Incremental isometric embedding of high-dimensional data using connected neighborhood graphs.

Area of Science:

• Computational Mathematics
• Machine Learning
• Data Science

Background:

• Nonlinear dimensionality reduction methods often rely on neighborhood graphs to embed high-dimensional data onto a lower-dimensional manifold.
• Incremental updates to these graphs are crucial for handling streaming data, which can be under-sampled or unevenly distributed.
• Existing methods face challenges in efficiently updating neighborhood structures and shortest path distances in dynamic datasets.

Purpose of the Study:

• To develop an incremental algorithm for dimensionality reduction that can efficiently update neighborhood graphs and shortest path distances.
• To enhance the Isomap algorithm to handle under-sampled or unevenly distributed data streams.
• To maintain accurate low-dimensional representations of high-dimensional data in dynamic environments.

Main Methods:

• Algorithms for updating k-edge-connected and k-connected neighborhood graphs upon data point addition or deletion.
• A method for efficiently updating all-pair shortest distances on the dynamic neighborhood graph.
• Integration of these updates with incremental classical multidimensional scaling (MDS) using iterative subspace approximation.

Main Results:

• The proposed incremental Isomap algorithm demonstrates efficiency in updating low-dimensional data configurations.
• The method successfully handles under-sampled and unevenly distributed data, preserving the underlying data structure.
• Experimental results on synthetic and real-world datasets validate the algorithm's performance and robustness.

Conclusions:

• The developed incremental Isomap approach provides an effective solution for dimensionality reduction in dynamic data streams.
• The enhancements allow for robust handling of data distribution variations, crucial for real-world applications.
• The algorithm offers an efficient and accurate method for maintaining manifold structures during incremental learning.