Lecture 14 : SVD , Power method , and Planted Graph problems ( + eigenvalues of random matrices )

Recall this theorem from last time. Theorem 1 (Singular Value Decomposition and best rank-k-approximation) An m × n real matrix has t ≤ min {m,n} nonnegative real numbers σ1, σ2, . . . , σt (called singular values) and two sets of unit vectors U = {u1, u2, . . . , ut} which are in <m and V = v1, v2, . . . , vt ∈ <n (all vectors are column vectors) where U, V are orthogonormal sets and ui M = σivi and Mvi = σiu T i . (1) (When M is symmetric, each ui = vi and the σi’s are eigenvalues and can be negative.) Furthermore, M can be represented as