One-to-many mappings, continuity constraints and latent variable models

We approach the problem of multivariate regression using latent variable models, which infer a low-dimensional representation of an observed, high-dimensional process. Defining functional relationships between variables may be conveniently done by picking informative points from the corresponding conditional distribution. However, this is problematic when this conditional distribution is multimodal, since there are in principle multiple candidates for the representative point, i.e., the mapping is one-to-many. We show, both with a toy example and with real-world data-the acoustic-to-articulatory mapping problem-that: 1) the modes of the conditional distribution contain information to potentially invert many-to-one as well as one-to-one mappings; 2) this information may be successfully used if some extra information is available, in particular continuity constraints for sequential data, for which we introduce a quantitative measure. We sketch algorithms for mode-finding in Gaussian mixtures and for performing smooth multivariate regression.