Multiple kernel nonnegative matrix factorization

Kernel nonnegative matrix factorization (KNMF) is a recent kernel extension of NMF, where matrix factorization is carried out in a reproducing kernel Hilbert space (RKHS) with a feature mapping φ(·). Given a data matrix X ∈ ℝ^m×n, KNMF seeks a decomposition, φ(X) ≈ UV^T, where the basis matrix takes the form U = φ(X)W and parameters W ∈ ℝ^n×r and V ∈ ℝ₊^n×r are estimated without explicit knowledge of φ(·). As in most of kernel methods, the performance of KNMF also heavily depends on the choice of kernel. In order to alleviate the kernel selection problem when a single kernel is used, we present multiple kernel NMF (MKNMF) where two learning problems are jointly solved in unsupervised manner: (1) learning the best convex combination of kernel matrices; (2) learning parameters W and V. We formulate multiple kernel learning in MKNMF as a linear programming and estimate W and V using multiplicative updates as in KNMF. Experiments on benchmark face datasets confirm the high performance of MKNMF over several existing variants of NMF, in the task of feature extrac tion for face classification.