5,258 research outputs found
Modeling and interpolation of the ambient magnetic field by Gaussian processes
Anomalies in the ambient magnetic field can be used as features in indoor
positioning and navigation. By using Maxwell's equations, we derive and present
a Bayesian non-parametric probabilistic modeling approach for interpolation and
extrapolation of the magnetic field. We model the magnetic field components
jointly by imposing a Gaussian process (GP) prior on the latent scalar
potential of the magnetic field. By rewriting the GP model in terms of a
Hilbert space representation, we circumvent the computational pitfalls
associated with GP modeling and provide a computationally efficient and
physically justified modeling tool for the ambient magnetic field. The model
allows for sequential updating of the estimate and time-dependent changes in
the magnetic field. The model is shown to work well in practice in different
applications: we demonstrate mapping of the magnetic field both with an
inexpensive Raspberry Pi powered robot and on foot using a standard smartphone.Comment: 17 pages, 12 figures, to appear in IEEE Transactions on Robotic
Variational approach for learning Markov processes from time series data
Inference, prediction and control of complex dynamical systems from time
series is important in many areas, including financial markets, power grid
management, climate and weather modeling, or molecular dynamics. The analysis
of such highly nonlinear dynamical systems is facilitated by the fact that we
can often find a (generally nonlinear) transformation of the system coordinates
to features in which the dynamics can be excellently approximated by a linear
Markovian model. Moreover, the large number of system variables often change
collectively on large time- and length-scales, facilitating a low-dimensional
analysis in feature space. In this paper, we introduce a variational approach
for Markov processes (VAMP) that allows us to find optimal feature mappings and
optimal Markovian models of the dynamics from given time series data. The key
insight is that the best linear model can be obtained from the top singular
components of the Koopman operator. This leads to the definition of a family of
score functions called VAMP-r which can be calculated from data, and can be
employed to optimize a Markovian model. In addition, based on the relationship
between the variational scores and approximation errors of Koopman operators,
we propose a new VAMP-E score, which can be applied to cross-validation for
hyper-parameter optimization and model selection in VAMP. VAMP is valid for
both reversible and nonreversible processes and for stationary and
non-stationary processes or realizations
High-performance Kernel Machines with Implicit Distributed Optimization and Randomization
In order to fully utilize "big data", it is often required to use "big
models". Such models tend to grow with the complexity and size of the training
data, and do not make strong parametric assumptions upfront on the nature of
the underlying statistical dependencies. Kernel methods fit this need well, as
they constitute a versatile and principled statistical methodology for solving
a wide range of non-parametric modelling problems. However, their high
computational costs (in storage and time) pose a significant barrier to their
widespread adoption in big data applications.
We propose an algorithmic framework and high-performance implementation for
massive-scale training of kernel-based statistical models, based on combining
two key technical ingredients: (i) distributed general purpose convex
optimization, and (ii) the use of randomization to improve the scalability of
kernel methods. Our approach is based on a block-splitting variant of the
Alternating Directions Method of Multipliers, carefully reconfigured to handle
very large random feature matrices, while exploiting hybrid parallelism
typically found in modern clusters of multicore machines. Our implementation
supports a variety of statistical learning tasks by enabling several loss
functions, regularization schemes, kernels, and layers of randomized
approximations for both dense and sparse datasets, in a highly extensible
framework. We evaluate the ability of our framework to learn models on data
from applications, and provide a comparison against existing sequential and
parallel libraries.Comment: Work presented at MMDS 2014 (June 2014) and JSM 201
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