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

arXiv: https://arxiv.org/abs/1806.00728 · HTML: https://mlsanity.com/paper/1806.00728/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2018.

## Built on by (most cited first, 3 of 3)

- [Observation-Augmented Contextual Multi-Armed Bandits for Robotic Search and Exploration](https://mlsanity.com/paper/2312.12583/) (2023)
- [HARPS: An Online POMDP Framework for Human-Assisted Robotic Planning and Sensing](https://mlsanity.com/paper/2110.10324/) (2021)
- [Spatial Language Likelihood Grounding Network for Bayesian Fusion of Human-Robot Observations](https://mlsanity.com/paper/2507.19947/) (2025)

## Abstract (arXiv)

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.