Hilbert Curve Projection Distance for Distribution Comparison.

Area of Science:

• Machine Learning
• Probability Theory
• Data Analysis

Background:

• Distribution comparison is crucial for machine learning tasks like classification and generative modeling.
• Existing metrics can suffer from high computational complexity or limitations in high-dimensional spaces.

Purpose of the Study:

• To propose a novel, low-complexity metric for measuring the distance between probability distributions.
• To introduce the Hilbert curve projection (HCP) distance and analyze its properties.

Main Methods:

• Projecting high-dimensional probability distributions using the Hilbert curve to create a coupling.
• Calculating the transport distance in the original space based on the derived coupling.
• Developing variants using subspace projections to mitigate the curse of dimensionality.

Main Results:

• The Hilbert curve projection (HCP) distance is demonstrated to be a proper metric for probability measures with bounded supports.
• The empirical HCP distance converges to its population counterpart at a rate of O(n^{-1/2max{d,p}}).
• HCP distance variants effectively address the curse of dimensionality.

Conclusions:

• HCP distance offers an effective, low-complexity alternative to Wasserstein distance.
• It overcomes limitations of sliced Wasserstein distance, showing promise for practical applications.