Band 36
Composite NUV Priors and Applications
Aus der Reihe
Series in Signal and Information Processing
-
- Englisch ausgewählt
64,00 €
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Beschreibung
Produktdetails
Einband
Paperback
Erscheinungsdatum
19.08.2022
Herausgeber
Hans-Andrea LoeligerVerlag
Hartung-GorreSeitenzahl
248
Maße (L/B/H)
21/14,8/1,8 cm
Gewicht
401 g
Auflage
2022
Sprache
Englisch
ISBN
978-3-86628-768-6
Normal with unknown variance (NUV) priors are a central idea of sparse
Bayesian learning and allow variational representations of non-Gaussian
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
sparsity-promoting 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., discrete-level constraints). Such
constraints cannot adequately be represented by linear-Gaussian models
and generally require special treatment. To handle such constraints
within a linear-Gaussian 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 M-ary: {0, 1, . . . ,M−1}). Furthermore, we
derive composite NUV representations of linear inequality constraints,
which enforce a model variable to be lower-bounded, upper-bounded,
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. Kalman-type 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 Bryson-Frazier (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 linear-Gaussian model with unknown parameters. We propose an
efficient iterative algorithm based on Gaussian message passing with
closed-form 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 digital-to-analog
conversion, discrete-phase 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.
Bayesian learning and allow variational representations of non-Gaussian
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
sparsity-promoting 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., discrete-level constraints). Such
constraints cannot adequately be represented by linear-Gaussian models
and generally require special treatment. To handle such constraints
within a linear-Gaussian 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 M-ary: {0, 1, . . . ,M−1}). Furthermore, we
derive composite NUV representations of linear inequality constraints,
which enforce a model variable to be lower-bounded, upper-bounded,
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. Kalman-type 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 Bryson-Frazier (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 linear-Gaussian model with unknown parameters. We propose an
efficient iterative algorithm based on Gaussian message passing with
closed-form 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 digital-to-analog
conversion, discrete-phase 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.
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