2021

Simulating Diffusion Bridges with Score Matching

Heng, Jeremy, De Bortoli, Valentin, Doucet, Arnaud et al.

Understand

We consider the problem of simulating diffusion bridges, which are diffusion processes that are conditioned to initialize and terminate at two given states.

The simulation of diffusion bridges has applications in diverse scientific fields and plays a crucial role in the statistical inference of discretely-observed diffusions.
This is known to be a challenging problem that has received much attention in the last two decades.
This article contributes to this rich body of literature by presenting a new avenue to obtain diffusion bridge approximations.

Built on

Time reversal of diffusions
Haussmann, U. G. and E. Pardoux (1986) · 1986
Earlier work this paper cites.
Integration by parts and time reversal for diffusion processes
Millet, A., D. Nualart, and M. Sanz (1989) · 1989
Earlier work this paper cites.
The simulation of pinned diffusions
Clark, J. M. C. (1990) · 1990
Earlier work this paper cites.
Stochastic differential equations
Kloeden, P. E. and E. Platen (1992) · 1992
Earlier work this paper cites.
Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes
Pedersen, A. R. (1995) · 1995
Earlier work this paper cites.
Multidimensional Diffusion Processes
Stroock, D. W. and S. S. Varadhan (1997) · 1997
Earlier work this paper cites.
Nonparametric estimation of state-price densities implicit in financial asset prices
Aït-Sahalia, Y. and A. W. Lo (1998) · 1998
Earlier work this paper cites.
Diffusions, Markov processes and Martingales: Volume 2: Itô Calculus
Rogers, L. C. G. and D. Williams (2000) · 2000
Earlier work this paper cites.
Likelihood inference for discretely observed nonlinear diffusions
Elerian, O., S. Chib, and N. Shephard (2001) · 2001
Earlier work this paper cites.
MCMC analysis of diffusion models with application to finance
Eraker, B. (2001) · 2001
Earlier work this paper cites.
On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm
Roberts, G. O. and O. Stramer (2001) · 2001
Earlier work this paper cites.
Transition path sampling: Throwing ropes over rough mountain passes, in the dark
Bolhuis, P. G., D. Chandler, C. Dellago, and P. L. Geissler (2002) · 2002
Earlier work this paper cites.
Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes
Durham, G. B. and A. R. Gallant (2002) · 2002
Earlier work this paper cites.
Conditional path sampling of SDEs and the Langevin MCMC method
Stuart, A. M., J. Voss, and P. Wilberg (2004) · 2004
Earlier work this paper cites.
Gradient Flows: in Metric Spaces and in the Space of Probability Measures
Ambrosio, L., N. Gigli, and G. Savaré (2005) · 2005
Earlier work this paper cites.
Exact simulation of diffusions
Beskos, A. and G. O. Roberts (2005) · 2005
Earlier work this paper cites.
Estimation of non-normalized statistical models by score matching
Hyvärinen, A. (2005) · 2005
Earlier work this paper cites.
Retrospective exact simulation of diffusion sample paths with applications
Beskos, A., O. Papaspiliopoulos, and G. O. Roberts (2006) · 2006
Earlier work this paper cites.
Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion)
Beskos, A., O. Papaspiliopoulos, G. O. Roberts, and P. Fearnhead (2006) · 2006
Earlier work this paper cites.

