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