Complex complex landscapes

TL;DR

This paper analyzes the saddle points of the complex $p$-spin model, revealing the number of solutions, spectral properties of the Hessian, and phase transitions, extending understanding of rugged landscapes into the complex domain.

Contribution

It provides an exact calculation of the average number of solutions and characterizes the spectral transition of the Hessian in the complex $p$-spin landscape.

Findings

Number of solutions saturates Bézout bound

Hessian spectrum exhibits a transition with a gap

Real solutions are a subset of complex solutions

Abstract

We study the saddle-points of the $p$-spin model -- the best understood example of a `complex' (rugged) landscape -- when its $N$ variables are complex. These points are the solutions to a system of $N$ random equations of degree $p-1$. We solve for $\overline{\mathcal N}$, the number of solutions averaged over randomness in the $N\to\infty$ limit. We find that it saturates the B\'ezout bound $\log\overline{\mathcal N}\sim N\log(p-1)$. The Hessian of each saddle is given by a random matrix of the form $C^\dagger C$, where $C$ is a complex symmetric Gaussian matrix with a shift to its diagonal. Its spectrum has a transition where a gap develops that generalizes the notion of `threshold level' well-known in the real problem. The results from the real problem are recovered in the limit of real parameters. In this case, only the square-root of the total number of solutions are real. In terms of the complex energy, the solutions are divided into sectors where the saddles have different topological properties.