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

2. 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.
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3. 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
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4. From Compressed Sensing
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5. Matrix Norm
Frobenius Norm
Operator Norm
Nuclear Norm
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6. Convex Envelopes of Rank and Cardinality Functions
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7. Additive of Rank and Nuclear Norm
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8. Nuclear Norm Minimization
This problem admits the primal-dual convex formulation
The primal-dual pair of semidefinite programs
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9. Restricted Isometry and Recovery of Low Rank Matrices
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10. Nearly Isometric Families
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12. Main Results
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13. Main Results
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14. 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
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15. Numerical Experiments
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17. Question?
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