Multiple-Instance Learning: Radon-Nikodym Approach to Distribution Regression Problem

TL;DR

This paper introduces a one-step distribution regression method using Radon-Nikodym theory, transforming distribution-to-value problems into vector-to-value problems via moments, enabling practical and stable computations.

Contribution

It presents a novel Radon-Nikodym based approach for distribution regression, including a stable polynomial basis for numerical implementation.

Findings

Effective estimation of $y$ from distribution moments.

Probabilistic outcomes derived from eigenvalue spectrum.

Practical implementation with a stable polynomial basis.

Abstract

For distribution regression problem, where a bag of $x$--observations is mapped to a single $y$ value, a one--step solution is proposed. The problem of random distribution to random value is transformed to random vector to random value by taking distribution moments of $x$ observations in a bag as random vector. Then Radon--Nikodym or least squares theory can be applied, what give $y(x)$ estimator. The probability distribution of $y$ is also obtained, what requires solving generalized eigenvalues problem, matrix spectrum (not depending on $x$) give possible $y$ outcomes and depending on $x$ probabilities of outcomes can be obtained by projecting the distribution with fixed $x$ value (delta--function) to corresponding eigenvector. A library providing numerically stable polynomial basis for these calculations is available, what make the proposed approach practical.