Related papers: Bregman Distances in Inverse Problems and Partial …
The purpose of this paper is twofold. On a technical side, we propose an extension of the Hausdorff distance from metric spaces to spaces equipped with asymmetric distance measures. Specifically, we focus on the family of Bregman…
We study an iterative regularization method of optimal control problems with control constraints. The regularization method is based on generalized Bregman distances. We provide convergence results under a combination of a source condition…
Recently, a new distance has been introduced for the graphs of two point-to-set operators, one of which is maximally monotone. When both operators are the subdifferential of a proper lower semicontinuous convex function, this distance…
The Bayesian approach to inverse problems provides a practical way to solve ill-posed problems by augmenting the observation model with prior information. Due to the measure-theoretic underpinnings, the approach has raised theoretical…
The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to…
In this article we deal with one-dimensional inverse problems concerning the Burgers equation and some related nonlinear systems (involving heat effects and/or variable density). In these problems, the goal is to find the size of the…
The paper focuses on the sparse approximation of signals using overcomplete representations, such that it preserves the (prior) structure of multi-dimensional signals. The underlying optimization problem is tackled using a multi-dimensional…
We consider the problem of estimating the inverse covariance matrix by maximizing the likelihood function with a penalty added to encourage the sparsity of the resulting matrix. We propose a new approach based on the split Bregman method to…
In this paper, we generalize the notions of centroids and barycenters to the broad class of information-theoretic distortion measures called Bregman divergences. Bregman divergences are versatile, and unify quadratic geometric distances…
Bregman proximal-type algorithms (BPs), such as mirror descent, have become popular tools in machine learning and data science for exploiting problem structures through non-Euclidean geometries. In this paper, we show that BPs can get…
In dynamic imaging, a key challenge is to reconstruct image sequences with high temporal resolution from strong undersampling projections due to a relatively slow data acquisition speed. In this paper, we propose a variational model using…
The constrained minimization (respectively maximization) of directed distances and of related generalized entropies is a fundamental task in information theory as well as in the adjacent fields of statistics, machine learning, artificial…
In this paper we discuss the explicit solution of certain extremal problems in Bergman spaces. In order to do this, we develop methods to calculate the Bergman projections of various functions. As a special case, we deal with canonical…
Distance transformation is an image processing technique used for many different applications. Related to a binary image, the general idea is to determine the distance of all background points to the nearest object point (or vice versa). In…
The MM principle is a device for creating optimization algorithms satisfying the ascent or descent property. The current survey emphasizes the role of the MM principle in nonlinear programming. For smooth functions, one can construct an…
Distances are pervasive in machine learning. They serve as similarity measures, loss functions, and learning targets; it is said that a good distance measure solves a task. When defining distances, the triangle inequality has proven to be a…
Graph-based methods provide a powerful tool set for many non-parametric frameworks in Machine Learning. In general, the memory and computational complexity of these methods is quadratic in the number of examples in the data which makes them…
Non-linear filtering approaches allow to obtain decompositions of images with respect to a non-classical notion of scale, induced by the choice of a convex, absolutely one-homogeneous regularizer. The associated inverse scale space flow can…
Matrix Factorization is a popular non-convex optimization problem, for which alternating minimization schemes are mostly used. They usually suffer from the major drawback that the solution is biased towards one of the optimization…
We develop a Bregman proximal gradient method for structure learning on linear structural causal models. While the problem is non-convex, has high curvature and is in fact NP-hard, Bregman gradient methods allow us to neutralize at least…