相关论文: Convergence Rates for Oversmoothing Banach Space R…

We present a framework for smooth optimization of explicitly regularized objectives for (structured) sparsity. These non-smooth and possibly non-convex problems typically rely on solvers tailored to specific models and regularizers. In…

A learning approach to selecting regularization parameters in multi-penalty Tikhonov regularization is investigated. It leads to a bilevel optimization problem, where the lower level problem is a Tikhonov regularized problem parameterized…

We provide a modified augmented Lagrange method coupled with a Tikhonov regularization for solving ill-posed state-constrained elliptic optimal control problems with sparse controls. We consider a linear quadratic optimal control problem…

The goal of this work is to obtain optimal rates for the convergence problem in mean field control. Our analysis covers cases where the solutions to the limiting problem may not be unique nor stable. Equivalently the value function of the…

We investigate a Tikhonov regularization scheme specifically tailored for shallow neural networks within the context of solving a classic inverse problem: approximating an unknown function and its derivatives within a unit cubic domain…

A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space…

For a Banach space $B$ of functions which satisfies for some $m>0$ $$ \max(\|F+G\|_B,\|F-G\|_B) \ge (\|F\|^s_B + m\|G\|^s_B)^{1/s}, \forall F,G\in B \ (*) $$ a significant improvement for lower estimates of the moduli of smoothness…

In this paper, we establish explicit convergence rates for the stochastic smooth approximations of infimal convolutions introduced and developed in \cite{MR4581306,MR4923371}. In particular, we quantify the convergence of the associated…

Recent advances in machine learning have led to increased interest in reproducing kernel Banach spaces (RKBS) as a more general framework that extends beyond reproducing kernel Hilbert spaces (RKHS). These works have resulted in the…

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…

In regularized risk minimization, the associated optimization problem becomes particularly difficult when both the loss and regularizer are nonsmooth. Existing approaches either have slow or unclear convergence properties, are restricted to…

In this paper we investigate the problem of identifying the source term in an elliptic system from a single noisy measurement couple of the Neumann and Dirichlet data. A variational method of Tikhonov-type regularization with specific…

Landweber-type methods are prominent for solving ill-posed inverse problems in Banach spaces and their convergence has been well-understood. However, how to derive their convergence rates remains a challenging open question. In this paper,…

In this work we consider, in a Banach space framework, the regularization of linear ill-posed problems. Our focus is on the recovery of solutions that have a logarithmic source representation. Such cases typically occur in exponentially…

In the era of deep learning, understanding over-fitting phenomenon becomes increasingly important. It is observed that carefully designed deep neural networks achieve small testing error even when the training error is close to zero. One…

We study integration and $L_2$-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh,…

This paper studies the problem of approximating a function $f$ in a Banach space $X$ from measurements $l_j(f)$, $j=1,\dots,m$, where the $l_j$ are linear functionals from $X^*$. Most results study this problem for classical Banach spaces…

We shall study in this paper the Lipschitz type stabilities and convergence rates of Tikhonov regularization for the recovery of the radiativities in elliptic and parabolic systems with Dirichlet boundary conditions. The Lipschitz type…

We study the problem of learning high dimensional regression models regularized by a structured-sparsity-inducing penalty that encodes prior structural information on either input or output sides. We consider two widely adopted types of…

This paper is concerned with minimization of a fourth-order linearized Canham-Helfrich energy subject to Dirichlet boundary conditions on curves inside the domain. Such problems arise in the modeling of the mechanical interaction of…