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Efficient Pareto Frontier Algorithms for Computing Structured Signal Representations = = Effiziente Pareto-Frontier-Algorithmen fur Berechnung Strukturierter Signaldarstellungen.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Efficient Pareto Frontier Algorithms for Computing Structured Signal Representations =/
Reminder of title:	Effiziente Pareto-Frontier-Algorithmen fur Berechnung Strukturierter Signaldarstellungen.
remainder title:	Effiziente Pareto-Frontier-Algorithmen fur Berechnung Strukturierter Signaldarstellungen.
Author:	Vural, Metin.
Description:	1 online resource (155 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 83-11, Section: B.
Contained By:	Dissertations Abstracts International83-11B.
Subject:	Sparsity. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29071207click for full text (PQDT)
ISBN:	9798426868472

Efficient Pareto Frontier Algorithms for Computing Structured Signal Representations = = Effiziente Pareto-Frontier-Algorithmen fur Berechnung Strukturierter Signaldarstellungen.
Vural, Metin.

Efficient Pareto Frontier Algorithms for Computing Structured Signal Representations =Effiziente Pareto-Frontier-Algorithmen fur Berechnung Strukturierter Signaldarstellungen.Effiziente Pareto-Frontier-Algorithmen fur Berechnung Strukturierter Signaldarstellungen. - 1 online resource (155 pages)

Source: Dissertations Abstracts International, Volume: 83-11, Section: B.

Thesis (Ph.D.)--Technische Universitaet Berlin (Germany), 2022.

Includes bibliographical references

ℓp-norm minimization plays a signifcant role in a variety of disciplines. It is not only important for the signal recovery in compressed sensing but also benefcial for fnding meaningful signal representations as for the sparse and anti-sparse coding related applications. Therefore, minimizing ℓp-norms in an efcient manner sparked interest in a variety of works.This thesis is concerned with the noise-constrained ℓp-norm minimization for 1 ≤ p ≤ ∞. Although there are various optimization problem formulations that may be used to minimize an ℓp-norm, constraining the noise can ofer a more meaningful optimization problem defnition since when there is a known noise tolerance in an application, one can simply canalise it into the optimization problem and formulate exactly what to solve. Thus, it is often easier to set the noise tolerance from the optimization perspective. Despite this, there is a lack of computationally efcient algorithms in the literature for the noise-constrained ℓp-norm minimization problem because its feasible area can be complicated. Diferent optimization problem formulations can provide equivalent solutions and some of them might be easier to solve than the others. Therefore, it might be tempting to solve a computationally efcient problem in order to have the solution to another one. In this thesis, we solved constrained ℓp-norm regularization to reach the solution of the noise-constrained ℓp-norm problem. We introduce optimality tracing based ℓp-norm minimization approaches with simple root fnding iterations for 1 ≤ p ≤ ∞. The optimality trade-of between both objectives, the ℓp-norm and a loss function that measures the data misft, is formulated as a nonlinear equation root fnding problem. We present and employ several simple, derivative-free and cost-efcient nonlinear equation root fnding methods to trace this optimality over a Pareto frontier. Some of these root fnding methods do not require diferentiable loss functions and are applicable for both convex and nonconvex data misfts and extend such problems to a broader class of applications. We also introduce a warm-start strategy of taking linear least-squares solution with the one that has minimum ℓ2-norm which is named method of frames (MOF) as an input to require fewer iterations. This warm-start may provide fexible and meaningful starting point initialization for many applications where MOF already exists and can be improved with a better understanding of fnitedimensional geometry, e.g. n-widths. The impact of the overcomplete matrix on the convergence rate of some of the presented approaches is demonstrated for matrices fulflling the Uniform Uncertainty Principle and Uncertainty Principle. These properties were formerly introduced to analyze the performance of random matrices for ℓ1 and ℓ∞-norm related applications respectively.In the last part of the thesis, i.e. in Chapter 7, ℓp-norm minimization related applications are probed with using several loss functions such as least-squares, Huber and a nonconvex penalty Student's t. ℓ1-norm is minimized with a typical compressed sensing example. Also, a generic test benchmark is utilized for the comparison of the nonlinear equation root fnders for ℓ1-norm minimization. A new communication scheme is introduced by minimizing ℓ∞-norm. Outlier detection problem is studied with the minimized ℓ∞-norm, and a prior is ofered for the minimized ℓ∞-norm with its performance on peak-to-average power ratio (PAPR). Noise-constrained nuclear norm is minimized as well for the Euclidean distance matrix completion problem with the application of wireless sensor network localization.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2023

