Browsing by Author "Müller, Klaus-Robert"
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- ItemAlgebraic geometric comparison of probability distributions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Király, Franz J.; von Bünau, Paul; Meinecke, Frank C.; Blythe, Duncan A.J.; Müller, Klaus-RobertWe propose a novel algebraic framework for treating probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of Algebraic Geometry, which we demonstrate in a compact proof for an identifiability criterion.
- ItemIn search on non-Gaussian components of a high-dimensional distribution(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Blanchard, Gilles; Kawanabe, Motoaki; Sugiyama, Masashi; Spokoiny, Vladimir; Müller, Klaus-RobertFinding non-Gaussian components of high-dimensional data is an important preprocessing step for efficient information processing. This article proposes a new em linear method to identify the "non-Gaussian subspace'' within a very general semi-parametric framework. Our proposed method, called NGCA (Non-Gaussian Component Analysis), is essentially based on the fact that we can construct a linear operator which, to any arbitrary nonlinear (smooth) function, associates a vector which belongs to the low dimensional non-Gaussian target subspace up to an estimation error. By applying this operator to a family of different nonlinear functions, one obtains a family of different vectors lying in a vicinity of the target space. As a final step, the target space itself is estimated by applying PCA to this family of vectors. We show that this procedure is consistent in the sense that the estimaton error tends to zero at a parametric rate, uniformly over the family, Numerical examples demonstrate the usefulness of our method
- ItemNew Inference Concepts for Analysing Complex Data(Zürich : EMS Publ. House, 2004) Müller, Klaus-Robert; Spokoiny, VladimirThe main purpose of this workshop was to assemble international leaders from statistics and machine learning to identify important research problems, to cross-fertilize between the disciplines, and to ultimately start coordinated research efforts toward better solutions. The workshop focused on discussing modern methods for analysis complex high dimensional data with applications to econometrics, finance, biomedicine, genomics etc.
- ItemStatistics meets Machine Learning(Zürich : EMS Publ. House, 2020) Dümbgen, Lutz; Müller, Klaus-Robert; Samworth, RichardTheory and application go hand in hand in most areas of statistics. In a world flooded with huge amounts of data waiting to be analyzed, classified and transformed into useful outputs, the designing of fast, robust and stable algorithms has never been as important as it is today. On the other hand, irrespective of whether the focus is put on estimation, prediction, classification or other purposes, it is equally crucial to provide clear guarantees that such algorithms have strong theoretical guarantees. Many statisticians, independently of their original research interests, have become increasingly aware of the importance of the numerical needs faced in numerous applications including gene expression profiling, health care, pattern and speech recognition, data security, marketing personalization, natural language processing, to name just a few. The goal of this workshop is twofold: (a) exchange knowledge on successful algorithmic approaches and discuss some of the existing challenges, and (b) to bring together researchers in statistics and machine learning with the aim of sharing expertise and exploiting possible differences in points of views to obtain a better understanding of some of the common important problems.