Nonnegative matrix factorization with combined kernels for small data representation

Kernel nonnegative matrix factorization (KNMF) has emerged as a promising nonlinear data representation method, especially for applications with small sample sizes. Existing methods are usually based on a single kernel function, representing samples by the global or local features learned by the matrix factorization algorithm. In this paper, a combined kernel method is proposed and applied to data representation for small-sample face recognition in particular. Based on the combined kernel, which is a linear combination of the fractional power inner-product kernel and a newly defined Gaussian-type kernel, the new KNMF method is able to extract both global and local nonlinear features from the inputs. An efficient gradient decent algorithm is derived to solve the combined kernel nonnegative matrix factorization (CKNMF) problem, and a rigorous convergence proof is presented. The proposed method is experimentally evaluated on small image datasets, and the results demonstrate its superior performance than the state-of-the-art KNMF methods.