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🎤 Vocab
SDE - Stochastic Differential Equation
MCMC - Markov Chain Monte Carlo
- “class of algorithms used to draw samples from a probability distribution”
Langevin - A specific type of SDE - Mirror Langevin SDE in dual space?
- Riemannian Langevin dynamics in primal space?
- Langevin MCMC
Brownian Motion
KL Divergence?
❗ Information
Talk Info
Speaker:
Qijia Jiang, Assistant Professor, Department of Statistics, UC Davis
Title:
“Applied Mathematics Beyond the Textbook”
Abstract:
I plan to share a few vignettes of my research evolving around the general themes of {optimization, MCMC sampling, diffusion model, Neural PDE solver} over the years. Throughout, I hope to reflect a bit on how machine learning is changing the way scientific computation is done, and point out the many connections across different sub-fields of Applied Math, as well as exciting synergies between Applied Math and the broader natural sciences disciplines.
Overview
Optimization <- Information theoretic complexity studies
infinite dim opt in general metric space -> MCMC sampling
“Applied math beyond the textbook”? Diffusion model Autodiff PDE solver
Many opportunities to exploit connections between topics and disciplines
MCMC struggles with multi-modality in the target distribution
- Borrow ideas from generative modeling: optimal stochastic control / optimal transport to steer a trajectory from
using machine learning
Paradigm Shift
Many high profile successes: (insert photo)
- Not just dumping more data, also ingenuity on how to implement
inductive?bias
Excited
Particularly excited about
- Theoretically: the connection between optimization, sampling, physics-inspired dynamical system (e.g., HMC, momentum), mean-filed game goes much deeper
- Computationally: bring powerful function fitting NN-architecture to solve more traditional tasks in sampling, control, PDE etc., is changing many areas of science
- Applications: in climate modeling (PDE), single-cell genomics (optimal transport), drug discovery & material design (sampling, generative modeling)
✒️ -> Scratch Notes
Prox step: Way to go between iterates. Steps toward optimization
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🔗 -> Links
Resources
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Connections
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