3,737 research outputs found
Periodic Splines and Gaussian Processes for the Resolution of Linear Inverse Problems
This paper deals with the resolution of inverse problems in a periodic
setting or, in other terms, the reconstruction of periodic continuous-domain
signals from their noisy measurements. We focus on two reconstruction
paradigms: variational and statistical. In the variational approach, the
reconstructed signal is solution to an optimization problem that establishes a
tradeoff between fidelity to the data and smoothness conditions via a quadratic
regularization associated to a linear operator. In the statistical approach,
the signal is modeled as a stationary random process defined from a Gaussian
white noise and a whitening operator; one then looks for the optimal estimator
in the mean-square sense. We give a generic form of the reconstructed signals
for both approaches, allowing for a rigorous comparison of the two.We fully
characterize the conditions under which the two formulations yield the same
solution, which is a periodic spline in the case of sampling measurements. We
also show that this equivalence between the two approaches remains valid on
simulations for a broad class of problems. This extends the practical range of
applicability of the variational method
Functional Regression
Functional data analysis (FDA) involves the analysis of data whose ideal
units of observation are functions defined on some continuous domain, and the
observed data consist of a sample of functions taken from some population,
sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the
development of this field, which has accelerated in the past 10 years to become
one of the fastest growing areas of statistics, fueled by the growing number of
applications yielding this type of data. One unique characteristic of FDA is
the need to combine information both across and within functions, which Ramsay
and Silverman called replication and regularization, respectively. This article
will focus on functional regression, the area of FDA that has received the most
attention in applications and methodological development. First will be an
introduction to basis functions, key building blocks for regularization in
functional regression methods, followed by an overview of functional regression
methods, split into three types: [1] functional predictor regression
(scalar-on-function), [2] functional response regression (function-on-scalar)
and [3] function-on-function regression. For each, the role of replication and
regularization will be discussed and the methodological development described
in a roughly chronological manner, at times deviating from the historical
timeline to group together similar methods. The primary focus is on modeling
and methodology, highlighting the modeling structures that have been developed
and the various regularization approaches employed. At the end is a brief
discussion describing potential areas of future development in this field
Modelling Periodic Measurement Data Having a Piecewise Polynomial Trend Using the Method of Variable Projection
This paper presents a new method for modelling periodic signals having an
aperiodic trend, using the method of variable projection. It is a major
extension to the IEEE-standard 1057 by permitting the background to be time
varying; additionally, any number of harmonics of the periodic portion can be
modelled. This paper focuses on using B-Splines to implement a piecewise
polynomial model for the aperiodic portion of the signal. A thorough algebraic
derivation of the method is presented, as well as a comparison to using global
polynomial approximation. It is proven that B-Splines work better for modelling
a more complicated aperiodic portion when compared to higher order polynomials.
Furthermore, the piecewise polynomial model is capable of modelling the local
signal variations produced by the interaction of a control system with a
process in industrial applications. The method of variable projection reduces
the problem to a one-dimensional nonlinear optimization, combined with a linear
least-squares computation. An added benefit of using the method of variable
projection is the possibility to calculate the covariances of the linear
coefficients of the model, enabling the calculation of confidence and
prediction intervals. The method is tested on both real measurement data
acquired in industrial processes, as well as synthetic data. The method shows
promising results for the precise characterization of periodic signals embedded
in highly complex aperiodic backgrounds. Finally, snippets of the m-code are
provided, together with a toolbox for B-Splines, which permit the
implementation of the complete computation
Actually Sparse Variational Gaussian Processes
Gaussian processes (GPs) are typically criticised for their unfavourable scaling in both computational and memory requirements. For large datasets, sparse GPs reduce these demands by conditioning on a small set of inducing variables designed to summarise the data. In practice however, for large datasets requiring many inducing variables, such as low-lengthscale spatial data, even sparse GPs can become computationally expensive, limited by the number of inducing variables one can use. In this work, we propose a new class of inter-domain variational GP, constructed by projecting a GP onto a set of compactly supported B-spline basis functions. The key benefit of our approach is that the compact support of the B-spline basis functions admits the use of sparse linear algebra to significantly speed up matrix operations and drastically reduce the memory footprint. This allows us to very efficiently model fast-varying spatial phenomena with tens of thousands of inducing variables, where previous approaches failed
