Damped Newton algorithms for matrix factorization with missing data

The problem of low-rank matrix factorization in the presence of missing data has seen significant attention in recent computer vision research. The approach that dominates the literature is EM-like alternation of closed-form solutions for the two factors of the matrix. An obvious alternative is nonlinear optimization of both factors simultaneously, a strategy which has seen little published research. This paper provides a comprehensive comparison of the two strategies by evaluating previously published factorization algorithms as well as some second order methods not previously presented for this problem. We conclude that, although alternation approaches can be very quick, their propensity to glacial convergence in narrow valleys of the cost function means that average-case performance is worse than second-order strategies. Further, we demonstrate the importance of two main observations: one, that schemes based on closed-form solutions alone are not suitable and that non-linear optimization strategies are faster, more accurate and provide more flexible frameworks for continued progress; and two, that basic objective functions are not adequate and that regularization priors must be incorporated, a process that is easier with nonlinear methods.