4,903 research outputs found
A weak characterization of slow variables in stochastic dynamical systems
We present a novel characterization of slow variables for continuous Markov
processes that provably preserve the slow timescales. These slow variables are
known as reaction coordinates in molecular dynamical applications, where they
play a key role in system analysis and coarse graining. The defining
characteristics of these slow variables is that they parametrize a so-called
transition manifold, a low-dimensional manifold in a certain density function
space that emerges with progressive equilibration of the system's fast
variables. The existence of said manifold was previously predicted for certain
classes of metastable and slow-fast systems. However, in the original work, the
existence of the manifold hinges on the pointwise convergence of the system's
transition density functions towards it. We show in this work that a
convergence in average with respect to the system's stationary measure is
sufficient to yield reaction coordinates with the same key qualities. This
allows one to accurately predict the timescale preservation in systems where
the old theory is not applicable or would give overly pessimistic results.
Moreover, the new characterization is still constructive, in that it allows for
the algorithmic identification of a good slow variable. The improved
characterization, the error prediction and the variable construction are
demonstrated by a small metastable system
Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
Transfer operators such as the Perron--Frobenius or Koopman operator play an
important role in the global analysis of complex dynamical systems. The
eigenfunctions of these operators can be used to detect metastable sets, to
project the dynamics onto the dominant slow processes, or to separate
superimposed signals. We extend transfer operator theory to reproducing kernel
Hilbert spaces and show that these operators are related to Hilbert space
representations of conditional distributions, known as conditional mean
embeddings in the machine learning community. Moreover, numerical methods to
compute empirical estimates of these embeddings are akin to data-driven methods
for the approximation of transfer operators such as extended dynamic mode
decomposition and its variants. One main benefit of the presented kernel-based
approaches is that these methods can be applied to any domain where a
similarity measure given by a kernel is available. We illustrate the results
with the aid of guiding examples and highlight potential applications in
molecular dynamics as well as video and text data analysis
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