Finding Low-Rank Matrices - From Matrix Completion to Recent Trends

Explore the fascinating world of low-rank matrices in this 54-minute lecture by Maryam Fazel from the University of Washington. Delve into matrix completion, recent trends, and their applications in recommendation systems and signal recovery. Learn about the Restricted Isometry Property, random matrix theory, and the geometry of low-rank matrices. Discover the power of nuclear norm recovery and its effectiveness in matrix completion problems. Investigate matrix decomposition, atomic norms, and statistical error measures. Gain insights into the Pareto optimal front and lower bounds on MSE risk. Engage with this comprehensive overview of low-rank matrix theory and its practical implications in various fields of study.

Intro
Recovery/estimation and hidden structure
Structure and randomness
Recommendation problem
Sparse phase retrieval
How can it work?
Restricted Isometry Property
Random matrix theory
Signal recovery
A simple 2D view
Meanings of rank
Low-rank geometry
Nuclear norm recovery
Aside: Matrix recovery algorithms
Nuclear norm works
Matrix completion
Matrix decomposition or demixing
When does it work?
General atomic norms
Pareto optimal front
A statistical error measure
Lower bound on MSE risk
Discussion