Gaussian determinantal processes: A new model for directionality in data.

Area of Science:

• Machine Learning
• Statistical Modeling
• Data Analysis

Background:

• Determinantal point processes (DPPs) are increasingly used for modeling negative dependence or repulsion in data.
• A comprehensive parametric statistical theory for DPPs remains underdeveloped.
• Understanding interpretable parametric modulation in DPPs is crucial for broader applications.

Purpose of the Study:

• To investigate a parametric family of Gaussian DPPs with interpretable parameter modulation.
• To analyze the impact of parameter modulation on the directional repulsion structure of observed points.
• To establish Gaussian DPPs as a viable alternative to principal component analysis (PCA) for dimension reduction.

Main Methods:

• Developed a parametric family of Gaussian DPPs.
• Analyzed the directional effects of parameter modulation on point repulsion.
• Utilized a spiked model, analogous to covariance matrix analysis, for statistical investigation.
• Compared the proposed DPP model with PCA for dimension reduction.

Main Results:

• Parameter modulation introduces directionality in the repulsion structure of Gaussian DPPs.
• Principal directions of repulsion correspond to maximal or long-ranged dependency.
• The proposed Gaussian DPP model serves as an effective dimension reduction tool, favoring directions of maximal data spread.
• Theoretical analysis provides a framework for studying PCA-like phenomena in DPPs.

Conclusions:

• Parametric Gaussian DPPs offer a novel approach to modeling directional repulsion in data.
• This model provides a statistically grounded alternative to PCA, particularly for dimension reduction.
• Further research in random matrix theory and stochastic geometry is warranted based on these findings.