Bayesian Learning via Q-Exponential Process

Regularization is one of the most fundamental topics in optimization, statistics and machine learning. To get sparsity in estimating a parameter u∈ℝd, an ℓq penalty term, Vert uVertq, is usually added to the objective function. What is the probabilistic distribution corresponding to such ℓq penalty? What is the correct stochastic process corresponding to Vert uVertq when we model functions u∈ Lq? This is important for statistically modeling large dimensional objects, e.g. images, with penalty to preserve certainty properties, e.g. edges in the image. In this work, we generalize the q-exponential distribution (with density proportional to) exp(- frac12|u|q) to a stochastic process named Q-exponential (Q-EP) process that corresponds to the Lq regularization of functions. The key step is to specify consistent multivariate q-exponential distributions by choosing from a large family of elliptic contour distributions. The work is closely related to Besov process which is usually defined by the expanded series. Q-EP can be regarded as a definition of Besov process with explicit probabilistic formulation and direct control on the correlation length. From the Bayesian perspective, Q-EP provides a flexible prior on functions with sharper penalty (q<2) than the commonly used Gaussian process (GP). We compare GP, Besov and Q-EP in modeling functional data, reconstructing images, and solving inverse problems and demonstrate the advantage of our proposed methodology.