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

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.

Tasks

Reproductions