Softplus Regressions and Convex Polytopes

To construct flexible nonlinear predictive distributions, the paper introduces a family of softplus function based regression models that convolve, stack, or combine both operations by convolving countably infinite stacked gamma distributions, whose scales depend on the covariates. Generalizing logistic regression that uses a single hyperplane to partition the covariate space into two halves, softplus regressions employ multiple hyperplanes to construct a confined space, related to a single convex polytope defined by the intersection of multiple half-spaces or a union of multiple convex polytopes, to separate one class from the other. The gamma process is introduced to support the convolution of countably infinite (stacked) covariate-dependent gamma distributions. For Bayesian inference, Gibbs sampling derived via novel data augmentation and marginalization techniques is used to deconvolve and/or demix the highly complex nonlinear predictive distribution. Example results demonstrate that softplus regressions provide flexible nonlinear decision boundaries, achieving classification accuracies comparable to that of kernel support vector machine while requiring significant less computation for out-of-sample prediction.