Abstract
Projections are conventional methods of dimensionality reduction for information visualization used to transform high-dimensional data into low dimensional space. If the projection method restricts the output space to two dimensions, the result is a scatter plot. The goal of this scatter plot is to visualize the relative relationships between high-dimensional data points that build up distance and density-based structures. However, the Johnson–Lindenstrauss lemma states that the two-dimensional similarities in the scatter plot cannot coercively represent high-dimensional structures. Here, a simplified emergent self-organizing map uses the projected points of such a scatter plot in combination with the dataset in order to compute the generalized U-matrix. The generalized U-matrix defines the visualization of a topographic map depicting the misrepresentations of projected points with regards to a given dimensionality reduction method and the dataset.•The topographic map provides accurate information about the high-dimensional distance and density based structures of high-dimensional data if an appropriate dimensionality reduction method is selected.•The topographic map can uncover the absence of distance-based structures.•The topographic map reveals the number of clusters in a dataset as the number of valleys.
Highlights
Restricting the Output space of a Projection method results in projection errors because the twodimensional similarities in the scatter plot cannot coercively represent high-dimensional distances. This is stated by the Johnson–Lindenstrauss lemma [5] and visualized in two examples
Example 1: Linear Separable Structures Example 2: Linear Non-Separable Structures Example 3: High-Dimensional Structures versus the Absence of Such Structures
Summary
Uncovering High-dimensional Structures of Projections from Dimensionality Reduction Methods. Of Hematology, Oncology and Immunology, Philipps-University of Marburg, Baldingerstraße, D-35043 Marburg b Databionics Research Group, Philipps-University of Marburg, Hans-Meerwein-Straße 6, Marburg D-35032, Germany abstract. Method name: Topographic Map Generation Using the Generalized Umatrix Keywords: Dimensionality reduction, Projection methods, Data visualization, Unsupervised neural networks, Self-organizing maps Article history: Received 22 January 2020; Accepted 4 October 2020; Available online 10 October 2020.
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