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In machine learning or statistics, it is often desirable to reduce thedimensionality of a sample of data points in a high dimensional space$\mathbb{R}^d$. This paper introduces a dimensionality reduction method wherethe embedding coordinates are the eigenvectors of a positive semi-definitekernel obtained as the solution of an infinite dimensional analogue of asemi-definite program. This embedding is adaptive and non-linear. We discussthis problem both with weak and strong smoothness assumptions about the learnedkernel. A main feature of our approach is the existence of an out-of-sampleextension formula of the embedding coordinates in both cases. Thisextrapolation formula yields an extension of the kernel matrix to adata-dependent Mercer kernel function. Our empirical results indicate that thisembedding method is more robust with respect to the influence of outliers,compared with a spectral embedding method.

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