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Hamiltonian Score Matching and Generative Flows (2410.20470v1)

Published 27 Oct 2024 in cs.LG and stat.ML

Abstract: Classical Hamiltonian mechanics has been widely used in machine learning in the form of Hamiltonian Monte Carlo for applications with predetermined force fields. In this work, we explore the potential of deliberately designing force fields for Hamiltonian ODEs, introducing Hamiltonian velocity predictors (HVPs) as a tool for score matching and generative models. We present two innovations constructed with HVPs: Hamiltonian Score Matching (HSM), which estimates score functions by augmenting data via Hamiltonian trajectories, and Hamiltonian Generative Flows (HGFs), a novel generative model that encompasses diffusion models and flow matching as HGFs with zero force fields. We showcase the extended design space of force fields by introducing Oscillation HGFs, a generative model inspired by harmonic oscillators. Our experiments validate our theoretical insights about HSM as a novel score matching metric and demonstrate that HGFs rival leading generative modeling techniques.

Summary

  • The paper introduces Hamiltonian Score Matching and Generative Flows by leveraging Hamiltonian mechanics to develop Hamiltonian Velocity Predictors for score estimation.
  • It establishes the Hamiltonian score discrepancy as a novel metric, offering a lower variance alternative to traditional denoising score matching objectives.
  • Using Hamiltonian ODEs and parameterized force fields, the approach achieves competitive performance in image generation and opens new avenues for modeling complex distributions.

An Expert Overview of "Hamiltonian Score Matching and Generative Flows"

The paper "Hamiltonian Score Matching and Generative Flows" introduces a novel approach leveraging the principles of Hamiltonian mechanics for score matching and generative modeling. The authors, Peter Holderrieth, Yilun Xu, and Tommi Jaakkola, propose new models referred to as Hamiltonian Velocity Predictors (HVPs) for score estimation and generative flow modeling, specifically introducing Hamiltonian Score Matching (HSM) and Hamiltonian Generative Flows (HGFs). This research sits at the intersection of classical physics and machine learning, aiming to extend the applicability of Hamiltonian mechanics beyond existing models like Hamiltonian Monte Carlo (HMC).

Hamiltonian Framework and Applications in Machine Learning

The theoretical foundation of this work is derived from Hamiltonian mechanics, which has been instrumental in physical sciences, providing a framework for studying the dynamics of systems. This mathematical structure has been widely adopted in machine learning, particularly for sampling algorithms such as HMC, which exploit the Hamiltonian pathways to formulate fast-mixing Markov chains for efficient sampling from complex probability densities.

In this paper, the authors explore the idea of engineering force fields deliberately within the Hamiltonian ordinary differential equations (ODEs) to facilitate both score matching and generative modeling. This innovative approach is executed through two primary contributions: Hamiltonian Score Matching (HSM) and Hamiltonian Generative Flows (HGFs).

Hamiltonian Score Matching (HSM)

HSM is presented as a score matching method where scores are approximated by replicating the conditions that preserve the Boltzmann-Gibbs distribution. The authors derive conditions under which a force field can precisely be considered a score field based on its ability to maintain the Boltzmann-Gibbs distribution invariant throughout Hamiltonian dynamics. By extending this preservation property, the authors introduce the Hamiltonian score discrepancy (HSD), which acts as a novel metric to gauge the quality of score approximations.

A significant theoretical contribution of this paper is the demonstration of the HSD's equivalence to traditional score matching objectives under certain conditions. The HSM framework allows one to minimize the HSD, offering a potentially lower variance alternative to denoising score matching for learning the true data distribution without injecting noise.

Hamiltonian Generative Flows (HGFs)

In proceeding to generative modeling, HGFs embody a new class of models that capitalize on the extended design space of force fields in Hamiltonian systems. By focusing on parameterized Hamiltonian ODEs, the work integrates established models such as diffusion models and flow matching into the HGF framework when considered with zero force fields. A distinguishing attribute of HGFs is their ability to generate state trajectories through parameterized force fields, using learned velocity predictors for backward integration toward the original data distribution.

The concept of Oscillation HGFs, utilizing harmonic oscillator dynamics as a force field, is particularly highlighted as it offers inherent scale-invariance, beneficial for enhanced training stability and sample quality. This specific type of HGF shows promising results, achieving competitive performance relative to state-of-the-art diffusion models in image generation tasks.

Implications and Future Directions

The implications of this research are multifaceted. Practically, leveraging Hamiltonian mechanics for designing generative models opens novel avenues for improving the efficiency and quality of models typically employed in fields such as image and molecular generation. Theoretical implications are equally profound, expanding the understanding of how classical mechanics principles can be translated into algorithmic innovations in machine learning.

Future developments might orient toward better integration of domain-specific force fields into HGFs, particularly for applications requiring adherence to known physical laws. Furthermore, tackling datasets lying on manifolds or exploring scalable bi-level optimization techniques for HSM represents promising areas for subsequent research.

In conclusion, "Hamiltonian Score Matching and Generative Flows" offers a rich theoretical and practical framework that extends the application of Hamiltonian dynamics into machine learning. By proposing a novel score matching metric and a generative modeling schema, it paves the way for future advancements at the intersection of classical physics and artificial intelligence.

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