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S

Series in Signal and Information Processing, Vol. 36
edited by HansAndrea Loeliger
Raphael Urs Keusch
Composite NUV Priors
and Applications
1^{st }Edition 2022. XXVI, 248 pages. € 64,00.
ISBN 9783866287686
Abstract
Normal
with unknown variance (NUV) priors are a central idea of sparse
Bayesian
learning and allow variational representations of
nonGaussian
priors. More specifically, such variational representations can be seen
as parameterized Gaussians, wherein
the parameters are generally unknown.
The
advantage is apparent: for fixed parameters, NUV priors are
Gaussian, and hence computationally compatible with
Gaussian models.
Moreover,
working with (linear)Gaussian models is particularly
attractive
since the Gaussian distribution is
closed under affine transformations,
marginalization, and
conditioning. Interestingly, the variational
representation proves to be
rather universal than restrictive: many common
sparsitypromoting priors (among
them, in particular, the Laplace
prior) can be represented in this
manner.
In
estimation problems, parameters or variables of the underlying model
are often subject to constraints
(e.g., discretelevel constraints). Such
constraints cannot
adequately be represented by linearGaussian models
and generally require special
treatment. To handle such constraints
within a linearGaussian setting, we
extend the idea of NUV priors beyond
its original use for sparsity. In
particular, we study compositions
of existing NUV priors, referred to
as composite NUV priors, and show
that many commonly used model
constraints can be represented in this
way.
In
Part I, we derive composite NUV representations of discretizing constraints,
which enforce a model variable to take
on values in a finite
set (e.g., binary: {0, 1}, or Mary: {0, 1, . . . ,M−1}). Furthermore, we
derive composite NUV representations of
linear inequality constraints,
which enforce a model variable to be
lowerbounded, upperbounded,
or both. In addition, we derive a
composite NUV representation of an
exclusion constraint,
which enforces a model variable to stay outside of
an exclusion region.
In
Part II, we review the standard linear state space representation to
model physical systems. Linear state
space models (LSSMs) are defined
only by a few parameters, bring
flexible modeling capabilities, and pave
the way for efficient algorithms thanks
to their linearity and recursive
structure. Kalmantype algorithms are commonly used to perform
inference
in Gaussian LSSMs. We will use a
Gaussian message passing scheme
based on factor graphs which offers
several improvements and can be seen
as a generalization of the standard
Kalman filter/smoother. In particular,
we will apply the modified
BrysonFrazier (MBF) smoother (augmented
with input estimation), which is
numerically stable and avoids
matrix inversions.
The
expressive power of composite NUV priors and their computational
compatibility with Gaussian
models allow us to reformulate a variety of
(constrained) optimization problems as statistical estimation
problems
in a linearGaussian model with
unknown parameters. We propose an
efficient iterative
algorithm based on Gaussian message passing with
closedform update rules
for the unknown parameters. An asset of the
algorithm is the linear
computational complexity in time (per iteration).
Consequently,
the method is able to efficiently handle long time horizons,
which is generally the bottleneck of
other algorithms.
Finally,
in Part III and IV, we demonstrate the applicability of the proposed
method using pertinent problems from
signal processing and constrained
control. We consider
problems ranging from digitaltoanalog
conversion,
discretephase beamforming, trajectory planning, to obstacle
avoidance, power
converter control, and more. The results are promising
and suggest that the proposed method
is a versatile toolbox to handle
various challenging
practical applications.
Keywords: Normal with unknown variance
(NUV); sparse Bayesian learning;
composite NUV priors; Gaussian message passing;
iteratively reweighted least squares (IRLS); constrained optimization; control
as inference.
About the author:
Raphael
Keusch was born in Muri (AG), Switzerland, in 1989
and grew
up in Buttwil
(AG), Switzerland. He received his diploma as an electronics
technician from Roche
Diagnostics Ltd., Rotkreuz, Switzerland,
in 2009. Subsequently, he enrolled in
the electrical engineering and information
technology program at
ETH Zurich, Switzerland, from which he
received his BSc and
MSc degrees in 2014 and 2016, respectively. During
his master’s degree, he spent a
semester as an exchange student at the
KTH Royal Institute of Technology, Stockholm, Sweden.
After graduation,
he worked as a signal processing
engineer for Sensirion AG, Stäfa,
Switzerland.
Since 2018, he has been a PhD candidate and a full research
assistant at the Signal
and Information Processing Laboratory (ISI) at
ETH Zurich under the supervision of Prof. HansAndrea Loeliger. His
research interests
include statistical signal processing, control, machine
learning and
electronics.
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