Mode of access: World Wide Web

ISBN: 9798426868472Subjects--Topical Terms:

3680690
Sparsity.
Index Terms--Genre/Form:

542853
Electronic books.

Efficient Pareto Frontier Algorithms for Computing Structured Signal Representations = = Effiziente Pareto-Frontier-Algorithmen fur Berechnung Strukturierter Signaldarstellungen.
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245 1 0 $a Efficient Pareto Frontier Algorithms for Computing Structured Signal Representations = $b Effiziente Pareto-Frontier-Algorithmen fur Berechnung Strukturierter Signaldarstellungen.
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500 $a Source: Dissertations Abstracts International, Volume: 83-11, Section: B.
500 $a Advisor: Stanczak, SLawomir; Sezgin, Aydin; Aravkin, Aleksandr; Raisch, Jorg.
502 $a Thesis (Ph.D.)--Technische Universitaet Berlin (Germany), 2022.
504 $a Includes bibliographical references
520 $a ℓp-norm minimization plays a signifcant role in a variety of disciplines. It is not only important for the signal recovery in compressed sensing but also benefcial for fnding meaningful signal representations as for the sparse and anti-sparse coding related applications. Therefore, minimizing ℓp-norms in an efcient manner sparked interest in a variety of works.This thesis is concerned with the noise-constrained ℓp-norm minimization for 1 ≤ p ≤ ∞. Although there are various optimization problem formulations that may be used to minimize an ℓp-norm, constraining the noise can ofer a more meaningful optimization problem defnition since when there is a known noise tolerance in an application, one can simply canalise it into the optimization problem and formulate exactly what to solve. Thus, it is often easier to set the noise tolerance from the optimization perspective. Despite this, there is a lack of computationally efcient algorithms in the literature for the noise-constrained ℓp-norm minimization problem because its feasible area can be complicated. Diferent optimization problem formulations can provide equivalent solutions and some of them might be easier to solve than the others. Therefore, it might be tempting to solve a computationally efcient problem in order to have the solution to another one. In this thesis, we solved constrained ℓp-norm regularization to reach the solution of the noise-constrained ℓp-norm problem. We introduce optimality tracing based ℓp-norm minimization approaches with simple root fnding iterations for 1 ≤ p ≤ ∞. The optimality trade-of between both objectives, the ℓp-norm and a loss function that measures the data misft, is formulated as a nonlinear equation root fnding problem. We present and employ several simple, derivative-free and cost-efcient nonlinear equation root fnding methods to trace this optimality over a Pareto frontier. Some of these root fnding methods do not require diferentiable loss functions and are applicable for both convex and nonconvex data misfts and extend such problems to a broader class of applications. We also introduce a warm-start strategy of taking linear least-squares solution with the one that has minimum ℓ2-norm which is named method of frames (MOF) as an input to require fewer iterations. This warm-start may provide fexible and meaningful starting point initialization for many applications where MOF already exists and can be improved with a better understanding of fnitedimensional geometry, e.g. n-widths. The impact of the overcomplete matrix on the convergence rate of some of the presented approaches is demonstrated for matrices fulflling the Uniform Uncertainty Principle and Uncertainty Principle. These properties were formerly introduced to analyze the performance of random matrices for ℓ1 and ℓ∞-norm related applications respectively.In the last part of the thesis, i.e. in Chapter 7, ℓp-norm minimization related applications are probed with using several loss functions such as least-squares, Huber and a nonconvex penalty Student's t. ℓ1-norm is minimized with a typical compressed sensing example. Also, a generic test benchmark is utilized for the comparison of the nonlinear equation root fnders for ℓ1-norm minimization. A new communication scheme is introduced by minimizing ℓ∞-norm. Outlier detection problem is studied with the minimized ℓ∞-norm, and a prior is ofered for the minimized ℓ∞-norm with its performance on peak-to-average power ratio (PAPR). Noise-constrained nuclear norm is minimized as well for the Euclidean distance matrix completion problem with the application of wireless sensor network localization.
