The Shape of Data: Intrinsic Distance for Data Distributions

TL;DR

IMD is a new metric for comparing data distributions based on their geometry:

• Fast — $$O(n)$$ in the number of data samples
• Extrinsic — does not rely on any positional information
• Multiscale — approximates and compares all moments of distributions

The ability to represent and compare machine learning models is crucial in order to quantify subtle model changes, evaluate generative models, and gather insights on neural network architectures. Existing techniques for comparing data distributions focus on global data properties such as mean and covariance. We develop a first-of-its-kind intrinsic and multi-scale method for characterizing and comparing data manifolds, using a lower-bound of the spectral variant of the Gromov-Wasserstein inter-manifold distance, which compares all data moments.