Positive semi-definite embedding for dimensionality reduction and out-of-sample extensions

In machine learning or statistics, it is often desirable to reduce the dimensionality of high dimensional data. We propose to obtain the low dimensional embedding coordinates as the eigenvectors of a positive semi-definite kernel matrix. This kernel matrix is the solution of a semi-definite program promoting a low rank solution and defined with the help of a diffusion kernel. Besides, we also discuss an infinite dimensional analogue of the same semi-definite program. From a practical perspective, a main feature of our approach is the existence of a non-linear out-of-sample extension formula of the embedding coordinates that we call a projected Nystr\"om approximation. This extension formula yields an extension of the kernel matrix to a data-dependent Mercer kernel function. Although the semi-definite program may be solved directly, we propose another strategy based on a rank constrained formulation solved thanks to a projected power method algorithm followed by a singular value decomposition. This strategy allows for a reduced computational time.