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Sparse additive modeling is a class of effective methods for performing high-dimensional nonparametric regression. In this work we show how shape constraints such as convexity/concavity and their extensions, can be integrated into additive…

As a popular tool for producing meaningful and interpretable models, large-scale sparse learning works efficiently when the underlying structures are indeed or close to sparse. However, naively applying the existing regularization methods…

The standard approach to compressive sampling considers recovering an unknown deterministic signal with certain known structure, and designing the sub-sampling pattern and recovery algorithm based on the known structure. This approach…

Analysis sparsity is a common prior in inverse problem or machine learning including special cases such as Total Variation regularization, Edge Lasso and Fused Lasso. We study the geometry of the solution set (a polyhedron) of the analysis…

Low rank regularization, in essence, involves introducing a low rank or approximately low rank assumption for matrix we aim to learn, which has achieved great success in many fields including machine learning, data mining and computer…

Regularization is a common tool in variational inverse problems to impose assumptions on the parameters of the problem. One such assumption is sparsity, which is commonly promoted using lasso and total variation-like regularization.…

Using machine learning to solve combinatorial optimization (CO) problems is challenging, especially when the data is unlabeled. This work proposes an unsupervised learning framework for CO problems. Our framework follows a standard…

Conventional algorithms for sparse signal recovery and sparse representation rely on $l_1$-norm regularized variational methods. However, when applied to the reconstruction of $\textit{sparse images}$, i.e., images where only a few pixels…

This paper considers the problem of recovering either a low rank matrix or a sparse vector from observations of linear combinations of the vector or matrix elements. Recent methods replace the non-convex regularization with $\ell_1$ or…

Implicit neural representations are a promising new avenue of representing general signals by learning a continuous function that, parameterized as a neural network, maps the domain of a signal to its codomain; the mapping from spatial…

Sparse channel estimation for massive multiple-input multiple-output systems has drawn much attention in recent years. The required pilots are substantially reduced when the sparse channel state vectors can be reconstructed from a few…

Finding efficient and provable methods to solve non-convex optimization problems is an outstanding challenge in machine learning and optimization theory. A popular approach used to tackle non-convex problems is to use convex relaxation…

We investigate fast methods that allow to quickly eliminate variables (features) in supervised learning problems involving a convex loss function and a $l_1$-norm penalty, leading to a potentially substantial reduction in the number of…

Due to the non-convex nature of training Deep Neural Network (DNN) models, their effectiveness relies on the use of non-convex optimization heuristics. Traditional methods for training DNNs often require costly empirical methods to produce…

In high-dimensional statistical inference in which the number of parameters to be estimated is larger than that of the holding data, regularized linear estimation techniques are widely used. These techniques have, however, some drawbacks.…

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel…

We analyze a class of norms defined via an optimal interpolation problem involving the composition of norms and a linear operator. This construction, known as infimal postcomposition in convex analysis, is shown to encompass various of…

The goal of this paper is to contrast and survey the major advances in two of the most commonly used high-dimensional techniques, namely, the Lasso and horseshoe regularization. Lasso is a gold standard for predictor selection while…

Convex optimization with sparsity-promoting convex regularization is a standard approach for estimating sparse signals in noise. In order to promote sparsity more strongly than convex regularization, it is also standard practice to employ…

High-dimensional sparse modeling with censored survival data is of great practical importance, as exemplified by modern applications in high-throughput genomic data analysis and credit risk analysis. In this article, we propose a class of…