Quantitative Analysis of Nonlinear Embedding

A lot of nonlinear embedding techniques have been developed to recover the intrinsic low-dimensional manifolds embedded in the high-dimensional space. However, the quantitative evaluation criteria are less studied in literature. The embedding quality is usually evaluated by visualization which is subjective and qualitative. The few existing evaluation methods to estimate the embedding quality, neighboring preservation rate for example, are not widely applicable. In this paper, we propose several novel criteria for quantitative evaluation, by considering the global smoothness and co-directional consistence of the nonlinear embedding algorithms. The proposed criteria are geometrically intuitive, simple, and easy to implement with a low computational cost. Experiments show that our criteria capture some new geometrical properties of the nonlinear embedding algorithms, and can be used as a guidance to deal with the embedding of the out-of-samples.