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In this paper, we connect some recent papers on smoothing of energy landscapes and scored-based generative models of machine learning to classical work in stochastic control.
Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung resp. den Sätzen über das Wärmegleichgewicht. Wiener Berichte 76, 373-435, 1877
L Boltzmann · 1909
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Über die Umkehrung der Naturgesetze
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E. Schrödinger, Sur la théorie relativiste de l’électron et l’interpretation de la mécanique quantique, Ann. Inst. H. Poincaré
1932
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On the probability of large deviations of random magnitudes (in Russian)
Ivan N Sanov · 1961
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Dynamical theories of Brownian motion
Edward Nelson · 1967
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Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control, Springer-Verlag, Berlin (1975)
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W. H. Fleming, Exit probabilities and optimal stochastic control, Appl. Math. Optim
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W. H. Fleming, Logarithmic transformation and stochastic control, in Advances in Filtering and Optimal Stochastic Control
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W. H. Fleming, Stochastic calculus of variations and mechanics, J. Optim. Th. Appl
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W. Fleming and S. Sheu, Stochastic variational formula for fundamental solutions of parabolic PDE, Appl. Math. and Optimization
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H. Föllmer, Time reversal on Wiener space, in Stochastic Processes - Mathematics and Physics
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Random fields and diffusion processes
Hans Föllmer · 1988
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I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus
1988
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E. Nelson, Stochastic mechanics and random fields, in Ècole d’Ètè de Probabilitès de Saint-Flour XV-XVII
1988
Cited alongside, same era.
P. Dai Pra and M. Pavon, Variational path-integral representations for the density of a diffusion process, Stochastics
1989
Cited alongside, same era.
M. Pavon, Stochastic control and nonequilibrium thermodynamical systems, Appl. Math. and Optimiz
1989
Cited alongside, same era.
M.Pavon and A.Wakolbinger, On free energy, stochastic control, and Schroedinger processes, in Modeling, Estimation and Control of Systems with Uncertainty
1991
Cited alongside, same era.
Schrödinger bridges from 1931 to 1991
A Wakolbinger · 1992
Cited alongside, same era.
Topics in optimal transportation
Cédric Villani · 2003
P. Chaudhari, A. Choromanska, S. Soatto, Y. LeCun, C. Baldassi, C. Borgs, J. Chayes, L. Sagun, R. Zecchina, Entropy-SGD: Biasing Gradient Descent Into Wide Valleys, ICLR 2017
2017
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Qianxiao Li, Cheng Tai, Weinan E, Stochastic Modified Equations and Adaptive Stochastic Gradient Algorithms, Proc. of the 34th Int. Conf. on Machine Learning
2017
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P. Chaudhari and S. Soatto, S., 2018, February. Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks. In 2018 Information Theory and Applications Workshop (ITA) (pp. 1-10). IEEE
2018
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P. Chaudhari, A. Oberman, S. Osher, S. Soatto and G. Carlier, Deep relaxation: partial differential equations for optimizing deep neural networks, Research in the Mathematical Sciences
2018
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Cited alongside, same era.
J. P. Agnelli, M. Cadeiras, E. G. Tabak, C. V. Turner and E. Vanden-Eijnden, Clustering and classification through normalizing flows in feature space, SIAM J. Multiscale Model. Simul., 8
2010
Cited alongside, same era.
Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, Score-based generative modeling through stochastic differential equations, ArXiv e-prints, arXiv: 2011.13456
2011
Cited alongside, same era.
A survey of the Schrödinger problem and some of its connections with optimal transport
Christian Léonard · 2014
Cited alongside, same era.
C. Baldassi, A. Ingrosso, C. Lucibello, L. Saglietti, and R. Zecchina, Subdominant Dense Clusters Allow for Simple Learning and High Computational Performance in Neural Networks with Discrete Synapses, Phys. Rev. Lett
2015
Cited alongside, same era.
Y. Chen, T. Georgiou and M. Pavon, Optimal steering of inertial particles diffusing anisotropically with losses, Proc. Amer. Control Conf
2015
Cited alongside, same era.
D. Rezendeand S. Mohamed, Variational inference with normalizing flows, in International Conference on Machine Learning
2015
Cited alongside, same era.
J. Ho, X. Chen, A. Srinivas, Y. Duan, and P. Abbeel, Flow++: Improving flow-based generative models with variational dequantization and architecture design, in International Conference on Machine Learning
2019
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G. Peyré and M. Cuturi, Computational Optimal Transport, Foundations and Trends in Machine Learning
2019
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J. Ho, A. Jain and P. Abbeel, Denoising diffusion probabilistic models, Advances in Neural Information Processing Systems
2020
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H. Wu, J. Köhler, and F. Noe, Stochastic normalizing flows, in Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M. F., and Lin, H. (eds.), Advances in Neural Information Processing Systems
2020
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Y. Chen, T.T. Georgiou and M. Pavon, Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schrödinger bridge, SIAM Review
2021
Later among the works it cites.
Y. Chen, T.T. Georgiou and M. Pavon, Optimal Transport in Systems and Control, Annual Review of Control, Robotics, and Autonomous Systems
2021
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M. Pavon, E. G. Tabak and G. Trigila, The data-driven Schrödinger bridge, ArXiv e-prints, arXiv: 1806.01364, Comm. Pure Appl. Math
2021
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Q. Zhang and Y. Chen, Diffusion Normalizing Flow, ArXiv e-prints, arXiv: 2110.07579v1, 35th Conference on Neural Information Processing Systems (NeurIPS 2021), Sydney, Australia
2021
Later among the works it cites.