Multiple--Instance Learning: Christoffel Function Approach to Distribution Regression Problem
This provides a novel, probability-based method for distribution regression, addressing a specific bottleneck in multiple-instance learning with potential domain-specific applications.
The paper tackles the distribution regression problem in multiple-instance learning by proposing a two-step Christoffel function approach, which models bag distributions and outcome variables in closed form, enabling numerical evaluation and practical implementation with a stable polynomial library.
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.