Non-Convex Portfolio Optimization via Energy-Based Models: A Comparative Analysis Using the Thermodynamic HypergRaphical Model Library (THRML) for Index Tracking

ArXiv ID: 2601.07792 “View on arXiv”

Authors: Javier Mancilla, Theodoros D. Bouloumis, Frederic Goguikian

Abstract

Portfolio optimization under cardinality constraints transforms the classical Markowitz mean-variance problem from a convex quadratic problem into an NP-hard combinatorial optimization problem. This paper introduces a novel approach using THRML (Thermodynamic HypergRaphical Model Library), a JAX-based library for building and sampling probabilistic graphical models that reformulates index tracking as probabilistic inference on an Ising Hamiltonian. Unlike traditional methods that seek a single optimal solution, THRML samples from the Boltzmann distribution of high-quality portfolios using GPU-accelerated block Gibbs sampling, providing natural regularization against overfitting. We implement three key innovations: (1) dynamic coupling strength that scales inversely with market volatility (VIX), adapting diversification pressure to market regimes; (2) rebalanced bias weights prioritizing tracking quality over momentum for index replication; and (3) sector-aware post-processing ensuring institutional-grade diversification. Backtesting on a 100-stock S and P 500 universe from 2023 to 2025 demonstrates that THRML achieves 4.31 percent annualized tracking error versus 5.66 to 6.30 percent for baselines, while simultaneously generating 128.63 percent total return against the index total return of 79.61 percent. The Diebold-Mariano test confirms statistical significance with p less than 0.0001 across all comparisons. These results position energy-based models as a promising paradigm for portfolio construction, bridging statistical mechanics and quantitative finance.

Keywords: Portfolio Optimization, Ising Model, Probabilistic Graphical Models, GPU Acceleration, Index Tracking

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper uses advanced statistical mechanics (Ising Hamiltonians, Boltzmann distributions) and NP-hard optimization, indicating high math complexity. It also provides concrete backtesting results with specific metrics, statistical significance tests, and implementation details (JAX library, GPU sampling), demonstrating strong empirical rigor.

  flowchart TD
    A["Research Goal: Improve Index Tracking via Non-Convex Portfolio Optimization"] --> B["Methodology: Energy-Based Model using THRML"]
    B --> C["Inputs: S&P 500 100-Stock Universe 2023-2025"]
    C --> D["Computational Process: GPU-Accelerated Probabilistic Inference"]
    D --> E["Dynamic Coupling & Sector-Aware Post-Processing"]
    E --> F["Outcomes: 4.31% Tracking Error vs 5.66-6.30% Baseline"]
    F --> G["Outcomes: 128.63% Total Return vs 79.61% Index Return"]
    G --> H["Statistical Significance: Diebold-Mariano p < 0.0001"]