Structured sparsity-inducing norms : statistical and algorithmic properties with applications to neuroimaging

Abstract : Numerous fields of applied sciences and industries have been recently witnessing a process of digitisation. This trend has come with an increase in the amount digital data whose processing becomes a challenging task. In this context, parsimony, also known as sparsity, has emerged as a key concept in machine learning and signal processing. It is indeed appealing to exploit data only via a reduced number of parameters. This thesis focuses on a particular and more recent form of sparsity, referred to as structured sparsity. As its name indicates, we shall consider situations where we are not only interested in sparsity, but where some structural prior knowledge is also available. The goal of this thesis is to analyze the concept of structured sparsity, based on statistical, algorithmic and applied considerations. To begin with, we introduce a family of structured sparsity-inducing norms whose statistical aspects are closely studied. In particular, we show what type of prior knowledge they correspond to. We then turn to sparse structured dictionary learning, where we use the previous norms within the framework of matrix factorization. From an optimization viewpoint, we derive several efficient and scalable algorithmic tools, such as working-set strategies and proximal-gradient techniques. With these methods in place, we illustrate on numerous real-world applications from various fields, when and why structured sparsity is useful. This includes, for instance, restoration tasks in image processing, the modelling of text documents as hierarchy of topics, the inter-subject prediction of sizes of objects from fMRI signals, and background-subtraction problems in computer vision.
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Rodolphe Jenatton. Structured sparsity-inducing norms : statistical and algorithmic properties with applications to neuroimaging. General Mathematics [math.GM]. École normale supérieure de Cachan - ENS Cachan, 2011. English. ⟨NNT : 2011DENS0048⟩. ⟨tel-00668379⟩



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