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From Physics to Probability: Hamiltonian Mechanics for Generative Modeling and MCMC
- Hamiltonian mechanics provides a framework based on energy to describe physical systems, offering insights into conservation laws and symmetries.
- By using generalized coordinates and momenta, Hamiltonian mechanics simplifies complex system dynamics, aiding in finding patterns.
- The framework transforms Newtonian second-order equations into two first-order equations, revealing hidden geometric structures in phase space.
- Hamilton's equations govern the evolution of systems in phase space, providing a geometric understanding of their behavior and symmetries.
- Symplecticity, a sacred invariant in Hamiltonian systems, ensures the preservation of area in phase space under Hamiltonian flows.
- Symplectic numerical integrators like Leapfrog Verlet maintain phase space volume conservation and are crucial for stable simulations in Hamiltonian systems.
- Hamiltonian Monte Carlo (HMC) leverages Hamiltonian dynamics to sample efficiently from complex probability distributions, useful in Bayesian statistics and Machine Learning.
- HMC bridges physics and ML by interpreting distributions in phase space and using Hamiltonian principles for exploration and sampling.
- Using symplecticity in machine learning models, exploring HMC for complex distributions, and combining Hamiltonian mechanics with ML techniques are potential research directions.
- Hamiltonian GANs represent a potential application of Hamiltonian formalism for generating physically plausible videos with neural networks.
- Hamiltonian mechanics offers stability, energy conservation, and efficiency for simulations and sampling in complex systems, essential for various scientific and ML applications.
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