Roughness-informed machine learning – A call for fractal and fractional calculi

Mohammad Partohaghighi, Roummel Marcia, Bruce J. West, YangQuan Chen,
Roughness-informed machine learning – A call for fractal and fractional calculi,
Journal of Information and Intelligence,
2025,
,
ISSN 2949-7159,
https://doi.org/10.1016/j.jiixd.2025.09.001.
(https://www.sciencedirect.com/science/article/pii/S2949715925000460)
Abstract: This paper presents a unified framework for roughness-informed machine learning, dividing roughness into four categories: statistical, geometric, manifold, and topological. Statistical roughness, analyzed with tools like WeightWatcher, utilizes heavy-tailed weight distributions. Geometric roughness, measured by a novel roughness index, quantifies oscillatory patterns in loss landscapes. Manifold roughness, captured by the two-scale effective dimension, integrates local geometry (via fisher information matrix) with global parameter space complexity. Topological roughness, derived from persistence diagrams, evaluates structural complexity of learned functions. Experiments on MNIST, CIFAR-10, CIFAR-100, a damped harmonic oscillator, Fractional Order ODE, and wave equation demonstrate the framework's effectiveness: statistical roughness enhances federated learning convergence, geometric roughness improves training stability, manifold roughness optimizes generalization through noise injection, and topological roughness ensures smoother, physically accurate solutions. The framework advances model design, optimization, and generalization, with links to fractal and fractional calculus.
Keywords: Machine learning; Optimization; Federated learning; Roughness; Fractal calculus; Fractional calculus