Efficient tracking of the dominant eigenspace of a normalized kernel matrix.

Area of Science:

• Machine Learning
• Computational Mathematics

Background:

• Kernel-based methods are powerful for nonlinear problems but computationally intensive for large datasets.
• Calculating the dominant eigenspace of kernel matrices is crucial for these methods.
• Existing techniques are limited to smaller data volumes due to high computational demands.

Purpose of the Study:

• To develop a computationally efficient kernel-based method for large datasets.
• To propose a numerically stable algorithm for tracking the dominant eigenspace of normalized kernel matrices.
• To enable the application of kernel methods to significantly larger data scales.

Main Methods:

• A novel tracking algorithm for the dominant eigenspace of normalized kernel matrices.
• The algorithm employs sequential updating (adding data points) and downdating (removing data points) of the kernel matrix.
• Focus on numerical stability and computational efficiency per iteration.

Main Results:

• The proposed algorithm achieves a very good approximation of the dominant eigenspace.
• Minimal operational and memory requirements per iteration step.
• Demonstrated effectiveness on representative case studies.

Conclusions:

• The developed kernel-based method is highly effective for large datasets.
• The algorithm provides a scalable and efficient solution for dominant eigenspace computation.
• This advancement broadens the applicability of kernel methods in machine learning.