Probabilistic Geometric Principal Component Analysis with application to neural data.

Area of Science:

• Neuroscience
• Data Science
• Machine Learning

Background:

• Dimensionality reduction is crucial in science, particularly neuroscience.
• Probabilistic Principal Component Analysis (PPCA) offers a probabilistic approach but is limited to linear models and Euclidean spaces.
• Neuroscience data often exhibit nonlinear manifold structures, not well-described by linear methods.

Purpose of the Study:

• To develop Probabilistic Geometric Principal Component Analysis (PGPCA) for dimensionality reduction on nonlinear manifold data.
• To generalize PPCA by incorporating nonlinear manifold geometry for improved data description.
• To provide a method that can learn data distributions both on and around a manifold.

Main Methods:

• Developed Probabilistic Geometric Principal Component Analysis (PGPCA), a novel dimensionality reduction algorithm.
• Incorporated knowledge of a given nonlinear manifold, fitted from the data.
• Derived a geometric coordinate system alongside the Euclidean one to capture deviations and noise.
• Developed a data-driven Expectation-Maximization (EM) algorithm for PGPCA parameter learning.

Main Results:

• PGPCA effectively models data distributions around specified nonlinear manifolds.
• PGPCA outperforms standard PPCA on data with nonlinear manifold structures.
• Demonstrated PGPCA's ability to capture deviations from the manifold and noise using a geometric coordinate system.
• Showcased PGPCA's capability to perform dimensionality reduction and learn distributions on and around the manifold.

Conclusions:

• PGPCA generalizes PPCA, offering superior performance for high-dimensional data residing on nonlinear manifolds, especially in neuroscience.
• The derived geometric coordinate system enhances the description of data deviations and noise.
• PGPCA provides a valuable tool for analyzing complex neural data, improving dimensionality reduction efficacy.