6,575 research outputs found
Diffusion Maps Kalman Filter for a Class of Systems with Gradient Flows
In this paper, we propose a non-parametric method for state estimation of
high-dimensional nonlinear stochastic dynamical systems, which evolve according
to gradient flows with isotropic diffusion. We combine diffusion maps, a
manifold learning technique, with a linear Kalman filter and with concepts from
Koopman operator theory. More concretely, using diffusion maps, we construct
data-driven virtual state coordinates, which linearize the system model. Based
on these coordinates, we devise a data-driven framework for state estimation
using the Kalman filter. We demonstrate the strengths of our method with
respect to both parametric and non-parametric algorithms in three tracking
problems. In particular, applying the approach to actual recordings of
hippocampal neural activity in rodents directly yields a representation of the
position of the animals. We show that the proposed method outperforms competing
non-parametric algorithms in the examined stochastic problem formulations.
Additionally, we obtain results comparable to classical parametric algorithms,
which, in contrast to our method, are equipped with model knowledge.Comment: 15 pages, 12 figures, submitted to IEEE TS
Robust Independent Component Analysis via Minimum Divergence Estimation
Independent component analysis (ICA) has been shown to be useful in many
applications. However, most ICA methods are sensitive to data contamination and
outliers. In this article we introduce a general minimum U-divergence framework
for ICA, which covers some standard ICA methods as special cases. Within the
U-family we further focus on the gamma-divergence due to its desirable property
of super robustness, which gives the proposed method gamma-ICA. Statistical
properties and technical conditions for the consistency of gamma-ICA are
rigorously studied. In the limiting case, it leads to a necessary and
sufficient condition for the consistency of MLE-ICA. This necessary and
sufficient condition is weaker than the condition known in the literature.
Since the parameter of interest in ICA is an orthogonal matrix, a geometrical
algorithm based on gradient flows on special orthogonal group is introduced to
implement gamma-ICA. Furthermore, a data-driven selection for the gamma value,
which is critical to the achievement of gamma-ICA, is developed. The
performance, especially the robustness, of gamma-ICA in comparison with
standard ICA methods is demonstrated through experimental studies using
simulated data and image data.Comment: 7 figure
Deep Learning in a Generalized HJM-type Framework Through Arbitrage-Free Regularization
We introduce a regularization approach to arbitrage-free factor-model
selection. The considered model selection problem seeks to learn the closest
arbitrage-free HJM-type model to any prespecified factor-model. An asymptotic
solution to this, a priori computationally intractable, problem is represented
as the limit of a 1-parameter family of optimizers to computationally tractable
model selection tasks. Each of these simplified model-selection tasks seeks to
learn the most similar model, to the prescribed factor-model, subject to a
penalty detecting when the reference measure is a local martingale-measure for
the entire underlying financial market. A simple expression for the penalty
terms is obtained in the bond market withing the affine-term structure setting,
and it is used to formulate a deep-learning approach to arbitrage-free affine
term-structure modelling. Numerical implementations are also performed to
evaluate the performance in the bond market.Comment: 23 Pages + Reference
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