Identifying drift, diffusion, and causal structure from temporal snapshots
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Rahmani, Hossein
issued: 2025-10-17T21:18:02Z
issued: 2025
related degree: Master of Science - MSc
degree grantor: University of British Columbia
description: Stochastic differential equations (SDEs) are a fundamental tool for modelling dynamic processes, including gene regulatory networks (GRNs), contaminant transport, financial markets, and image generation. However, learning the underlying SDE from data is a challenging task, especially if individual trajectories are not observable. Motivated by research in single-cell datasets, we present the first comprehensive approach for jointly identifying the drift and diffusion of an SDE from its temporal marginals. First, a detailed explanation of the trajectory inference problem is given, and its connection to entropic optimal transport formulated as a convex variational problem is established. Assuming linear drift and additive diffusion, it is then proved that these parameters are identifiable from marginals if and only if the initial distribution lacks any generalized rotational symmetries. Further, it is shown that the causal graph of any SDE with additive diffusion can be recovered from the SDE parameters, thereby linking the problem to causal discovery. To complement this theory, entropy-regularized optimal transport is adapted to handle anisotropic diffusion, and APPEX (Alternating Projection Parameter Estimation from X0) is introduced as an iterative algorithm designed to estimate the drift, diffusion, and trajectories solely from temporal marginals. It is demonstrated that APPEX iteratively decreases KullbackâLeibler divergence to the true solution, and its effectiveness is illustrated on simulated data from linear additive noise SDEs.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/92657
note: Identifying drift, diffusion, and causalstructure from temporal snapshotsbyHossein RahmaniB.Sc., Sharif University of Technology, 2023A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2025© Hossein Rahmani 2025The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Identifying drift, diffusion, and causal structure from temporalsnapshotssubmitted by Hossein Rahmani in partial fulfillment of the requirementsfor the degree of Master of Science in Mathematics.Examining Committee:Elina Robeva, Associate Professor, Department of Mathematics, UBCSupervisorGeoffrey Schiebinger, Associate professor, Department of Mathematics, UBCSupervisory Committee MemberiiAbstractStochastic differential equations (SDEs) are a fundamental tool for modellingdynamic processes, including gene regulatory networks (GRNs), contami-nant transport, financial markets, and image generation. However, learningthe underlying SDE from data is a challenging task, especially if individualtrajectories are not observable. Motivated by research in single-cell datasets,we present the first comprehensive approach for jointly identifying the driftand diffusion of an SDE from its temporal marginals. First, a detailed ex-planation of the trajectory inference problem is given, and its connectionto entropic optimal transport formulated as a convex variational problem isestablished. Assuming linear drift and additive diffusion, it is then provedthat these parameters are identifiable from marginals if and only if the ini-tial distribution lacks any generalized rotational symmetries. Further, itis shown that the causal graph of any SDE with additive diffusion can berecovered from the SDE parameters, thereby linking the problem to causaldiscovery. To complement …
has type: Thesis/Dissertation
date issued: 2025-11
is shown at: 10.14288/1.0450482
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Robeva, Elina
type: Text
identifier URI: http://hdl.handle.net/2429/92657