A generalized diffusion frame for parsimonious representation of functions on data defined manifolds

Area of Science:

• Machine Learning
• Data Analysis
• Functional Analysis

Background:

• Semi-supervised learning often models high-dimensional data as low-dimensional manifolds.
• Projections onto eigenspaces of diffusion maps are standard techniques.
• Diffusion wavelets offer approximate projections using heat kernel iterates.

Purpose of the Study:

• To generalize diffusion map projections to quasi-metric measure spaces.
• To develop a function representation method using general operators and their iterates.
• To characterize local function smoothness and enable automatic feature detection.

Main Methods:

• Consideration of a quasi-metric measure space X and a general operator T.
• Development of function representations as linear combinations of operator iterates.
• Analysis of local norm behavior for smoothness characterization.

Main Results:

• A novel representation of functions obviating eigenvalue/eigenfunction computation.
• Characterization of local function smoothness via representation terms.
• Demonstration of automatic feature detection and parsimonious representations.

Conclusions:

• The developed theory provides a general framework for function representation in learning.
• The method offers advantages in computational efficiency and data representation.
• Applicable to smooth compact manifolds with operators commuting with the heat operator.