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WIREs Data Mining Knowl Discov
Impact Factor: 2.111

Data visualization by nonlinear dimensionality reduction

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In this overview, commonly used dimensionality reduction techniques for data visualization and their properties are reviewed. Thereby, the focus lies on an intuitive understanding of the underlying mathematical principles rather than detailed algorithmic pipelines. Important mathematical properties of the technologies are summarized in the tabular form. The behavior of representative techniques is demonstrated for three benchmarks, followed by a short discussion on how to quantitatively evaluate these mappings. In addition, three currently active research topics are addressed: how to devise dimensionality reduction techniques for complex non‐vectorial data sets, how to easily shape dimensionality reduction techniques according to the users preferences, and how to device models that are suited for big data sets. WIREs Data Mining Knowl Discov 2015, 5:51–73. doi: 10.1002/widm.1147

Data sets used for the demonstration of the dimensionality reduction techniques, sphere (a), swisshole (b) and mnist (c).
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Results for t‐SNE for the mnist data set (a) and the effects of the integration of auxiliary information using the Fisher matrix (c), as well as its extension toward the full data set of about 60,000 data points starting from standard t‐SNE (b) or Fisher‐t‐SNE (d), respectively.
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Results of the quality evaluation for the three data sets sphere (a), swisshole (b), and mnist (c) using the popular parametric and nonparametric mapping techniques.
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Results for popular nonparametric mapping techniques for the three data sets sphere (a, d, g), swisshole (b, e, h) and mnist (c, f, i) using the methods MVU (a, b, c) and t‐SNE (d, e, f) as well as the supervised linear projection technique LMNN (g, h, i) taking the labels of the data as indicated by the color codes as auxiliary information.
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Results for popular nonparametric mapping techniques for the three data sets sphere (a, d, g, j), swisshole (b, e, h, k) and mnist (c, f, i, l) using the methods (from top to bottom): Sammon's mapping, Isomap, LLE, and Laplacian eigenmaps.
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Visualization of data using the self‐organizing map. (a) A typical rectangular SOM grid, (b) trained SOM for the sphere data depicted in the data space, (c) unfolding of the SOM to achieve a two‐dimensional visualization, and (d) SOM grid for the mnist data set.
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Results for kernel PCA (a, b, c) and manifold charting (d, e, f) for the three data sets sphere (a, d), swisshole (b, e) and mnist (c, f).
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Demonstration of PCA. (a) The principal components of a two‐dimensional data set, (b) PCA projection of the sphere data set, (c) PCA projection of the swisshole data set, and (d) PCA projection of the mnist data set.
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Technologies > Machine Learning
Technologies > Structure Discovery and Clustering
Technologies > Visualization

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