Nonlinear low-dimensional regression using auxiliary coordinates

When doing regression with inputs and outputs that are high-dimensional, it often makes sense to reduce the dimensionality of the inputs before mapping to the outputs. Much work in statistics and machine learning, such as reduced-rank regression, slice inverse regression and their variants, has focused on linear dimensionality reduction, or on estimating the dimensionality reduction first and then the mapping. We propose a method where both the dimensionality reduction and the mapping can be nonlinear and are estimated jointly. Our key idea is to define an objective function where the low-dimensional coordinates are free parameters, in addition to the dimensionality reduction and the mapping. This has the effect of decoupling many groups of parameters from each other, affording a far more effective optimization than if using a deep network with nested mappings, and to use a good initialization from slice inverse regression or spectral methods. Our experiments with image and robot applications show our approach to improve over direct regression and various existing approaches.