531 research outputs found
Data-driven Efficient Solvers and Predictions of Conformational Transitions for Langevin Dynamics on Manifold in High Dimensions
We work on dynamic problems with collected data that
distributed on a manifold . Through the
diffusion map, we first learn the reaction coordinates where is a manifold isometrically embedded into an
Euclidean space for . The reaction coordinates
enable us to obtain an efficient approximation for the dynamics described by a
Fokker-Planck equation on the manifold . By using the reaction
coordinates, we propose an implementable, unconditionally stable, data-driven
upwind scheme which automatically incorporates the manifold structure of
. Furthermore, we provide a weighted convergence analysis of
the upwind scheme to the Fokker-Planck equation. The proposed upwind scheme
leads to a Markov chain with transition probability between the nearest
neighbor points. We can benefit from such property to directly conduct
manifold-related computations such as finding the optimal coarse-grained
network and the minimal energy path that represents chemical reactions or
conformational changes. To establish the Fokker-Planck equation, we need to
acquire information about the equilibrium potential of the physical system on
. Hence, we apply a Gaussian Process regression algorithm to
generate equilibrium potential for a new physical system with new parameters.
Combining with the proposed upwind scheme, we can calculate the trajectory of
the Fokker-Planck equation on based on the generated equilibrium
potential. Finally, we develop an algorithm to pullback the trajectory to the
original high dimensional space as a generative data for the new physical
system.Comment: 59 pages, 16 figure
On boundary detection
Given a sample of a random variable supported by a smooth compact manifold
, we propose a test to decide whether the boundary of
is empty or not with no preliminary support estimation. The test statistic
is based on the maximal distance between a sample point and the average of its
-nearest neighbors. We prove that the level of the test can be estimated,
that, with probability one, its power is one for large enough, and that
there exists a consistent decision rule. Heuristics for choosing a convenient
value for the parameter and identifying observations close to the
boundary are also given. We provide a simulation study of the test
An Efficient and Continuous Voronoi Density Estimator
We introduce a non-parametric density estimator deemed Radial Voronoi Density
Estimator (RVDE). RVDE is grounded in the geometry of Voronoi tessellations and
as such benefits from local geometric adaptiveness and broad convergence
properties. Due to its radial definition RVDE is moreover continuous and
computable in linear time with respect to the dataset size. This amends for the
main shortcomings of previously studied VDEs, which are highly discontinuous
and computationally expensive. We provide a theoretical study of the modes of
RVDE as well as an empirical investigation of its performance on
high-dimensional data. Results show that RVDE outperforms other non-parametric
density estimators, including recently introduced VDEs.Comment: 12 page
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