Optimal Prediction of Multivalued Functions from Point Samples

TL;DR

This paper introduces an optimal prediction method for multivalued functions based on point samples, utilizing convex optimization within Hilbert and continuous function spaces, applicable in worst-case scenarios.

Contribution

It develops a convex optimization-based affine prediction procedure that is optimal for multivalued functions within specified model sets.

Findings

The prediction procedure is optimal in the worst-case for convex model sets.

The method applies to reproducing kernel Hilbert spaces and continuous function spaces.

The approach is formulated as a convex optimization problem.

Abstract

Predicting the value of a function $f$ at a new point given its values at old points is an ubiquitous scientific endeavor, somewhat less developed when $f$ produces multiple values that depend on one another, e.g. when it outputs likelihoods or concentrations. Considering the points as fixed (not random) entities and focusing on the worst-case, this article uncovers a prediction procedure that is optimal relatively to some model-set information about $f$. When the model sets are convex, this procedure turns out to be an affine map constructed by solving a convex optimization program. The theoretical result is specified in the two practical frameworks of (reproducing kernel) Hilbert spaces and of spaces of continuous functions.