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.
Highlights
Spin glasses are the paradigm of many-variable “complex landscapes,” a category that includes neural networks and optimization problems like constraint satisfaction [1]
Where J is a symmetric tensor whose elements are real Gaussian variables and z ∈ RN is constrained to the sphere zT z = N
This problem has been studied in the algebra [4] and probability literature [5,6]. It has been attacked from several angles: the replica trick to compute the Boltzmann–Gibbs distribution [2], a Kac–Rice [7,8,9] procedure to compute the number of saddle points of the energy function [10], and gradient-descent dynamics starting from a high-energy configuration [11]
Summary
Jaron Kent-Dobias and Jorge Kurchan Laboratoire de Physique de l’Ecole Normale Supérieure, 75005 Paris, France (Received December 2020; accepted 1 April 2021; published April 2021). 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 N , the number of solutions averaged over randomness in the N → ∞ limit. 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
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