Summary of Generative Modelling with Tensor Train Approximations Of Hamilton–jacobi–bellman Equations, by David Sommer et al.
Generative Modelling with Tensor Train approximations of Hamilton–Jacobi–Bellman equations
by David Sommer, Robert Gruhlke, Max Kirstein, Martin Eigel, Claudia Schillings
First submitted to arxiv on: 23 Feb 2024
Categories
- Main: Machine Learning (stat.ML)
- Secondary: Machine Learning (cs.LG); Statistics Theory (math.ST)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary The proposed paper addresses the challenge of sampling from probability densities, which is crucial in fields like Uncertainty Quantification (UQ) and Generative Modelling (GM). In GM, reverse-time diffusion processes relying on log-densities from Ornstein-Uhlenbeck forward processes are popular tools. However, these log-densities can be obtained by solving a Hamilton-Jacobi-Bellman (HJB) equation from stochastic optimal control. Instead of using indirect methods like policy iteration and neural networks, the authors propose direct time integration to solve the HJB equation. This approach utilizes compressed polynomials represented in Tensor Train (TT) format for spatial discretization. The method is sample-free, agnostic to normalization constants, and can avoid the curse of dimensionality due to TT compression. The authors provide a complete derivation of the HJB equation’s action on TT polynomials and demonstrate the performance of the proposed integration method on a nonlinear sampling task in 20 dimensions. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary The paper solves a problem that is important for understanding uncertainty in complex systems. Right now, scientists use indirect methods to get log-densities from probability densities, which can be tricky and slow. The authors came up with a new way to do it using something called Tensor Train format. This method is special because it doesn’t need samples, doesn’t care about normalization constants, and can handle big problems by compressing the data. |
Keywords
* Artificial intelligence * Diffusion * Probability
