Presented by: Mingyuan Zhou Duke University, ECE February 18, 2011 Weighted Low-Rank Approximation Nathan Srebro and Tommi Jaakkola ICML 2003 Presented by: Mingyuan Zhou Duke University, ECE February 18, 2011

Outline Introduction Low rank matrix factorization Missing values and an EM procedure Low rank logistic regression Experimental results Conclusions

Introduction Factor model Weighted norms Efficient optimization methods

Low rank matrix factorization Objective function Solutions ( = 1)

Low rank matrix factorization Solutions

Low rank matrix factorization Since are unlikely to be diagonalizable for all rows, The critical points of the weighted low-rank approximation problem lack the eigenvector structure of the unweighted case. Another implication of this is that the incremental structure of unweighted low-rank approximations is lost: an optimal rank-k factorization cannot necessarily be extended to an optimal rank-(k + 1) factorization.

Low rank matrix factorization

Missing values and an EM procedure Initializing X with A or 0 Initializing X with 0 and let

Missing values and an EM procedure

Low rank logistic regression

Experimental results

Experimental results

Conclusions