47,048 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
Intrinsic Gaussian processes on complex constrained domains
We propose a class of intrinsic Gaussian processes (in-GPs) for
interpolation, regression and classification on manifolds with a primary focus
on complex constrained domains or irregular shaped spaces arising as subsets or
submanifolds of R, R2, R3 and beyond. For example, in-GPs can accommodate
spatial domains arising as complex subsets of Euclidean space. in-GPs respect
the potentially complex boundary or interior conditions as well as the
intrinsic geometry of the spaces. The key novelty of the proposed approach is
to utilise the relationship between heat kernels and the transition density of
Brownian motion on manifolds for constructing and approximating valid and
computationally feasible covariance kernels. This enables in-GPs to be
practically applied in great generality, while existing approaches for
smoothing on constrained domains are limited to simple special cases. The broad
utilities of the in-GP approach is illustrated through simulation studies and
data examples
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