Similar

Simulation of conditioned diffusion and application to parameter estimation
Delyon, B. and Y. Hu (2006) · 2006
Cited alongside, same era.
On simulated likelihood of discretely observed diffusion processes and comparison to closed-form approximation
Stramer, O. and J. Yan (2007) · 2007
Cited alongside, same era.
MCMC methods for diffusion bridges
Beskos, A., G. O. Roberts, A. Stuart, and J. Voss (2008) · 2008
Cited alongside, same era.
Bayesian inference for nonlinear multivariate diffusion models observed with error
Golightly, A. and D. J. Wilkinson (2008) · 2008
Cited alongside, same era.
Particle Markov chain Monte Carlo methods (with discussion)
Andrieu, C., A. Doucet, and R. Holenstein (2010) · 2010
Cited alongside, same era.
On generating Monte Carlo samples of continuous diffusion bridges
Lin, M., R. Chen, and P. Mykland (2010) · 2010
Cited alongside, same era.
A connection between score matching and denoising autoencoders
Vincent, P. (2011) · 2011
Cited alongside, same era.
Quantifying the Waddington landscape and biological paths for development and differentiation
Wang, J., K. Zhang, L. Xu, and E. Wang (2011) · 2011
Cited alongside, same era.
Importance sampling techniques for estimation of diffusion models
Papaspiliopoulos, O. and G. O. Roberts (2012) · 2012
Cited alongside, same era.
Data augmentation for diffusions
Papaspiliopoulos, O., G. O. Roberts, and O. Stramer (2013) · 2013
Cited alongside, same era.
Simple simulation of diffusion bridges with application to likelihood inference for diffusions
Bladt, M. and M. Sørensen (2014) · 2014
Cited alongside, same era.
Adam: A method for stochastic optimization
Original
Kingma, D. P. and J. Ba (2014) · 2014
Cited alongside, same era.

Then

Attention is all you need
Vaswani, A., N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin (2017) · 2017
Later among the works it cites.
Improved bridge constructs for stochastic differential equations
Whitaker, G. A., A. Golightly, R. J. Boys, and C. Sherlock (2017) · 2017
Later among the works it cites.
Unbiased smoothing using particle independent Metropolis–Hastings
Middleton, L., G. Deligiannidis, A. Doucet, and P. E. Jacob (2019) · 2019
Later among the works it cites.
Exact sampling of polymer conformations using Brownian bridges
Wang, S., D. Ramkrishna, and V. Narsimhan (2020) · 2020
Later among the works it cites.
A piecewise deterministic Monte Carlo method for diffusion bridges
Bierkens, J., S. Grazzi, F. van der Meulen, and M. Schauer (2021) · 2021
Closest in time.
Diffusion Schrödinger bridge with applications to score-based generative modeling
De Bortoli, V., J. Thornton, J. Heng, and A. Doucet (2021) · 2021
Closest in time.
How to train your energy-based models
Original
Song, Y. and D. P. Kingma (2021) · 2021
Closest in time.
Score-based generative modeling through stochastic differential equations
Song, Y., J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole (2021) · 2021
Closest in time.
Solving Schrödinger bridges via maximum likelihood
Vargas, F., P. Thodoroff, A. Lamacraft, and N. Lawrence (2021) · 2021
Closest in time.
Diffusion bridges for stochastic Hamiltonian systems and shape evolutions
Arnaudon, A., F. van der Meulen, M. Schauer, and S. Sommer (2022) · 2022
Closest in time.
Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions
Chen, S., S. Chewi, J. Li, Y. Li, A. Salim, and A. Zhang (2022) · 2022
Closest in time.
Likelihood training of Schrödinger bridge using forward-backward SDEs theory
Chen, T., G.-H. Liu, and E. Theodorou (2022) · 2022
Closest in time.
Manifold Markov chain Monte Carlo methods for Bayesian inference in diffusion models
Graham, M. M., A. H. Thiery, and A. Beskos (2022) · 2022
Closest in time.
Time reversal of diffusion processes under a finite entropy condition
Cattiaux, P., G. Conforti, I. Gentil, and C. Léonard (2023) · 2023
Closest in time.
Score matching for bridges without time-reversals
Original
Baker, E. L., M. Schauer, and S. Sommer (2024) · 2024
Closest in time.
Conditioning non-linear and infinite-dimensional diffusion processes
Baker, E. L., G. Yang, M. Severinsen, C. Hipsley, and S. Sommer (2024) · 2024
Closest in time.
Nearly d d -linear convergence bounds for diffusion models via stochastic localization
Benton, J., V. De Bortoli, A. Doucet, and G. Deligiannidis (2024) · 2024
Closest in time.
Parameter inference via differentiable diffusion bridge importance sampling
Original
Boserup, N., G. Yang, M. L. Severinsen, C. A. Hipsley, and S. Sommer (2024) · 2024
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…