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Deformations of Boltzmann Distributions
Bálint Máté · François Fleuret
Consider a one-parameter family of Boltzmann distributions $p_t(x) = \tfrac{1}{Z_t}e^{-S_t(x)}$. This work studies the problem of sampling from $p_{t_0}$ by first sampling from $p_{t_1}$ and then applying a transformation $\Psi_{t_1}^{t_0}$ so that the transformed samples follow $p_{t_0}$. We derive an equation relating $\Psi$ and the corresponding family of unnormalized log-likelihoods $S_t$. The utility of this idea is demonstrated on the $\phi^4$ lattice field theory by extending its defining action $S_0$ to a family of actions $S_t$ and finding a $\tau$ such that normalizing flows perform better at learning the Boltzmann distribution $p_\tau$ than at learning $p_0$.

#### Author Information

##### François Fleuret (University of Geneva)

François Fleuret got a PhD in Mathematics from INRIA and the University of Paris VI in 2000, and an Habilitation degree in Mathematics from the University of Paris XIII in 2006. He is Full Professor in the department of Computer Science at the University of Geneva, and Adjunct Professor in the School of Engineering of the École Polytechnique Fédérale de Lausanne. He has published more than 80 papers in peer-reviewed international conferences and journals. He is Associate Editor of the IEEE Transactions on Pattern Analysis and Machine Intelligence, serves as Area Chair for NeurIPS, AAAI, and ICCV, and in the program committee of many top-tier international conferences in machine learning and computer vision. He was or is expert for multiple funding agencies. He is the inventor of several patents in the field of machine learning, and co-founder of Neural Concept SA, a company specializing in the development and commercialization of deep learning solutions for engineering design. His main research interest is machine learning, with a particular focus on computational aspects and sample efficiency.