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Empirical studies on the landscape of neural networks have shown thatlow-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-convexneural network model framed as a continuous constraint satisfaction problem. Weintroduce a general analytical method for computing energy barriers in thesimplex with vertex configurations sampled from the equilibrium. We find thatin the over-parameterized regime the solution manifold displays simpleconnectivity properties. There exists a large geodesically convex componentthat is attractive for a wide range of optimization dynamics. Inside thisregion we identify a subset of atypical high-margin solutions that aregeodesically connected with most other solutions, giving rise to a star-shapedgeometry. We analytically characterize the organization of the connected spaceof solutions and show numerical evidence of a transition, at larger constraintdensities, where the aforementioned simple geodesic connectivity breaks down.

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