Actually Sparse Variational Gaussian Processes
Gaussian processes (GPs) are typically criticised for their unfavourable
scaling in both computational and memory requirements. For large datasets,
sparse GPs reduce these demands by conditioning on a small set of inducing
variables designed to summarise the data. In practice however, for large
datasets requiring many inducing variables, such as low-lengthscale spatial
data, even sparse GPs can become computationally expensive, limited by the
number of inducing variables one can use. In this work, we propose a new class
of inter-domain variational GP, constructed by projecting a GP onto a set of
compactly supported B-spline basis functions. The key benefit of our approach
is that the compact support of the B-spline basis functions admits the use of
sparse linear algebra to significantly speed up matrix operations and
drastically reduce the memory footprint. This allows us to very efficiently
model fast-varying spatial phenomena with tens of thousands of inducing
variables, where previous approaches failed.Comment: 14 pages, 5 figures, published in AISTATS 202
Poisson inverse problems
In this paper we focus on nonparametric estimators in inverse problems for
Poisson processes involving the use of wavelet decompositions. Adopting an
adaptive wavelet Galerkin discretization, we find that our method combines the
well-known theoretical advantages of wavelet--vaguelette decompositions for
inverse problems in terms of optimally adapting to the unknown smoothness of
the solution, together with the remarkably simple closed-form expressions of
Galerkin inversion methods. Adapting the results of Barron and Sheu [Ann.
Statist. 19 (1991) 1347--1369] to the context of log-intensity functions
approximated by wavelet series with the use of the Kullback--Leibler distance
between two point processes, we also present an asymptotic analysis of
convergence rates that justifies our approach. In order to shed some light on
the theoretical results obtained and to examine the accuracy of our estimates
in finite samples, we illustrate our method by the analysis of some simulated
examples.Comment: Published at http://dx.doi.org/10.1214/009053606000000687 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Self-similar prior and wavelet bases for hidden incompressible turbulent motion
This work is concerned with the ill-posed inverse problem of estimating
turbulent flows from the observation of an image sequence. From a Bayesian
perspective, a divergence-free isotropic fractional Brownian motion (fBm) is
chosen as a prior model for instantaneous turbulent velocity fields. This
self-similar prior characterizes accurately second-order statistics of velocity
fields in incompressible isotropic turbulence. Nevertheless, the associated
maximum a posteriori involves a fractional Laplacian operator which is delicate
to implement in practice. To deal with this issue, we propose to decompose the
divergent-free fBm on well-chosen wavelet bases. As a first alternative, we
propose to design wavelets as whitening filters. We show that these filters are
fractional Laplacian wavelets composed with the Leray projector. As a second
alternative, we use a divergence-free wavelet basis, which takes implicitly
into account the incompressibility constraint arising from physics. Although
the latter decomposition involves correlated wavelet coefficients, we are able
to handle this dependence in practice. Based on these two wavelet
decompositions, we finally provide effective and efficient algorithms to
approach the maximum a posteriori. An intensive numerical evaluation proves the
relevance of the proposed wavelet-based self-similar priors.Comment: SIAM Journal on Imaging Sciences, 201
On the Uniqueness of Inverse Problems with Fourier-domain Measurements and Generalized TV Regularization
We study the super-resolution problem of recovering a periodic
continuous-domain function from its low-frequency information. This means that
we only have access to possibly corrupted versions of its Fourier samples up to
a maximum cut-off frequency. The reconstruction task is specified as an
optimization problem with generalized total-variation regularization involving
a pseudo-differential operator. Our special emphasis is on the uniqueness of
solutions. We show that, for elliptic regularization operators (e.g., the
derivatives of any order), uniqueness is always guaranteed. To achieve this
goal, we provide a new analysis of constrained optimization problems over Radon
measures. We demonstrate that either the solutions are always made of Radon
measures of constant sign, or the solution is unique. Doing so, we identify a
general sufficient condition for the uniqueness of the solution of a
constrained optimization problem with TV-regularization, expressed in terms of
the Fourier samples.Comment: 20 page
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