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S

Series in Signal and Information Processing, Vol. 30
edited by HansAndrea Loeliger
Nour Zalmaï
A
State Space World for Detecting and
Estimating
Events and Learning
Sparse
Signal
Decompositions
1. Auflage/1^{st} edition 2017. 260 Seiten/pages,
€
64,00. ISBN 9783866285941
Many signals can be labeled
with a small set of events such that each event is categorized according to its
surrounding signal shapes. In this thesis, we provide a general approach based
on linear state space models to learn sparse signal decompositions from
singlechannel and multi channel discretetime measurements. The proposed
approach provides a sparse multichannel representation of a given signal,
which can be interpreted as a signal labeling. This thesis is organized in
three parts.
In the first part, several
important properties of linear state space models (LSSMs) are revisited.
Especially, signals generated with an autonomous LSSM are thoroughly
investigated and fully characterized. In particular, we show that the set of
such signals forms a ring and that the correlation function between any
autonomous LSSM signal and any discretetime signal can be recursively and efficiently
computed. These two properties along with the vast modeling capabilities of
LSSM signals are at the heart of this thesis.
In the second part, we
develop a general approach to detect events in (singlechannel or
multichannel) discretetime signals and estimate the parameters of such
events. Since the number of events is assumed to be substantially smaller than
the number of samples, the set of detected events is interpreted as a sparse
representation of the given signal. An event locally creates characteristic
signals which are modeled with a twosided autonomous LSSM; the rightsided
model accounts for the signals observed after that event while the leftsided
model accounts for the signals before that event. Thus, the problem of event
detection and estimation is substituted by fitting at any given time a LSSM signal
to observations. For this purpose, new cost functions are defined: a
LSSMweighted squared error cost and a LSSMweighted polynomial cost. These
cost functions have the attractive property of being recursively computed. In
addition, closedform solutions for several minimization problems are
available. As far as event detection is concerned, several hypothesis tests
with a suitable notion of local likelihood are promoted. Surprisingly, event
detection in various conditions, such as in the presence of an unknown additive
or multiplicative interference signal, can be naturally dealt with. Finally,
various important practical applications are addressed in detail in order to
exemplify the potential of the proposed approach for event detection and
estimation.
In the third and last part,
we propose a general approach to learn sparse signal decompositions. We assume
that each signal component can be sparsely represented in the input domain of
some unknown LSSM. We model sparse inputs with zeromean Gaussian random
variables with unknown variances, as in the sparse Bayesian learning framework.
Then, all unknown parameters are estimated by maximum likelihood with an
expectation maximization (EM) algorithm where all parameters are jointly
updated with closedform expressions and all expectation quantities are
efficiently computed with a Gaussian message passing algorithm. This general
approach can deal with a large variety of sparse signal decomposition problems.
Among them, we address the problems of learning repetitive signal shapes,
learning classes of signal shapes, and decomposing a
signal with scaled, timeshifted, and timedilated versions of a signal shape.
All these concepts and methods are illustrated with practical examples.
Keywords: Linear state space models; event detection and
estimation; learning sparse signal decompositions; sparse Bayesian learning;
unsupervised feature extraction.
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