520 $a Die Minimierung von ℓp-Normen spielt in einer Vielzahl von Disziplinen eine bedeutende Rolle. Sie ist nicht nur wichtig fur die Signalwiederherstellung bei der komprimierten Abtastung, sondern auch nutzlich, um sinnvolle Signaldarstellungen zu finden, wie fur Anwendungen mit geringer und anti-sparse Codierung. Daher hat die effiziente Minimierung von ℓp-Normen das Interesse an einer Vielzahl von Arbeiten geweckt.Diese Dissertation beschaftigt sich mit der rauschbeschrankten ℓp-Normminimierung fur 1 ≤ p ≤ ∞. Obwohl es verschiedene Formulierungen von Optimierungsproblemen gibt, die verwendet werden konnen, um eine ℓp-Norm zu minimieren, kann die Beschrankung des Rauschens eine sinnvollere Definition des Optimierungsproblems bieten. Bei bekannter Rauschtoleranz in einer Anwendung kann man diese in das Optimierungsproblem kanalisieren/integrieren/einbauen und genau formulieren, was/welches Problem zu losen ist. Daher ist es aus Optimierungssicht oft einfacher, die Rauschtoleranz einzustellen. Trotzdem fehlt es in der Literatur an recheneffizienten Algorithmen fur das rauschbeschrankte ℓp-Norm-Minimierungsproblem, da sein zulassiger Bereich kompliziert sein kann. Verschiedene Formulierungen von Optimierungsproblemen konnen aquivalente Losungen liefern und einige von ihnen sind moglicherweise einfacher zu losen als andere. Daher kann es verlockend sein, ein recheneffizientes Problem zu losen, um die Losung fur ein anderes zu erhalten. In dieser Arbeit wurde die eingeschrankte ℓp-Norm-Regularisierung gelost, um die Losung des rauschbeschrankten ℓp-Norm-Problems zu erreichen. Die auf der Optimalitatsverfolgung basierenden ℓp-Norm-Minimierungsansatze mit einfachen Wurzelfindungsiterationen fur 1 ≤ p ≤ ∞ werden hierzu vorgestellt. Der Optimalitats-Trade-off zwischen beiden Zielen, der ℓp-Norm und einer Verlustfunktion, die die Datenfehlanpassung misst, wird als nichtlineares Gleichungswurzelfindungsproblem formuliert. Mehrere einfache, ableitungsfreie und kosteneffiziente Methoden zum Auffinden von nichtlinearen Gleichungswurzeln werden prasentiert und verwendet, um diese Optimalitat uber eine Pareto-Grenze zu verfolgen. Einige dieser Wurzelfindungsverfahren erfordern keine differenzierbaren Verlustfunktionen und sind sowohl fur konvexe als auch fur nichtkonvexe Datenfehlanpassungen anwendbar und erweitern solche Probleme auf eine breitere Klasse von Anwendungen. Wir fuhren auch eine Warmstart-Strategie ein, die eine lineare Losung der kleinsten Quadrate mit derjenigen mit minimaler ℓ2-Norm, die method of frames (MOF) heist, als Eingabe verwendet, um weniger Iterationen zu erfordern. Dieser Warmstart kann fur viele Anwendungen, bei denen MOF bereits existiert, eine flexible und sinnvolle Startpunktinitialisierung bieten und durch ein besseres Verstandnis der endlichdimensionalen Geometrie verbessert werden, z.B. nwidths. Der Einfluss der ubervollstandigen Matrix auf die Konvergenzrate einiger der vorgestellten Ansatze wird fur Matrizen gezeigt, die das Uniform Uncertainty Principle und das Uncertainty Principle erfullen. Diese Eigenschaften wurden fruher eingefuhrt, um die Leistung von Zufallsmatrizen fur ℓ1- bzw. ℓ∞-normbezogene Anwendungen zu analysieren.Im letzten Teil der Arbeit, d. h. in Kapitel 7, werden ℓp-Norm-Minimierungsbezogene Anwendungen mit verschiedenen Verlustfunktionen wie Least-Squares, Huber und einer nichtkonvexen Penalty untersucht Student's t. ℓ1-norm wird mit einem typischen Compressed-Sensing-Beispiel minimiert. Auserdem wird ein generischer Testbenchmark fur den Vergleich der nichtlinearen Gleichungswurzelfinder fur die ℓ1-Norm-Minimierung verwendet. Durch Minimierung der ℓ∞-Norm wird ein neues Kommunikationsschema eingefuhrt. Das Problem der Ausreisererkennung wird mit der minimierten ℓ∞-Norm untersucht und ein Prior wird fur die minimierte ℓ∞-Norm mit ihrer Leistung auf peak-to-average power ratio (PAPR) angeboten. Die rauschbeschrankte Nuklearnorm wird auch fur das Euklidische Distanzmatrix-Erganzungsproblem mit der Anwendung der drahtlosen Sensornetzwerklokalisierung minimiert.
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650 4 $a Norms. $3 3562802
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650 4 $a Signal processing. $3 533904
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