Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization Presenter: Xia Li.

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Presentation transcript:

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization Presenter: Xia Li

Introduction An affine rank minimization problem Minimization of the l1 norm is a well known heuristic for the cardinality minimization problem. L1 heuristic can be a priori guaranteed to yield the optimal solution. The results from the compressed sensing literature might be extended to provide guarantees about the nuclear norm heuristic for the more general rank minimization problem. 11/15/2018

Outline Introduction From Compressed Sensing to Rank Minimization Restricted Isometry and Recovery of Low-Rank Matrices Algorithms for Nuclear Norm Minimization Necessary and Sufficient Conditions for Success of the Nuclear Norm Heuristic for Rank Minimization Discussion and Future Developments 11/15/2018

From Compressed Sensing 11/15/2018

Matrix Norm Frobenius Norm Operator Norm Nuclear Norm 11/15/2018

Convex Envelopes of Rank and Cardinality Functions 11/15/2018

Additive of Rank and Nuclear Norm 11/15/2018

Nuclear Norm Minimization This problem admits the primal-dual convex formulation The primal-dual pair of semidefinite programs 11/15/2018

Restricted Isometry and Recovery of Low Rank Matrices 11/15/2018

Nearly Isometric Families 11/15/2018

11/15/2018

Main Results 11/15/2018

Main Results 11/15/2018

Algorithms for Nuclear Norm Minimization The trade-offs between computational speed and guarantees on the accuracy of the resulting solution. Algorithms for Nuclear Norm Minimization Interior Point Methods for Semidefinite Programming For small problems where a high-degree of numerical precision is required, interior point methods for semidefinite programming can be directly applied to solve affine nuclear minimization problems. Projected Subgradient Methods Low-rank Parametrization SDPLR and the Method of Multipliers 11/15/2018

Numerical Experiments 11/15/2018

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Question? 11/15/2018