# How to Learn a Star: Binary Classification with Starshaped Polyhedral Sets

arXiv: https://arxiv.org/abs/2505.01346 · HTML: https://mlsanity.com/paper/2505.01346/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 14 references that are in the map.
- First posted on arXiv 2 May 2025.

## Builds on (most cited first, 3 of 3)

- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [The VC-dimension of k-vertex d-polytopes](https://mlsanity.com/paper/2004.04841/) (2020)
- [Learning polytopes with fixed facet directions](https://mlsanity.com/paper/2201.03419/) (2022)

## Abstract (arXiv)

We consider binary classification restricted to a class of continuous piecewise linear functions whose decision boundaries are (possibly nonconvex) starshaped polyhedral sets, supported on a fixed polyhedral simplicial fan. We investigate the expressivity of these function classes and describe the combinatorial and geometric structure of the loss landscape, most prominently the sublevel sets, for two loss-functions: the 0/1-loss (discrete loss) and a log-likelihood loss function. In particular, we give explicit bounds on the VC dimension of this model, and concretely describe the sublevel sets of the discrete loss as chambers in a hyperplane arrangement. For the log-likelihood loss, we give sufficient conditions for the optimum to be unique, and describe the geometry of the optimum when varying the rate parameter of the underlying exponential probability distribution.