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1

The special atom space and Haar wavelets in higher dimensions

Kwessi Eddy, Souza de Geraldo, Djitte Ngalla, Ndiaye Mariama

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

131--151

EN

In this note, we will revisit the special atom space introduced in the early 1980s by Geraldo De Souza and Richard O’Neil. In their introductory work and in later additions, the space was mostly studied on the real line. Interesting properties and connections to spacessuch as Orlicz, Lipschitz, Lebesgue, and Lorentz spaces made these spaces ripe for exploration in higherdimensions. In this article, we extend this definition to the plane and space and show that almost all the interesting properties such as their Banach structure, Hölder’s-type inequalities, and duality are preserved. In particular, dual spaces of special atom spaces are natural extension of Lipschitz and generalized Lipschitz spaces of functions in higher dimensions. We make the point that this extension could allow for the study of a wide range of problems including a connection that leads to what seems to be a new definition of Haar functions, Haar wavelets, and wavelets on the plane and on the space.

2

Exponential rate of convergence independent of the dimension in a mean-field system of particles

Dyda B., Tugaut J.

Probability and Mathematical Statistics

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2017

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Vol. 37, Fasc. 1

145--161

EN

This article deals with a mean-field model. We consider a large number of particles interacting through their empirical law. We know that there is a unique invariant probability for this diffusion.We look at functional inequalities. In particular, we briefly show that the diffusion satisfies a Poincaré inequality. Then, we establish a so-called WJ-inequality, which is independent of the number of particles.

3

Scalable Clustering for Mining Local-Correlated Clusters in High Dimensions and Large Datasets

Lu K-C., Yang D-L.

Fundamenta Informaticae

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2010

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Vol. 98, nr 1

15-32

EN

Clustering is useful for mining the underlying structure of a dataset in order to support decision making since target or high-risk groups can be identified. However, for high dimensional datasets, the result of traditional clustering methods can be meaningless as clusters may only be depicted with respect to a small part of features. Taking customer datasets as an example, certain customers may correlate with their salary and education, and the others may correlate with their job and house location. If one uses all the features of a customer for clustering, these local-correlated clusters may not be revealed. In addition, processing high dimensions and large datasets is a challenging problem in decision making. Searching all the combinations of every feature with every record to extract local-correlated clusters is infeasible, which is in exponential scale in terms of data dimensionality and cardinality. In this paper, we propose a scalable 2-Leveled Approximated Hyper-Image-based Clustering framework, referred as 2L-HIC-A, for mining local-correlated clusters, where each level clustering process requires only one scan of the original dataset. Moreover, the data-processing time of 2L-HIC-A can be independent of the input data size. In 2L-HIC-A, various well-developed image processing techniques can be exploited for mining clusters. In stead of proposing a new clustering algorithm, our framework can accommodate other clustering methods for mining local-corrected clusters, and to shed new light on the existing clustering techniques.