Multiple--Instance Learning: Christoffel Function Approach to Distribution Regression Problem

TL;DR

This paper introduces a novel distribution regression method using Christoffel functions, enabling closed-form solutions and probabilistic outcome estimation in multiple-instance learning, with practical numerical stability.

Contribution

It presents a two-step Christoffel function approach for distribution regression, differing from error minimization methods by using probability-based knowledge representation.

Findings

Closed-form solution for distribution regression

Probabilistic outcome estimation via Gauss quadrature

Numerically stable polynomial basis library available

Abstract

A two--step Christoffel function based solution is proposed to distribution regression problem. On the first step, to model distribution of observations inside a bag, build Christoffel function for each bag of observations. Then, on the second step, build outcome variable Christoffel function, but use the bag's Christoffel function value at given point as the weight for the bag's outcome. The approach allows the result to be obtained in closed form and then to be evaluated numerically. While most of existing approaches minimize some kind an error between outcome and prediction, the proposed approach is conceptually different, because it uses Christoffel function for knowledge representation, what is conceptually equivalent working with probabilities only. To receive possible outcomes and their probabilities Gauss quadrature for second--step measure can be built, then the nodes give possible outcomes and normalized weights -- outcome probabilities. A library providing numerically stable polynomial basis for these calculations is available, what make the proposed approach practical.