# The star-shaped space of solutions of the spherical negative perceptron

arXiv: https://arxiv.org/abs/2305.10623 · HTML: https://mlsanity.com/paper/2305.10623/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 10 references that are in the map.
- First posted on arXiv 18 May 2023.

## Builds on (most cited first, 2 of 2)

- [Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity](https://mlsanity.com/paper/2106.15933/) (2021)
- [Typical and atypical solutions in non-convex neural networks with discrete and continuous weights](https://mlsanity.com/paper/2304.13871/) (2023)

## Built on by (most cited first, 8 of 8)

- [Exploring Neural Network Landscapes: Star-Shaped and Geodesic Connectivity](https://mlsanity.com/paper/2404.06391/) (2024)
- [Do Deep Neural Network Solutions Form a Star Domain?](https://mlsanity.com/paper/2403.07968/) (2024)
- [Exact full-RSB SAT/UNSAT transition in infinitely wide two-layer neural networks](https://mlsanity.com/paper/2410.06717/) (2024)
- [High-dimensional manifold of solutions in neural networks: insights from statistical physics](https://mlsanity.com/paper/2309.09240/) (2023)
- [Architecture-Aware Minimization (A2M): How to Find Flat Minima in Neural Architecture Search](https://mlsanity.com/paper/2503.10404/) (2025)
- [Sharp description of local minima in the loss landscape of high-dimensional two-layer ReLU neural networks](https://mlsanity.com/paper/2604.09412/) (2026)
- [Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval](https://mlsanity.com/paper/2602.17779/) (2026)
- [Dynamical loss functions shape landscape topography and improve learning in artificial neural networks](https://mlsanity.com/paper/2410.10690/) (2024)

## Abstract (arXiv)

Empirical studies on the landscape of neural networks have shown that low-energy configurations are often found in complex connected structures, where zero-energy paths between pairs of distant solutions can be constructed. Here we consider the spherical negative perceptron, a prototypical non-convex neural network model framed as a continuous constraint satisfaction problem. We introduce a general analytical method for computing energy barriers in the simplex with vertex configurations sampled from the equilibrium. We find that in the over-parameterized regime the solution manifold displays simple connectivity properties. There exists a large geodesically convex component that is attractive for a wide range of optimization dynamics. Inside this region we identify a subset of atypical high-margin solutions that are geodesically connected with most other solutions, giving rise to a star-shaped geometry. We analytically characterize the organization of the connected space of solutions and show numerical evidence of a transition, at larger constraint densities, where the aforementioned simple geodesic connectivity breaks down.