Secrets of Matrix Factorization: Supplementary Document

This document consists of details to which the original submission article has made references. These include a list of symbols, some matrix identities and calculus rules, a list of VarPro derivatives, symmetry analysis of the un-regularized VarPro Gauss-Newton matrix and MTSS computation details. Symbol Meaning Rp×q The space of real matrices of size p× q S The space of symmetric real matrices of size p× p M The measurement matrix ∈ Rm×n (may contain noise and missing data) W The weight matrix ∈ Rm×n U The first decomposition matrix ∈ Rm×r V The second decomposition matrix ∈ Rn×r u vec(U) ∈ R v vec(V>) ∈ R m The number of rows in M n The number of columns in M p The number of visible entries in M pj The number of visible entries in the j-th column of M f The objective ∈ R ε The vector objective ∈ R J Jacobian H Hessian V∗ arg minV f(U, V) ∈ Rn×r Π Projection matrix ∈ Rp×mn selecting nonzero entries of vec(W) W̃ Π diag(vec(W)) ∈ Rp×mn m̃ W̃ vec(M) ∈ R Ũ W̃(In ⊗ U) ∈ Rp×nr Ṽ W̃(V⊗ Im) ∈ Rp×mr Kmr The permutation matrix satisfying vec(U>) = Kmr vec(U) ∈ Rmr×mr Knr The permutation matrix satisfying vec(V>) = Knr vec(V) ∈ Rnr×nr R W (UV> − M) ∈ Rm×n Z (W R)⊗ Ir ∈ Rmr×nr Table 1: A list of symbols used in [1].