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Living on the Edge - Phase Transitions in Random Convex Programs
Published on Aug 26, 20135381 Views
Recent empirical research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerg
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Chapter list
Living on the Edge00:00
Phase Transitions00:48
What is a Phase Transition?00:50
Example: Sparse Linear Inversion02:23
Research Challenge...04:05
Random Convex Programs04:16
Warmup: Regularized Denoising06:29
Setup for Regularized Denoising06:43
The Risk of Regularized Denoising13:13
Statistical Dimension15:06
Geometry of Denoising15:43
The Statistical Dimension15:59
Basic Examples17:13
Circular Cones17:55
Descent Cones18:27
Descent Cone of l1 Norm at Sparse Vector18:53
Descent Cone of S1 Norm at Low-Rank Matrix19:33
Statistical Dimension & Phase Transitions20:03
Regularized Linear Inverse Problems24:05
Setup for Linear Inverse Problems26:55
Geometry of Linear Inverse Problems27:02
Linear Inverse Problems with Random Data30:32
Sparse Reconstruction via l1 Minimization31:27
Low-Rank Recovery via S1 Minimization32:01
Demixing Structured Signals32:56
Setup for Demixing Problems33:05
Geometry of Demixing Problems34:32
Demixing Problems with Random Incoherence35:52
Sparse + Sparse via l1 + l1 Minimization36:21
Low-Rank + Sparse via S1 + l1 Minimization37:15
Cone Programs with Random Constraints38:08
Cone Program with Random Constraints40:25
Example: Some Random SOCPs40:35
To learn more...40:39