Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization

2026-05-23

AI Summary

This work addresses the challenges posed by non-convex energy landscapes and poor convergence to discrete solutions in NP-hard combinatorial optimization problems—such as QUBO, binary sparse coding, and planted-solution Ising models—by introducing a physics-inspired continuous relaxation framework. The approach parameterizes binary variables as phase states on the complex unit circle, leveraging the implicit regularization induced by this representation to enable standard real-valued optimizers, like gradient descent, to naturally converge to discrete solutions. This study is the first to demonstrate that complex-phase encoding inherently yields implicit regularization and successfully integrates it into conventional real-valued optimization without altering the original problem structure. Experiments show zero error on 160×160 QUBO instances (σ=0.25), perfect recovery in underdetermined sparse coding outperforming OMP and LASSO (σ=0.15), and exact ground-state recovery in 8 out of 11 planted-solution benchmarks.

Abstract

We introduce a physics-inspired continuous relaxation framework that yields substantially improved solutions for NP-hard combinatorial optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), binary sparse coding, and planted-solution Ising models. By parameterizing discrete binary variables as continuous wave-like states on the complex unit circle, we inherently smooth highly non-convex energy landscapes. We show that representing binary variables as complex phases reveals an implicit regularization mechanism that promotes convergence toward discrete states. Extracting this mechanism yields significant improvements even within standard real-valued optimization frameworks, using this regularizer explicitly. Empirically, this regularization yields vastly higher ground-state convergence rates than standard real-valued alternatives. Our models achieved zero error in large-scale 160x160 QUBO tasks under severe noise (sigma=0.25), and outperformed traditional algorithms (OMP and LASSO) in underdefined sparse coding with perfect recovery at sigma=0.15. The solver's robustness was further validated by recovering exact ground-state configurations in 8 out of 11 rigorously engineered planted-solution benchmarks.

Problem

• Combinatorial Optimization
• QUBO
• Sparse Coding
• Ising Model
• NP-hard

Innovation

• implicit binarization
• complex phase dynamics
• combinatorial optimization
• continuous relaxation
• regularization mechanism