arXiv subjects
Publications and source records attributed to Wolfgang Stummer.
For some smooth special case of generalized $\varphi-$divergences as well as of new divergences (called scaled shift divergences), we derive approximations of the omnipresent (weighted) $\ell_{1}-$distance and (weighted) $\ell_{1}-$norm.
We propose a new random method to minimize deterministic continuous functions over subsets $\mathcal{S}$ of high-dimensional space $\mathbb{R}^K$ without assuming convexity. Our procedure alternates between a Global Search (GS) regime to identify candidates and a Concentrated Search (CS) regime to improve an eligible candidate in the constraint set $\mathcal{S}$. Beyond the alternation between those completely different regimes, the originality of our approach lies in leveraging high dimensionality. We demonstrate rigorous concentration properties under the $CS$ regime. In parallel, we also show that $GS$ reaches any point in $\mathcal{S}$ in finite time. Finally, we demonstrate the relevance of our new method by giving two concrete applications. The first deals with the reduction of the $\ell_{1}-$norm of a LASSO solution. Secondly, we compress a neural network by pruning weights while maintaining performance; our approach achieves significant weight reduction with minimal performance loss, offering an effective solution for network optimization.
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 intelligence, signal processing and pattern recognition. In our previous paper "A precise bare simulation approach to the minimization of some distances. I. Foundations", we obtained such kind of constrained optima by a new dimension-free precise bare (pure) simulation method, provided basically that (i) the underlying directed distance is of f-divergence type, and that (ii) this can be connected to a light-tailed probability distribution in a certain manner. In the present paper, we extend this approach such that constrained optimization problems of a very huge amount of directed distances and generalized entropies -- and beyond -- can be tackled by a newly developed dimension-free extended bare simulation method, for obtaining both optima as well as optimizers. Almost no assumptions (like convexity) on the set of constraints are needed, within our discrete setup of arbitrary dimension, and our method is precise (i.e., converges in the limit). For instance, we cover constrained optimizations of arbitrary f-divergences, Bregman distances, scaled Bregman distances and weighted Euclidean distances. The potential for wide-spread applicability is indicated, too; in particular, we deliver many recent references for uses of the involved distances/divergences in various different research fields (which may also serve as an interdisciplinary interface).
In information theory -- as well as in the adjacent fields of statistics, machine learning, artificial intelligence, signal processing and pattern recognition -- many flexibilizations of the omnipresent Kullback-Leibler information distance (relative entropy) and of the closely related Shannon entropy have become frequently used tools. To tackle corresponding constrained minimization (respectively maximization) problems by a newly developed dimension-free bare (pure) simulation method, is the main goal of this paper. Almost no assumptions (like convexity) on the set of constraints are needed, within our discrete setup of arbitrary dimension, and our method is precise (i.e., converges in the limit). As a side effect, we also derive an innovative way of constructing new useful distances/divergences. To illustrate the core of our approach, we present numerous solved cases. The potential for widespread applicability is indicated, too; in particular, we deliver many recent references for uses of the involved distances/divergences and entropies in various different research fields (which may also serve as an interdisciplinary interface).
Density-based directed distances -- particularly known as divergences -- between probability distributions are widely used in statistics as well as in the adjacent research fields of information theory, artificial intelligence and machine learning. Prominent examples are the Kullback-Leibler information distance (relative entropy) which e.g. is closely connected to the omnipresent maximum likelihood estimation method, and Pearson's chisquare-distance which e.g. is used for the celebrated chisquare goodness-of-fit test. Another line of statistical inference is built upon distribution-function-based divergences such as e.g. the prominent (weighted versions of) Cramer-von Mises test statistics respectively Anderson-Darling test statistics which are frequently applied for goodness-of-fit investigations; some more recent methods deal with (other kinds of) cumulative paired divergences and closely related concepts. In this paper, we provide a general framework which covers in particular both the above-mentioned density-based and distribution-function-based divergence approaches; the dissimilarity of quantiles respectively of other statistical functionals will be included as well. From this framework, we structurally extract numerous classical and also state-of-the-art (including new) procedures. Furthermore, we deduce new concepts of dependence between random variables, as alternatives to the celebrated mutual information. Some variational representations are discussed, too.
We present a toolkit of directed distances between quantile functions. By employing this, we solve some new optimal transport (OT) problems which e.g. considerably flexibilize some prominent OTs expressed through Wasserstein distances.
The paper introduces scaled Bregman distances of probability distributions which admit non-uniform contributions of observed events. They are introduced in a general form covering not only the distances of discrete and continuous stochastic observations, but also the distances of random processes and signals. It is shown that the scaled Bregman distances extend not only the classical ones studied in the previous literature, but also the information divergence and the related wider class of convex divergences of probability measures. An information processing theorem is established too, but only in the sense of invariance w.r.t. statistically sufficient transformations and not in the sense of universal monotonicity. Pathological situations where coding can increase the classical Bregman distance are illustrated by a concrete example. In addition to the classical areas of application of the Bregman distances and convex divergences such as recognition, classification, learning and evaluation of proximity of various features and signals, the paper mentions a new application in 3D-exploratory data analysis. Explicit expressions for the scaled Bregman distances are obtained in general exponential families, with concrete applications in the binomial, Poisson and Rayleigh families, and in the families of exponential processes such as the Poisson and diffusion processes including the classical examples of the Wiener process and geometric Brownian motion.
We compute exact values respectively bounds of "distances" - in the sense of (transforms of) power divergences and relative entropy - between two discrete-time Galton-Watson branching processes with immigration GWI for which the offspring as well as the immigration is arbitrarily Poisson-distributed (leading to arbitrary type of criticality). Implications for asymptotic distinguishability behaviour in terms of contiguity and entire separation of the involved GWI are given, too. Furthermore, we determine the corresponding limit quantities for the context in which the two GWI converge to Feller-type branching diffusion processes, as the time-lags between observations tend to zero. Some applications to (static random environment like) Bayesian decision making and Neyman-Pearson testing are presented as well.