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We introduce a learning-based algorithm to obtain a measurement matrix for compressive sensing related recovery problems. The focus lies on matrices with a constant modulus constraint which typically represent a network of analog phase…

In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without…

We present the applications of methods from nonlinear local harmonic analysis for calculations in nonlinear collective dynamics described by different forms of Vlasov-Maxwell-Poisson equations. Our approach is based on methods provided the…

Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this…

Two complementary approaches have been extensively used in signal and image processing leading to novel results, the sparse representation methodology and the variational strategy. Recently, a new sparsity based model has been proposed, the…

In many applications, data come with a natural ordering. This ordering can often induce local dependence among nearby variables. However, in complex data, the width of this dependence may vary, making simple assumptions such as a constant…

We consider the problem of sparse variable selection in nonparametric additive models, with the prior knowledge of the structure among the covariates to encourage those variables within a group to be selected jointly. Previous works either…

We consider the problem of reconstructing an infinite set of sparse, finite-dimensional vectors, that share a common sparsity pattern, from incomplete measurements. This is in contrast to the work [17], where the single vector signal can be…

The choice of the sensing matrix is crucial in compressed sensing. Random Gaussian sensing matrices satisfy the restricted isometry property, which is crucial for solving the sparse recovery problem using convex optimization techniques.…

This thesis studies two problems in modern statistics. First, we study selective inference, or inference for hypothesis that are chosen after looking at the data. The motiving application is inference for regression coefficients selected by…

We propose a new approach to mixed-frequency regressions in a high-dimensional environment that resorts to Group Lasso penalization and Bayesian techniques for estimation and inference. In particular, to improve the prediction properties of…

We propose a method to reconstruct sparse signals degraded by a nonlinear distortion and acquired at a limited sampling rate. Our method formulates the reconstruction problem as a nonconvex minimization of the sum of a data fitting term and…

In many modern applications, including analysis of gene expression and text documents, the data are noisy, high-dimensional, and unordered--with no particular meaning to the given order of the variables. Yet, successful learning is often…

This work is focussed on the inversion task of inferring the distribution over parameters of interest leading to multiple sets of observations. The potential to solve such distributional inversion problems is driven by increasing…

Composite optimization problems involve minimizing the composition of a smooth map with a convex function. Such objectives arise in numerous data science and signal processing applications, including phase retrieval, blind deconvolution,…

Conventional inverse optimization inputs a solution and finds the parameters of an optimization model that render a given solution optimal. The literature mostly focuses on inferring the objective function in linear problems when accepted…

Sparse modeling is a powerful framework for data analysis and processing. Traditionally, encoding in this framework is performed by solving an L1-regularized linear regression problem, commonly referred to as Lasso or Basis Pursuit. In this…

In this paper we study nonconvex penalization using Bernstein functions whose first-order derivatives are completely monotone. The Bernstein function can induce a class of nonconvex penalty functions for high-dimensional sparse estimation…

We highlight some recent new delevelopments concerning the sparse representation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus…

Compressed sensing is a signal processing technique whereby the limits imposed by the Shannon--Nyquist theorem can be exceeded provided certain conditions are imposed on the signal. Such conditions occur in many real-world scenarios, and…