An optimal control perspective on diffusion-based generative modeling Article Swipe
Julius Berner
,
Lorenz Richter
,
Karen Ullrich
·
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2211.01364
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2211.01364
We establish a connection between stochastic optimal control and generative models based on stochastic differential equations (SDEs), such as recently developed diffusion probabilistic models. In particular, we derive a Hamilton-Jacobi-Bellman equation that governs the evolution of the log-densities of the underlying SDE marginals. This perspective allows to transfer methods from optimal control theory to generative modeling. First, we show that the evidence lower bound is a direct consequence of the well-known verification theorem from control theory. Further, we can formulate diffusion-based generative modeling as a minimization of the Kullback-Leibler divergence between suitable measures in path space. Finally, we develop a novel diffusion-based method for sampling from unnormalized densities -- a problem frequently occurring in statistics and computational sciences. We demonstrate that our time-reversed diffusion sampler (DIS) can outperform other diffusion-based sampling approaches on multiple numerical examples.
Related Topics
Concepts
Stochastic differential equation
Optimal control
Diffusion
Stochastic control
Computer science
Divergence (linguistics)
Applied mathematics
Mathematical optimization
Generative model
Probabilistic logic
Sampling (signal processing)
Perspective (graphical)
Importance sampling
Mathematics
Generative grammar
Artificial intelligence
Statistics
Monte Carlo method
Philosophy
Physics
Linguistics
Computer vision
Thermodynamics
Filter (signal processing)
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2211.01364
- https://arxiv.org/pdf/2211.01364
- OA Status
- green
- Cited By
- 6
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4308163982
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W4308163982Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2211.01364Digital Object Identifier
- Title
-
An optimal control perspective on diffusion-based generative modelingWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2022Year of publication
- Publication date
-
2022-11-02Full publication date if available
- Authors
-
Julius Berner, Lorenz Richter, Karen UllrichList of authors in order
- Landing page
-
https://arxiv.org/abs/2211.01364Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2211.01364Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2211.01364Direct OA link when available
- Concepts
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Stochastic differential equation, Optimal control, Diffusion, Stochastic control, Computer science, Divergence (linguistics), Applied mathematics, Mathematical optimization, Generative model, Probabilistic logic, Sampling (signal processing), Perspective (graphical), Importance sampling, Mathematics, Generative grammar, Artificial intelligence, Statistics, Monte Carlo method, Philosophy, Physics, Linguistics, Computer vision, Thermodynamics, Filter (signal processing)Top concepts (fields/topics) attached by OpenAlex
- Cited by
-
6Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 1, 2024: 3, 2023: 1, 2022: 1Per-year citation counts (last 5 years)
- Related works (count)
-
10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.derive | 27 |
| abstract_inverted_index.direct | 66 |
| abstract_inverted_index.method | 102 |
| abstract_inverted_index.models | 10 |
| abstract_inverted_index.space. | 95 |
| abstract_inverted_index.theory | 52 |
| abstract_inverted_index.(SDEs), | 16 |
| abstract_inverted_index.between | 4, 90 |
| abstract_inverted_index.control | 7, 51, 74 |
| abstract_inverted_index.develop | 98 |
| abstract_inverted_index.governs | 32 |
| abstract_inverted_index.methods | 48 |
| abstract_inverted_index.models. | 23 |
| abstract_inverted_index.optimal | 6, 50 |
| abstract_inverted_index.problem | 110 |
| abstract_inverted_index.sampler | 124 |
| abstract_inverted_index.theorem | 72 |
| abstract_inverted_index.theory. | 75 |
| abstract_inverted_index.Finally, | 96 |
| abstract_inverted_index.Further, | 76 |
| abstract_inverted_index.equation | 30 |
| abstract_inverted_index.evidence | 61 |
| abstract_inverted_index.measures | 92 |
| abstract_inverted_index.modeling | 82 |
| abstract_inverted_index.multiple | 133 |
| abstract_inverted_index.recently | 19 |
| abstract_inverted_index.sampling | 104, 130 |
| abstract_inverted_index.suitable | 91 |
| abstract_inverted_index.transfer | 47 |
| abstract_inverted_index.densities | 107 |
| abstract_inverted_index.developed | 20 |
| abstract_inverted_index.diffusion | 21, 123 |
| abstract_inverted_index.equations | 15 |
| abstract_inverted_index.establish | 1 |
| abstract_inverted_index.evolution | 34 |
| abstract_inverted_index.examples. | 135 |
| abstract_inverted_index.formulate | 79 |
| abstract_inverted_index.modeling. | 55 |
| abstract_inverted_index.numerical | 134 |
| abstract_inverted_index.occurring | 112 |
| abstract_inverted_index.sciences. | 117 |
| abstract_inverted_index.approaches | 131 |
| abstract_inverted_index.connection | 3 |
| abstract_inverted_index.divergence | 89 |
| abstract_inverted_index.frequently | 111 |
| abstract_inverted_index.generative | 9, 54, 81 |
| abstract_inverted_index.marginals. | 42 |
| abstract_inverted_index.outperform | 127 |
| abstract_inverted_index.statistics | 114 |
| abstract_inverted_index.stochastic | 5, 13 |
| abstract_inverted_index.underlying | 40 |
| abstract_inverted_index.well-known | 70 |
| abstract_inverted_index.consequence | 67 |
| abstract_inverted_index.demonstrate | 119 |
| abstract_inverted_index.particular, | 25 |
| abstract_inverted_index.perspective | 44 |
| abstract_inverted_index.differential | 14 |
| abstract_inverted_index.minimization | 85 |
| abstract_inverted_index.unnormalized | 106 |
| abstract_inverted_index.verification | 71 |
| abstract_inverted_index.computational | 116 |
| abstract_inverted_index.log-densities | 37 |
| abstract_inverted_index.probabilistic | 22 |
| abstract_inverted_index.time-reversed | 122 |
| abstract_inverted_index.diffusion-based | 80, 101, 129 |
| abstract_inverted_index.Kullback-Leibler | 88 |
| abstract_inverted_index.Hamilton-Jacobi-Bellman | 29 |
| cited_by_percentile_year.max | 97 |
| cited_by_percentile_year.min | 89 |
| countries_distinct_count | 0 |
| institutions_distinct_count | 3 |
| citation_normalized_percentile.value | 0.78525515 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |