Geodesic-Based Data Visualization Distortion for Local Neighborhood Emphasis
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Solution Overview
Problem
Existing machine learning techniques for data visualization, such as Isomap and t-SNE, often distort representations when conditions are not met, and are expensive to implement, especially as the number of data points increases, limiting the understanding of local neighborhoods in high-dimensional data spaces.
Innovation Solution
The method involves constructing geodesics from a focus point to visualize data in a lower-dimensional space using primary eigenvectors, creating a vector field, and applying a positive semidefinite differential operator to emphasize local neighborhoods while allowing distortion of distant regions, thereby efficiently reducing data dimensions for visualization.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If machine learning techniques such as Isomap or t-SNE are used to reduce dimensions and visualize data, then understanding of local neighborhoods can be improved, but computational expense increases significantly as the number of data points increases
Solution Approach 1:
The patent segments the data visualization problem by identifying a focus point and constructing geodesics from that single point to all other data points, rather than computing relationships between all pairs of points. This segmentation reduces the computational scope while maintaining accurate local neighborhood representation around the focus point.
Solution Approach 2:
The patent applies local quality by accurately reconstructing only the local neighborhood around the focus point using geodesic-based dimensionality reduction, while allowing distant regions to be distorted or omitted. This selective approach maintains measurement precision where needed while reducing overall computational expense.
2Manufacturing precision
If techniques like Isomap are used to preserve isometries, then local structure is maintained, but the method is expensive to implement particularly as data points increase
Solution Approach 1:
The patent extracts the essential geometric information needed for accurate local representation by computing geodesics from a single focus point, rather than implementing the full Isomap algorithm which requires computing and maintaining complex manifold structures across all data points. This extraction maintains isometry preservation locally while reducing implementation complexity.
3Productivity
If data is visualized in lower-dimensional space, then visualization efficiency is improved, but distortion of distant neighborhoods increases
Solution Approach 1:
The patent applies local quality by making the visualization accurate and undistorted in the local neighborhood of the focus point, while accepting that distant regions may be distorted or omitted. This selective approach prioritizes information preservation where it matters most for the user's area of interest while maintaining visualization efficiency.
Data Source
AI summary
This application is directed to modifying display of data visualizations. A computer system visualizes on its display a set of data in a first data space. A focus point is identified in the first data space and used as a starting point for constructing a set of geodesics, each of which is configured to connect the focus point to a respective data point in the first data space. A plurality of primary eigenvectors are determined in the first data space and they are used to create a second data space having fewer dimensions than the first data space. For each data point, the computer system determines a plurality of projection values associated with the primary eigenvectors. The set of data is re-visualized in the second data space created using the projection values of each data point.