# The Wishart planted ensemble: A tunably-rugged pairwise Ising model with a first-order phase transition

## TL;DR

The paper introduces the Wishart planted ensemble, a tunably-hard Ising model with a first-order phase transition, useful for benchmarking optimization algorithms and understanding rugged energy landscapes.

## Contribution

It presents a new class of Ising models with tunable hardness, analytical insights into their thermodynamics, and characterization of their phase transition and energy landscape.

## Key findings

- Exhibits a first-order phase transition in temperature.
- Displays a wide range of algorithmic hardness with a peak in solution time.
- Analytical expressions for hardness peak and energy distribution are derived.

## Abstract

We propose the Wishart planted ensemble, a class of zero-field Ising models with tunable algorithmic hardness and specifiable (or planted) ground state. The problem class arises from a simple procedure for generating a family of random integer programming problems with specific statistical symmetry properties, but turns out to have intimate connections to a sign-inverted variant of the Hopfield model. The Hamiltonian contains only 2-spin interactions, with the coupler matrix following a type of Wishart distribution. The class exhibits a classical first-order phase transition in temperature. For some parameter settings the model has a locally-stable paramagnetic state, a feature which correlates strongly with difficulty in finding the ground state and suggests an extremely rugged energy landscape. We analytically probe the ensemble thermodynamic properties by deriving the Thouless-Anderson-Palmer equations and free energy and corroborate the results with a replica and annealed approximation analysis; extensive Monte Carlo simulations confirm our predictions of the first-order transition temperature. The class exhibits a wide variation in algorithmic hardness as a generation parameter is varied, with a pronounced easy-hard-easy profile and peak in solution time towering many orders of magnitude over that of the easy regimes. By deriving the ensemble-averaged energy distribution and taking into account finite-precision representation, we propose an analytical expression for the location of the hardness peak and show that at fixed precision, the number of constraints in the integer program must increase with system size to yield truly hard problems. The Wishart planted ensemble is interesting for its peculiar physical properties and provides a useful and analytically-transparent set of problems for benchmarking optimization algorithms.