Related papers: Nonlinear frames and sparse reconstructions in Ban…

Sparse additive models are an attractive choice in circumstances calling for modelling flexibility in the face of high dimensionality. We study the signal detection problem and establish the minimax separation rate for the detection of a…

This thesis consists of original contributions in the area of digital signal processing. The reconstruction of signals sparse (highly concentrated) in various transform domains is the primary problem analyzed in the thesis. The considered…

We consider a class of inverse problems defined by a nonlinear map from parameter or model functions to the data. We assume that solutions exist. The space of model functions is a Banach space which is smooth and uniformly convex; however,…

This paper is motivated by structured sparsity for deep neural network training. We study a weighted group L0-norm constraint, and present the projection and normal cone of this set. Using randomized smoothing, we develop zeroth and…

We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main…

Given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded, but has an unbounded…

We consider the recovery of signals from their observations, which are samples of a transform of the signals rather than the signals themselves, by using machine learning (ML). We will develop a theoretical framework to characterize the…

Fixed point iterations are a fundamental tool in numerical analysis and scientific computing for the approximation of solutions to nonlinear problems. Their convergence is often established via the Banach fixed point theorem, provided that…

We consider the inverse problem of recovering an unknown functional parameter $u$ in a separable Banach space, from a noisy observation $y$ of its image through a known possibly non-linear ill-posed map ${\mathcal G}$. The data $y$ is…

We develop a general theoretical and algorithmic framework for sparse approximation and structured prediction in $\mathcal{P}_2(\Omega)$ with Wasserstein barycenters. The barycenters are sparse in the sense that they are computed from an…

Many asymptotically minimax procedures for function estimation often rely on somewhat arbitrary and restrictive assumptions such as isotropy or spatial homogeneity. This work enhances the theoretical understanding of Bayesian additive…

We study a family of sparse estimators defined as minimizers of some empirical Lipschitz loss function -- which include the hinge loss, the logistic loss and the quantile regression loss -- with a convex, sparse or group-sparse…

This paper addresses the problem of robust process and sensor fault reconstruction for nonlinear systems. The proposed method augments the system dynamics with an approximated internal linear model of the combined contribution of known…

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained L1 minimization. In this paper, we…

We prove that a Hilbert space frame $\fti$ contains a Riesz basis if every subfamily $\ftj , J \subseteq I ,$ is a frame for its closed span. Secondly we give a new characterization of Banach spaces which do not have any subspace isomorphic…

We consider the task of computing an approximate minimizer of the sum of a smooth and non-smooth convex functional, respectively, in Banach space. Motivated by the classical forward-backward splitting method for the subgradients in Hilbert…

The Banach-Picard iteration is widely used to find fixed points of locally contractive (LC) maps. This paper extends the Banach-Picard iteration to distributed settings; specifically, we assume the map of which the fixed point is sought to…

Stochastic optimization powers the scalability of modern artificial intelligence, spanning machine learning, deep learning, reinforcement learning, and large language model training. Yet, existing theory remains largely confined to Hilbert…

In [2] we characterized in terms of a quadratic growth condition various metric regularity properties of the subdifferential of a lower semicontinuous convex function acting in a Hilbert space. Motivated by some recent results in [16] where…

A fruitful approach for solving signal deconvolution problems consists of resorting to a frame-based convex variational formulation. In this context, parallel proximal algorithms and related alternating direction methods of multipliers have…