A Unifying Representer Theorem for Inverse Problems and Machine Learning

Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augmenting the origin...

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Vydané v:	Foundations of computational mathematics Ročník 21; číslo 4; s. 941 - 960
Hlavný autor:	Unser, Michael
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.08.2021 Springer Nature B.V
Predmet:	Applications of Mathematics Banach spaces Computer Science Economics Inverse problems Linear and Multilinear Algebras Machine learning Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Optimization Regularization Theorems 68T05 Convex optimization Machine learning Representer theorem 65J20 Banach space Regularization 46N10 47A52 Inverse problem
ISSN:	1615-3375, 1615-3383
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Abstract	Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augmenting the original cost functional by an energy that penalizes solutions with undesirable behavior. The effect of regularization is very well understood when the penalty involves a Hilbertian norm. Another popular configuration is the use of an ℓ 1 -norm (or some variant thereof) that favors sparse solutions. In this paper, we propose a higher-level formulation of regularization within the context of Banach spaces. We present a general representer theorem that characterizes the solutions of a remarkably broad class of optimization problems. We then use our theorem to retrieve a number of known results in the literature such as the celebrated representer theorem of machine leaning for RKHS, Tikhonov regularization, representer theorems for sparsity promoting functionals, the recovery of spikes, as well as a few new ones.
AbstractList	Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augmenting the original cost functional by an energy that penalizes solutions with undesirable behavior. The effect of regularization is very well understood when the penalty involves a Hilbertian norm. Another popular configuration is the use of an $$\ell _1$$ ℓ 1 -norm (or some variant thereof) that favors sparse solutions. In this paper, we propose a higher-level formulation of regularization within the context of Banach spaces. We present a general representer theorem that characterizes the solutions of a remarkably broad class of optimization problems. We then use our theorem to retrieve a number of known results in the literature such as the celebrated representer theorem of machine leaning for RKHS, Tikhonov regularization, representer theorems for sparsity promoting functionals, the recovery of spikes, as well as a few new ones. Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augmenting the original cost functional by an energy that penalizes solutions with undesirable behavior. The effect of regularization is very well understood when the penalty involves a Hilbertian norm. Another popular configuration is the use of an ℓ1-norm (or some variant thereof) that favors sparse solutions. In this paper, we propose a higher-level formulation of regularization within the context of Banach spaces. We present a general representer theorem that characterizes the solutions of a remarkably broad class of optimization problems. We then use our theorem to retrieve a number of known results in the literature such as the celebrated representer theorem of machine leaning for RKHS, Tikhonov regularization, representer theorems for sparsity promoting functionals, the recovery of spikes, as well as a few new ones. Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augmenting the original cost functional by an energy that penalizes solutions with undesirable behavior. The effect of regularization is very well understood when the penalty involves a Hilbertian norm. Another popular configuration is the use of an ℓ 1 -norm (or some variant thereof) that favors sparse solutions. In this paper, we propose a higher-level formulation of regularization within the context of Banach spaces. We present a general representer theorem that characterizes the solutions of a remarkably broad class of optimization problems. We then use our theorem to retrieve a number of known results in the literature such as the celebrated representer theorem of machine leaning for RKHS, Tikhonov regularization, representer theorems for sparsity promoting functionals, the recovery of spikes, as well as a few new ones.
Author	Unser, Michael
Author_xml	– sequence: 1 givenname: Michael surname: Unser fullname: Unser, Michael email: michael.unser@epfl.ch organization: Biomedical Imaging Group, Ecole polytechnique fédérale de Lausanne (EPFL)
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Snippet	Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements....
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SubjectTerms	Applications of Mathematics Banach spaces Computer Science Economics Inverse problems Linear and Multilinear Algebras Machine learning Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Optimization Regularization Theorems
Title	A Unifying Representer Theorem for Inverse Problems and Machine Learning
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