Learning a kernel matrix for nonlinear dimensionality reduction

We investigate how to learn a kernel matrix for high dimensional data that lies on or near a low dimensional manifold. Noting that the kernel matrix implicitly maps the data into a nonlinear feature space, we show how to discover a mapping that “unfolds ” the un-derlying manifold from which the data was sampled. The kernel matrix is constructed by maximizing the variance in feature space subject to local constraints that preserve the angles and distances between nearest neigh-bors. The main optimization involves an in-stance of semidefinite programming—a fun-damentally different computation than pre-vious algorithms for manifold learning, such as Isomap and locally linear embedding. The optimized kernels perform better than poly-nomial and Gaussian kernels for problems in manifold learning, but worse for problems in large margin classification. We explain these results in terms of the geometric properties of different kernels and comment on various interpretations of other manifold learning al-gorithms as kernel methods. 1