Uncovering High-dimensional Structures of Projections from Dimensionality Reduction Methods.

Area of Science:

• Data Visualization
• Machine Learning
• Information Theory

Background:

• Dimensionality reduction techniques, like scatter plots, transform high-dimensional data into lower dimensions for visualization.
• The Johnson-Lindenstrauss lemma highlights limitations in preserving high-dimensional structures in low-dimensional projections.
• Existing methods struggle to accurately represent complex data relationships in 2D scatter plots.

Purpose of the Study:

• To develop a novel visualization method for assessing the accuracy of dimensionality reduction projections.
• To introduce a generalized U-matrix for creating topographic maps of data misrepresentations.
• To enhance the understanding of high-dimensional data structures and their preservation in lower dimensions.

Main Methods:

• Utilized a simplified emergent self-organizing map (SOM) approach.
• Computed the generalized U-matrix using projected data points and the original dataset.
• Developed a topographic map to visualize projection misrepresentations.

Main Results:

• The topographic map accurately reflects high-dimensional distance and density-based structures when an appropriate dimensionality reduction method is used.
• The method can identify datasets lacking discernible distance-based structures.
• The number of valleys in the topographic map corresponds to the number of clusters within the dataset.

Conclusions:

• The developed topographic map offers a reliable tool for evaluating dimensionality reduction techniques.
• This visualization aids in uncovering hidden structures and determining cluster counts in high-dimensional data.
• The approach addresses the limitations of traditional scatter plots in representing complex data relationships.