# Hamiltonian Metric Learning and Energy-Based Training: A Dissipative Geometric Framework for Optimization

arXiv: https://arxiv.org/abs/2610.04969 · HTML: https://mlsanity.com/paper/2610.04969/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 20 references that are in the map.
- First posted on arXiv 4 Oct 2026.

## Builds on (most cited first, 7 of 7)

- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamics](https://mlsanity.com/paper/2012.04728/) (2020)
- [Noether's Learning Dynamics: Role of Symmetry Breaking in Neural Networks](https://mlsanity.com/paper/2105.02716/) (2021)
- [Conformal Symplectic and Relativistic Optimization](https://mlsanity.com/paper/1903.04100/) (2019)
- [Short and Straight: Geodesics on Differentiable Manifolds](https://mlsanity.com/paper/2305.15228/) (2023)

## Abstract (arXiv)

Optimization in machine learning is usually expressed through iterative rules that update model parameters using information from the loss landscape. In this work, we study an alternative viewpoint in which parameter optimization is treated as the evolution of a dissipative dynamical system. The model parameters are regarded as generalized coordinates, the loss acts as a potential energy, and a positive-definite metric defines the local kinetic geometry of parameter space. Starting from a variational formulation, we derive the corresponding Hamiltonian dynamics, the geometric force generated by a position-dependent metric, and a metric-compatible Rayleigh dissipation law. The resulting continuous system satisfies a monotonic energy-dissipation relation, while its discrete form allows the influence of curvature on the optimization trajectory to be studied directly. We illustrate the framework using a controlled CIFAR-10 image-reconstruction problem for which the optimum is known analytically. With the image Hessian used as the metric, the curvature dependence of the quadratic modal dynamics is removed. We then reparameterize the same image as a matrix product state, producing a genuinely position-dependent metric and a nonzero geometric force. Poincar'e return maps provide a complementary phase-space view of the resulting contraction dynamics. These examples establish HAMLET as a geometric, energy-based framework for studying optimization as dissipative motion in parameter space. On a five-seed MNIST MLP benchmark, HAMLET attains the highest mean test accuracy (97.95%) and the lowest mean test negative log-likelihood (0.0696) among the three evaluated optimizers.