Implicit Multidimensional Projection of Local Subspaces

TL;DR

This paper introduces a visualization technique that analyzes local subspaces in multidimensional data projections using implicit function differentiation, providing deeper insights into data structure beyond traditional point-based methods.

Contribution

It presents a novel method for visualizing local subspace shapes and directions in multidimensional projections, enhancing understanding of global data structure.

Findings

Effective visualization of local subspaces using glyphs.

Accurate and efficient vector transformation via implicit differentiation.

Demonstrated usefulness on high-dimensional benchmark datasets.

Abstract

We propose a visualization method to understand the effect of multidimensional projection on local subspaces, using implicit function differentiation. Here, we understand the local subspace as the multidimensional local neighborhood of data points. Existing methods focus on the projection of multidimensional data points, and the neighborhood information is ignored. Our method is able to analyze the shape and directional information of the local subspace to gain more insights into the global structure of the data through the perception of local structures. Local subspaces are fitted by multidimensional ellipses that are spanned by basis vectors. An accurate and efficient vector transformation method is proposed based on analytical differentiation of multidimensional projections formulated as implicit functions. The results are visualized as glyphs and analyzed using a full set of specifically-designed interactions supported in our efficient web-based visualization tool. The usefulness of our method is demonstrated using various multi- and high-dimensional benchmark datasets. Our implicit differentiation vector transformation is evaluated through numerical comparisons; the overall method is evaluated through exploration examples and use cases.