Related papers: A Duality Approach to Regularized Learning Problem…
This article delves into the study of the theory of regularized learning in Banach spaces for linear-functional data. It encompasses discussions on representer theorems, pseudo-approximation theorems, and convergence theorems. Regularized…
We study the solutions of infinite dimensional linear inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary…
Regularisation theory in Banach spaces, and non--norm-squared regularisation even in finite dimensions, generally relies upon Bregman divergences to replace norm convergence. This is comparable to the extension of first-order optimisation…
There is growing body of learning problems for which it is natural to organize the parameters into matrix, so as to appropriately regularize the parameters under some matrix norm (in order to impose some more sophisticated prior knowledge).…
We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a…
Constructing or learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation problem, regularized learning…
In this paper, we introduce a novel two-point gradient method for solving the ill-posed problems in Banach spaces and study its convergence analysis. The method is based on the well known iteratively regularized Landweber iteration method…
In this paper, we discuss the construction, analysis and implementation of a novel iterative regularization scheme with general convex penalty term for nonlinear inverse problems in Banach spaces based on the homotopy perturbation…
We consider composite linear inverse problems where the signal to recover is modeled as a sum of two functions. We study a variational framework formulated as an optimization problem over the pairs of components using two regularization…
We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice. We carry out analysis with respect to existence and…
This article gives dual representations for convex integral functionals on the linear space of regular processes. This space turns out to be a Banach space containing many more familiar classes of stochastic processes and its dual can be…
The standard approach for dealing with the ill-posedness of the training problem in machine learning and/or the reconstruction of a signal from a limited number of measurements is regularization. The method is applicable whenever the…
We investigate the function-space optimality (specifically, the Banach-space optimality) of a large class of shallow neural architectures with multivariate nonlinearities/activation functions. To that end, we construct a new family of…
We study a general version of the adversarial online learning problem. We are given a decision set $\mathcal{X}$ in a reflexive Banach space $X$ and a sequence of reward vectors in the dual space of $X$. At each iteration, we choose an…
This paper proposes a unified framework for the investigation of constrained learning theory in reflexive Banach spaces of features via regularized empirical risk minimization. The focus is placed on Tikhonov-like regularization with…
Spaces of homogeneous polynomials on a Banach space are frequently equipped with quasinorms instead of norms. In this paper we develop a technique to replace the original quasi-norm by a norm in a dual preserving way, in the sense that the…
Proximal gradient methods are a popular tool for the solution of structured, nonsmooth minimization problems. In this work, we investigate an extension of the former to general Banach spaces and provide worst-case convergence rates for,…
We establish strong duality relations for functional two-step compositional risk-constrained learning problems with multiple nonconvex loss functions and/or learning constraints, regardless of nonconvexity and under a minimal set of…
In this paper, we propose and analyze a two-point gradient method for solving inverse problems in Banach spaces which is based on the Landweber iteration and an extrapolation strategy. The method allows to use non-smooth penalty terms,…
In the present work, we discuss variational regularization for ill-posed nonlinear problems with focus on an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation…