmajew and Bart Vandereycken · 2020
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Jesse van Oostrum · 2020
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Rank optimality for the Burer–Monteiro factorization
Irene Waldspurger and Alden Waters · 2020
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How many samples is a good initial point worth in low-rank matrix recovery?
Jialun Zhang and Richard Zhang · 2020
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A schatten-q low-rank matrix perturbation analysis via perturbation projection error bound
Yuetian Luo, Rungang Han, and Anru R Zhang · 2021
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Yuetian Luo, Xudong Li, and Anru R Zhang · 2021
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Yuetian Luo, Xudong Li, and Anru R Zhang · 2021
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Global convergence of gradient descent for asymmetric low-rank matrix factorization
Tian Ye and Simon S Du · 2021
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General low-rank matrix optimization: Geometric analysis and sharper bounds
Haixiang Zhang, Yingjie Bi, and Javad Lavaei · 2021
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The global optimization geometry of low-rank matrix optimization
Zhihui Zhu, Qiuwei Li, Gongguo Tang, and Michael B Wakin · 2021
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On the geometric analysis of a quartic–quadratic optimization problem under a spherical constraint
Haixiang Zhang, Andre Milzarek, Zaiwen Wen, and Wotao Yin · 2021
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Foivos Alimisis and Bart Vandereycken · 2022
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Local and global linear convergence of general low-rank matrix recovery problems
Yingjie Bi, Haixiang Zhang, and Javad Lavaei · 2022
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Nearly optimal bounds for the global geometric landscape of phase retrieval
Jian-Feng Cai, Meng Huang, Dong Li, and Yang Wang · 2022
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On the analysis of optimization with fixed-rank matrices: a quotient geometric view
Shuyu Dong, Bin Gao, Wen Huang, and Kyle A Gallivan · 2022
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Noisy low-rank matrix optimization: Geometry of local minima and convergence rate
Ziye Ma and Somayeh Sojoudi · 2022
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Factorization approach for low-complexity matrix completion problems: Exponential number of spurious solutions and failure of gradient methods
Baturalp Yalcin, Haixiang Zhang, Javad Lavaei, and Somayeh Sojoudi · 2022
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