23,147 research outputs found
Gaussian processes with linear operator inequality constraints
This paper presents an approach for constrained Gaussian Process (GP)
regression where we assume that a set of linear transformations of the process
are bounded. It is motivated by machine learning applications for
high-consequence engineering systems, where this kind of information is often
made available from phenomenological knowledge. We consider a GP over
functions on taking values in
, where the process is still Gaussian when
is a linear operator. Our goal is to model under the
constraint that realizations of are confined to a convex set of
functions. In particular, we require that , given
two functions and where pointwise. This formulation provides a
consistent way of encoding multiple linear constraints, such as
shape-constraints based on e.g. boundedness, monotonicity or convexity. We
adopt the approach of using a sufficiently dense set of virtual observation
locations where the constraint is required to hold, and derive the exact
posterior for a conjugate likelihood. The results needed for stable numerical
implementation are derived, together with an efficient sampling scheme for
estimating the posterior process.Comment: Published in JMLR: http://jmlr.org/papers/volume20/19-065/19-065.pd
Estimation in high dimensions: a geometric perspective
This tutorial provides an exposition of a flexible geometric framework for
high dimensional estimation problems with constraints. The tutorial develops
geometric intuition about high dimensional sets, justifies it with some results
of asymptotic convex geometry, and demonstrates connections between geometric
results and estimation problems. The theory is illustrated with applications to
sparse recovery, matrix completion, quantization, linear and logistic
regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change
Multilevel ensemble Kalman filtering for spatio-temporal processes
We design and analyse the performance of a multilevel ensemble Kalman filter
method (MLEnKF) for filtering settings where the underlying state-space model
is an infinite-dimensional spatio-temporal process. We consider underlying
models that needs to be simulated by numerical methods, with discretization in
both space and time. The multilevel Monte Carlo (MLMC) sampling strategy,
achieving variance reduction through pairwise coupling of ensemble particles on
neighboring resolutions, is used in the sample-moment step of MLEnKF to produce
an efficient hierarchical filtering method for spatio-temporal models. Under
sufficient regularity, MLEnKF is proven to be more efficient for weak
approximations than EnKF, asymptotically in the large-ensemble and
fine-numerical-resolution limit. Numerical examples support our theoretical
findings.Comment: Version 1: 39 pages, 4 figures.arXiv admin note: substantial text
overlap with arXiv:1608.08558 . Version 2 (this version): 52 pages, 6
figures. Revision primarily of the introduction and the numerical examples
sectio
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