On constrained sparse matrix factorization

Various linear subspace methods can be formulated in the notion of matrix factorization in which a cost function is minimized subject to some constraints. Among them, constraints on sparseness have received much attention recently. Some popular constraints such as non-negativity, lasso penalty, and (plain) orthogonality etc have been so far applied to extract sparse features. However, little work has been done to give theoretical and experimental analyses on the differences of the impacts of different constraints within a framework. In this paper, we analyze the problem in a more general framework called Constrained Sparse Matrix Factorization (CSMF). In CSMF, a particular case called CSMF with non-negative components (CSMFnc) is further discussed. Unlike NMF, CSMFnc allows not only additive but also subtractive combinations of non-negative sparse components. It is useful to produce much sparser features than those produced by NMF and meanwhile have better reconstruction ability, achieving a trade-off between sparseness and low MSE value. Moreover, for optimization, an alternating algorithm is developed and a gentle update strategy is further proposed for handling the alternating process. Experimental analyses are performed on the Swimmer data set and CBCL face database. In particular, CSMF can successfully extract all the proper components without any ghost on Swimmer, gaining a significant improvement over the compared well-known algorithms.