Related papers: $\ell_{1}^{2}-\eta\ell_{2}^{2}$ sparsity regulariz…
We focus on the minimization of the least square loss function either under a $k$-sparse constraint or with a sparse penalty term. Based on recent results, we reformulate the $\ell_0$ pseudo-norm exactly as a convex minimization problem by…
We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in $\mathbb{R}$. We prove that, after an appropriate translation, each of them is necessarily an even function which…
During the past years there has been an explosion of interest in learning methods based on sparsity regularization. In this paper, we discuss a general class of such methods, in which the regularizer can be expressed as the composition of a…
Flexible sparsity regularization means stably approximating sparse solutions of operator equations by using coefficient-dependent penalizations. We propose and analyse a general nonconvex approach in this respect, from both theoretical and…
It is well known that $\ell_1$ minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity, as a function of the ratio between the system dimensions,…
We propose a convex variational principle to find sparse representation of low-lying eigenspace of symmetric matrices. In the context of electronic structure calculation, this corresponds to a sparse density matrix minimization algorithm…
In this effort, we propose a convex optimization approach based on weighted $\ell_1$-regularization for reconstructing objects of interest, such as signals or images, that are sparse or compressible in a wavelet basis. We recover the…
We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + \alpha \partial \mathcal{R}(u) \ni f^\delta \end{equation*} for linear and nonlinear ill-posed problems with monotone operators…
A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that…
Classical sparsity promoting methods rely on the l0 norm, which treats all nonzero components as equally significant. In practical inverse problems, however, solutions often exhibit many small amplitude components that have little effect on…
With the capability of accurately representing a functional relationship between the inputs of a physical system's model and output quantities of interest, neural networks have become popular for surrogate modeling in scientific…
Regularization is a core component of modern inverse problems, as it helps establish the well-posedness of the solution of interest. Popular regularization approaches include variational regularization and iterative regularization. The…
We establish an equivalence between the $\ell_2$-regularized solution path for a convex loss function, and the solution of an ordinary differentiable equation (ODE). Importantly, this equivalence reveals that the solution path can be viewed…
We consider model selection and estimation for partial spline models and propose a new regularization method in the context of smoothing splines. The regularization method has a simple yet elegant form, consisting of roughness penalty on…
In the context of compressed sensing, the nonconvex $\ell_q$ minimization with $0<q<1$ has been studied in recent years. In this paper, by generalizing the sharp bound for $\ell_1$ minimization of Cai and Zhang, we show that the condition…
This paper studies $\ell_1$ regularization with high-dimensional features for support vector machines with a built-in reject option (meaning that the decision of classifying an observation can be withheld at a cost lower than that of…
In this paper we discuss the numerical solution of elliptic distributed optimal control problems with state or control constraints when the control is considered in the energy norm. As in the unconstrained case we can relate the…
It is well known that $\ell_1$ minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity, as a function of the ratio between the system dimensions,…
In this paper we look at a particular problem related to under-determined linear systems of equations with sparse solutions. $\ell_1$-minimization is a fairly successful polynomial technique that can in certain statistical scenarios find…
This paper considers a large class of linear operator equations, including linear boundary value problems for partial differential equations, and treats them as linear recovery problems for objects from their data. Well-posedness of the…