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Current methods for regularization in machine learning require quite specificmodel assumptions (e.g. a kernel shape) that are not derived from priorknowledge about the application, but must be imposed merely to make the methodwork. We show in this paper that regularization can indeed be achieved byassuming nothing but invariance principles (w.r.t. scaling, translation, androtation of input and output space) and the degree of differentiability of thetrue function.  Concretely, we derive a novel (non-Gaussian) stochastic process from theabove minimal assumptions, and we present a corresponding Bayesian inferencemethod for regression. The mean posterior turns out to be a polyharmonicspline, and the posterior process is a mixture of t-processes.  Compared with Gaussian process regression, the proposed method shows equalperformance and has the advantages of being (i) less arbitrary (no choice ofkernel) (ii) potentially faster (no kernel parameter optimization), and (iii)having better extrapolation behavior.  We believe that the proposed theory has central importance for the conceptualfoundations of regularization and machine learning and also has great potentialto enable practical advances in ML areas beyond regression.

Scale invariant process regression: Towards Bayesian ML with minimal assumptions