1. Nuclear Norm Heuristic for Rank Minimization
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
Necessary and Sufficient Conditions for Success of the Nuclear Norm Heuristic for Rank Minimization
Presenter: Zhen Hu

2. 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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3. 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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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. Necessary and Sufficient Conditions
The authors present a necessary and sufficient condition for the solution of the nuclear norm heuristic to coincide with the minimum rank solution in an affine space.
The condition characterizes a particular property of the null space of the linear map which defines the affine space.
The authors also present a reduction of the standard Linear Matrix Inequality (LMI) constrained rank minimization problem to a rank minimization problem with only equality constraints.
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18. Moreover, the authors show that when the linear map defining the constraint set is generated by sampling its entries independently from a Gaussian distribution, the null-space characterization holds with overwhelming probability provided the dimensions of the equality constraints are of appropriate size.
The authors provide numerical experiments demonstrating that even when matrix dimensions are small, the nuclear norm heuristic does indeed always recover the minimum rank solution when the number of constraints is sufficiently large.
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19. Necessary and Sufficient Conditions
The main optimization problem under study is
The main concern is when the optimal solution coincides with the optimal solution of
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21. Theorem 3.1
The following theorem generalizes this null-space criterion to a critical property that guarantees when the nuclear norm heuristic finds the minimum rank solution of A(X)=b for all values of the vector b.
A particular property of the null space of the linear map
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22. Lemma 4.1
For the proof of Theorem 3.1
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23. Theorem 3.2
Reduction to the affine case!
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24. Weak Bound
Probabilistic generation of constraints satisfying null space characterization
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25. Strong Bound
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26. Performance
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27. Discussion
Having illustrated the natural connections between affine rank minimization and affine cardinality minimization, we were able to draw on these parallels to determine scenarios where the nuclear norm heuristic was able to exactly solve the rank minimization problem.
Future developments
Factored measurements and alternative ensembles
Noisy measurements and low rank approximation
Incoherent ensembles and partially observed transforms
Alternative numerical methods
Geometric interpretations
Parsimonious models and optimization
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28. Question?
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