Data-Free/Data-Sparse Softmax Parameter Estimation with Structured Class Geometries

TL;DR

This paper introduces a geometric approach to estimate softmax model parameters in data-sparse scenarios by solving linear equations derived from class boundary polytopes, enabling data-free or data-efficient learning.

Contribution

It presents a novel geometric framework for softmax parameter estimation using convex polytopes, allowing closed-form solutions without extensive data or optimization.

Findings

Linear system solutions encode class boundary geometries.

Closed-form softmax parameters can be obtained from polytope specifications.

Some classification problems cannot be modeled with softmax boundaries due to geometric constraints.

Abstract

This note considers softmax parameter estimation when little/no labeled training data is available, but a priori information about the relative geometry of class label log-odds boundaries is available. It is shown that `data-free' softmax model synthesis corresponds to solving a linear system of parameter equations, wherein desired dominant class log-odds boundaries are encoded via convex polytopes that decompose the input feature space. When solvable, the linear equations yield closed-form softmax parameter solution families using class boundary polytope specifications only. This allows softmax parameter learning to be implemented without expensive brute force data sampling and numerical optimization. The linear equations can also be adapted to constrained maximum likelihood estimation in data-sparse settings. Since solutions may also fail to exist for the linear parameter equations derived from certain polytope specifications, it is thus also shown that there exist probabilistic classification problems over m convexly separable classes for which the log-odds boundaries cannot be learned using an m-class softmax model.