Finding the Asymptotically Optimal Baire Distance for Multi-Channel Data

Bradley, Patrick Erik; Braun, Andreas

doi:10.4236/am.2015.63046

Cited by 6 publications

(3 citation statements)

References 12 publications

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“…In general, related to such a multidimensional Baire distance is the Baire distance formulated for multi-channel data, i.e. hyperspectral images, and used for machine learning (Support Vector Machine, supervised classification) in [3].…”

Section: Multidimensional Baire Distancementioning

confidence: 99%

Linear Time Visualization and Search in Big Data using Pixellated Factor Space Mapping

Murtagh¹

2019

Preprint

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It is demonstrated how linear computational time and storage efficient approaches can be adopted when analyzing very large data sets. More importantly, interpretation is aided and furthermore, basic processing is easily supported. Such basic processing can be the use of supplementary, i.e. contextual, elements, or particular associations. Furthermore pixellated grid cell contents can be utilized as a basic form of imposed clustering. For a given resolution level, here related to an associated m-adic (m here is a non-prime integer) or p-adic (p is prime) number system encoding, such pixellated mapping results in partitioning. The association of a range of m-adic and p-adic representations leads naturally to an imposed hierarchical clustering, with partition levels corresponding to the m-adicbased and p-adic-based representations and displays. In these clustering embedding and imposed cluster structures, some analytical visualization and search applications are described.

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Section: Multidimensional Baire Distancementioning

confidence: 99%

Linear Time Visualization and Search in Big Data using Pixellated Factor Space Mapping

Murtagh¹

2019

Preprint

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“…However, in general, the ordering of features often relies on system-related or user-defined criteria, and there is for instance no natural ordering of features with respect to the topology of the given data, with respect to their contribution to the variability of considered data or with respect to their relevance for a classification task. To establish a suitable ordering via BOFR, orderings which minimize the mean Baire distance can be considered as proposed in [83]. It is shown there that for sufficiently small γ > 1, this minimizing ordering can be found via gradient descent.…”

Section: Unsupervised Fs: Baire-optimal Feature Ranking (Bofr)mentioning

confidence: 99%

Unsupervised Feature Selection Based on Ultrametricity and Sparse Training Data: A Case Study for the Classification of High-Dimensional Hyperspectral Data

2018

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In this paper, we investigate the potential of unsupervised feature selection techniques for classification tasks, where only sparse training data are available. This is motivated by the fact that unsupervised feature selection techniques combine the advantages of standard dimensionality reduction techniques (which only rely on the given feature vectors and not on the corresponding labels) and supervised feature selection techniques (which retain a subset of the original set of features). Thus, feature selection becomes independent of the given classification task and, consequently, a subset of generally versatile features is retained. We present different techniques relying on the topology of the given sparse training data. Thereby, the topology is described with an ultrametricity index. For the latter, we take into account the Murtagh Ultrametricity Index (MUI) which is defined on the basis of triangles within the given data and the Topological Ultrametricity Index (TUI) which is defined on the basis of a specific graph structure. In a case study addressing the classification of high-dimensional hyperspectral data based on sparse training data, we demonstrate the performance of the proposed unsupervised feature selection techniques in comparison to standard dimensionality reduction and supervised feature selection techniques on four commonly used benchmark datasets. The achieved classification results reveal that involving supervised feature selection techniques leads to similar classification results as involving unsupervised feature selection techniques, while the latter perform feature selection independently from the given classification task and thus deliver generally versatile features.

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“…Data tends towards being ultrametric with increasing dimension, and it becomes easier to find clusters with increasing dimension (Murtagh, 2009). In Bradley and Braun (2015), an asymptotically optimal ultrametric distance is computed for high-dimensional data.…”

Section: Introductionmentioning

confidence: 99%

On the Logistic Behaviour of the Topological Ultrametricity of Data

Bradley

2018

J Classif

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Recently, it has been observed that topological ultrametricity of data can be expressed as an integral over a function which describes local ultrametricity. It was then observed empirically that this function begins as a sharply decreasing function, in order to increase again back to one. After providing a method for estimating the falling part of the local ultrametricity of data, empirical evidence is given for its logistic behaviour in relation to the number of connected components of the Vietoris-Rips graphs involved. The result is a functional dependence between that number and the number of maximal cliques. Further, it turns out that the logistic parameters depend linearly on the datasize. These observations are interpreted in terms of the Erdős-Rényi model for random graphs. Thus the findings allow to define a percolationbased index for almost ultrametricity which can be estimated in O(N 2 log N ) time which is more efficient than most ultrametricity indices.

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Finding the Asymptotically Optimal Baire Distance for Multi-Channel Data

Cited by 6 publications

References 12 publications

Linear Time Visualization and Search in Big Data using Pixellated Factor Space Mapping

Linear Time Visualization and Search in Big Data using Pixellated Factor Space Mapping

Unsupervised Feature Selection Based on Ultrametricity and Sparse Training Data: A Case Study for the Classification of High-Dimensional Hyperspectral Data

On the Logistic Behaviour of the Topological Ultrametricity of Data

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