On the relation of slow feature analysis and Laplacian eigenmaps

Area of Science:

• Machine Learning
• Computational Neuroscience
• Data Science

Background:

• Laplacian eigenmaps (LEMs) are increasingly used for nonlinear dimensionality reduction.
• Applications include spectral clustering, semisupervised learning, and reinforcement learning.
• Slow feature analysis (SFA) is a biologically inspired algorithm for learning invariant representations.

Purpose of the Study:

• To establish a close relationship between Laplacian eigenmaps (LEMs) and slow feature analysis (SFA).
• To interpret SFA as a function approximation of LEMs.
• To propose a generalized SFA applicable to spectral clustering.

Main Methods:

• Interpreting SFA as a function approximation of LEMs.
• Utilizing the temporal structure of data to implicitly define neighborhoods for LEMs.
• Generalizing SFA for arbitrary neighborhood relations.

Main Results:

• Demonstrated that SFA can be viewed as an approximation of LEMs.
• Showcased the implicit definition of topological neighborhoods in LEMs through data's temporal structure.
• Successfully applied generalized SFA to spectral clustering tasks.

Conclusions:

• SFA and LEMs share a fundamental connection, offering a unifying perspective.
• Generalized SFA expands the applicability of SFA, particularly for spectral clustering.
• This work bridges unsupervised learning algorithms with applications in dimensionality reduction and clustering.