Learning One-to-Many Mapping With Locally Linear Maps Based on Manifold Structure

This letter proposes a new method to realize a nonlinear mapping of one-to-many correspondences. Assuming that a small number of training pairs are given with their actual correspondences, each tangent space is locally constructed on a submanifold around each labeled sample. Moreover, the linear transformation between paired tangent spaces is derived by solving an optimization problem, which is designed to bring locally linear maps into closer proximity in each class. Finally, a global nonlinear mapping is realized by combining these locally linear maps. In simulations of an S-curve to Swiss-roll, a lip to speech, and room impulse response to position of microphone mappings, the proposed method shows the remarkable mapping ability.