GhostUMAP2: Measuring and Analyzing $(r,d)$-Stability of UMAP

Area of Science:

• Data Science
• Machine Learning
• Dimensionality Reduction

Background:

• Uniform Manifold Approximation and Projection (UMAP) is a popular dimensionality reduction technique.
• The stochastic nature of UMAP's optimization can lead to unstable projection results.
• The impact of this stochasticity on UMAP's reliability is not well understood.

Purpose of the Study:

• To introduce a novel framework, (r,d)-stability, for analyzing the stability of UMAP projections.
• To quantify the impact of stochastic elements, such as initial positions and negative sampling, on UMAP results.
• To provide tools and guidelines for assessing and improving UMAP projection stability.

Main Methods:

• Introduction of 'ghosts' (duplicate data points) to represent potential positional variations due to stochasticity.
• Definition of (r,d)-stability based on the confinement of ghost projections within a specified radius.
• Development of an adaptive dropping scheme for efficient computation of ghost projections.

Main Results:

• The proposed (r,d)-stability framework effectively analyzes the impact of stochasticity in UMAP.
• An adaptive dropping scheme reduces runtime by up to 60% while preserving ~90% of unstable points.
• A visualization tool facilitates interactive exploration of data point projection stability.

Conclusions:

• The (r,d)-stability framework provides a robust method for evaluating UMAP projection reliability.
• The adaptive dropping scheme offers significant computational efficiency gains.
• The developed tools and guidelines aid researchers in achieving more stable and interpretable UMAP results.