Generalize the infeasible-boundary characterization beyond binary classification

Prove that, for an arbitrary number of classes C, only posterior-sample configurations in the set of deterministic configurations with at most one non-deterministic posterior sample can minimize aleatoric uncertainty at fixed total uncertainty below the maximum attainable by that set, thereby establishing the conjectured infeasible-boundary characterization beyond the binary-class case.

Background

The paper studies the attainable pairs of aleatoric uncertainty (AU) and total uncertainty (TU) generated by a finite collection of N class-probability vectors over C classes. The authors conjecture that, below a specified maximum TU, minimizing AU at fixed TU requires a configuration in which N−1 posterior samples are deterministic and at most one posterior sample is non-deterministic, with the latter lying on an edge between two simplex vertices.

The authors prove this characterization when C=2 by showing that any configuration with at least two non-deterministic predictions can be perturbed while preserving TU and strictly decreasing AU. They state that the general case for arbitrary C remains unresolved. Establishing it would characterize the infeasible boundary of the finite-sample AU–TU uncertainty region in the general multiclass setting.