Low rank matrix approximations

low rank matrix approximation

• special case of low rank approximation, where we use SVD to find optimal solution
• basically:
  • rank-constrained approximation problem: $mi{n}_{A_k\in {\mathbb{R}}^{m,n}}||A-A_k|{|}_F^2$ , such that $rank(A_k)=k$, given $1\le k\le r$
    • k is the parameter chosen based on how much info wanted to retain
    • k is small → info is less accurate, efficient computation and storage
    • k is large → info is accurate, more computation and storage
  • keeping only the first k largest singular values & the corresponding columns of U and V. This gives a new set of matrices $U_k,{\Sigma }_k,and{V}_k^T$
  • then, multiple these matrices to get the rank-k approximation of A
• help to minimize the Frobenius norm of the difference btw A and B (refer to Frobenius norm in relation of F-norm with singular values)
  • the ratio of the F-norm of B to F-norm of A is how much the info in A is captured by