相关论文: Relationship between the Bregman divergence and be…

We establish the convergence of the forward-backward splitting algorithm based on Bregman distances for the sum of two monotone operators in reflexive Banach spaces. Even in Euclidean spaces, the convergence of this algorithm has so far…

Preserving the robustness of the procedure has, at the present time, become almost a default requirement for statistical data analysis. Since efficiency at the model and robustness under misspecification of the model are often in conflict,…

This paper focuses on the Bregman divergence defined by the reciprocal function, called the inverse divergence. For the loss function defined by the monotonically increasing function $f$ and inverse divergence, the conditions for the…

The bias-variance decomposition is a central result in statistics and machine learning, but is typically presented only for the squared error. We present a generalization of the bias-variance decomposition where the prediction error is a…

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…

To extract the relevant features in a given dataset is a difficult task, recently resolved in the non-negative data case with the Non-negative Matrix factorization (NMF) method. The objective of this research work is to extend this method…

A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values,…

The convex transform order is one way to make precise comparison between the skewness of probability distributions on the real line. We establish a simple and complete characterisation of when one Beta distribution is smaller than another…

Circumcenters play an important role in the design and analysis of accelerating various iterative methods in optimization. In this work, we propose Bregman (pseudo-)circumcenters associated with finite sets. We show the existence and give…

Classic subdifferentials in variational analysis may fail to fully represent the Bregman proximal operator in the absence of convexity. In this paper, we fill this gap by introducing the left and right \emph{Bregman level proximal…

Logarithmic score and information divergence appear in both information theory, statistics, statistical mechanics, and portfolio theory. We demonstrate that all these topics involve some kind of optimization that leads directly to the use…

In this paper, we provide a simple convergence analysis of proximal gradient algorithm with Bregman distance, which provides a tighter bound than existing result. In particular, for the problem of minimizing a class of convex objective…

Neural networks are popular state-of-the-art models for many different tasks.They are often trained via back-propagation to find a value of the weights that correctly predicts the observed data. Although back-propagation has shown good…

We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a…

Bregman divergences $D_\phi$ are a class of divergences parametrized by a convex function $\phi$ and include well known distance functions like $\ell_2^2$ and the Kullback-Leibler divergence. There has been extensive research on algorithms…

We revisit the mathematical foundations of proper scoring rules (PSRs) and Bregman divergences and present their characteristic properties in a unified theoretical framework. In many situations it is preferable not to generate a PSR…

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…

In this short note, we give the convergence analysis of the policy in the recent famous policy mirror descent (PMD). We mainly consider the unregularized setting following [11] with generalized Bregman divergence. The difference is that we…

For a wide range of pairs of mixed norm spaces such that one space is contained in another, we characterize all cases when contractive norm inequalities hold. In particular, this yields such results for many pairs of weighted Bergman…

We describe the relation between simple logarithmic CFTs associated with closed and open strings, and their "infinite metric" limits, corresponding to the beta-gamma systems. This relation is studied on the level of the BRST complex: we…