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@PHDTHESIS{Schnoor:1012246,
author = {Schnoor, Ekkehard},
othercontributors = {Rauhut, Holger and Führ, Hartmut},
title = {{I}terative soft-thresholding from a statistical learning
perspective},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
publisher = {RWTH Aachen University},
reportid = {RWTH-2025-04943},
pages = {1 Online-Ressource : Illustrationen},
year = {2024},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University 2025; Dissertation, RWTH Aachen University, 2024},
abstract = {This dissertation explores connections between the areas of
compressive sensing and machine learning. It is centered
around the so-called iterative soft-thresholding algorithm
(ISTA), an iterative algorithm to solve the l1 -regularized
least squares problem also known as LASSO (least absolute
shrinkage and selection operator) that has various
applications in statistics and signal processing. We will
investigate two statistical learning problems that can be
regarded as two different interpretations of the same
underlying optimization problem and its solution through
ISTA. While both are different, in common they have a
generalization perspective, i.e., we aim for finding
performance guarantees at inference, that is when applying
the trained model to new data samples that have not been
used for training, but can be regarded as samples from the
same underlying (but in practice, typically unknown)
distribution. Thus, the contribution of this thesis lies in
providing novel investigations of the iterative
soft-thresholding algorithm from the viewpoint of
statistical learning theory. We heavily rely on tools from
high-dimensional probability theory to prove our results.
The first of the problems we consider deals with an
interpretation of ISTA as a neural network, a topic which
attracted attention with the rise of deep learning in the
past decade. As a first step to introduce trainable
parameters, we address a rather simple model, where a
dictionary is learned implicitly. Then, we extend our
results to a greatly generalized setup including a variety
of ISTA-inspired neural networks, ranging from recurrent
ones to architectures more similar to feedforward neural
networks. Based on estimates of the Rademacher complexity of
the corresponding hypothesis classes, we derive the first
generalization error bounds for such specific neural network
architectures and compare our theoretical findings to
numerical experiments. While previous works strongly focused
on generalization of deep learning in the context of
classification tasks, we provide theoretical results in the
context of inverse problems, which is much less explored in
the literature. The second problem considers the application
of LASSO in a classification context, where the solution
found through ISTA plays the role of a sparse linear
classifier. Under realistic assumptions on the training
data, we show that this induces a concentration on the
distribution over the corresponding hypothesis class. This
enables us to derive an algorithm to predict the
classification accuracy based solely on statistical
properties of the training data, which we confirm with the
help of numerical experiments.},
cin = {112710 / 110000},
ddc = {510},
cid = {$I:(DE-82)112710_20190828$ / $I:(DE-82)110000_20140620$},
pnm = {DFG project G:(GEPRIS)404374102 - Strukturiertes
Compressive Sensing mittels gelernten neuronalen Netzen
(SCoSNeL) (404374102) / SPP 1798: Compressed Sensing in der
Informationsverarbeitung (255450733)},
pid = {G:(GEPRIS)404374102 / G:(GEPRIS)255450733},
typ = {PUB:(DE-HGF)11},
doi = {10.18154/RWTH-2025-04943},
url = {https://publications.rwth-aachen.de/record/1012246},
}
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