Singular Vectors

Singular Value Decomposition (SVD) initiates CUR Matrix Decomposition by providing singular vectors essential for calculating leverage scores.

SVD returns left and right singular vectors for calculating column and row leverage scores. Perform SVD on the following matrix:

A ε R^mxn

The matrix is factorized as follows:

A=UΣV^T

where U = [u¹ u²...u^m] and V = [v¹ v²...vⁿ] are orthogonal matrices.

Σ is a diagonal m × n matrix with non-negative real numbers σ1,...,σ_ρ on the diagonal, where ρ = min {m,n} and σ_ξ is the ξ^th singular value of A.

Let u^ξ and v^ξ be the ξ^th left and right singular vector of A, the j^th column of A can thus be approximated by the top k singular vectors and corresponding singular values as:

Description of the illustration singular_vectors.eps

where v^ξ_j is the j^th coordinate of the ξ^th right singular vector.