# Principal Differences Analysis: Interpretable Characterization of Differences between Distributions

arXiv: https://arxiv.org/abs/1510.08956 · HTML: https://mlsanity.com/paper/1510.08956/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2015.

## Built on by (most cited first, 6 of 6)

- [Approximating 1-Wasserstein Distance with Trees](https://mlsanity.com/paper/2206.12116/) (2022)
- [Variable Selection for Kernel Two-Sample Tests](https://mlsanity.com/paper/2302.07415/) (2023)
- [A Manifold Two-Sample Test Study: Integral Probability Metric with Neural Networks](https://mlsanity.com/paper/2205.02043/) (2022)
- [Variable Selection in Maximum Mean Discrepancy for Interpretable Distribution Comparison](https://mlsanity.com/paper/2311.01537/) (2023)
- [Integral Probability Metrics Meet Neural Networks: The Radon-Kolmogorov-Smirnov Test](https://mlsanity.com/paper/2309.02422/) (2023)
- [An Empirical Study of Self-supervised Learning with Wasserstein Distance](https://mlsanity.com/paper/2310.10143/) (2023)

## Abstract (arXiv)

We introduce principal differences analysis (PDA) for analyzing differences between high-dimensional distributions. The method operates by finding the projection that maximizes the Wasserstein divergence between the resulting univariate populations. Relying on the Cramer-Wold device, it requires no assumptions about the form of the underlying distributions, nor the nature of their inter-class differences. A sparse variant of the method is introduced to identify features responsible for the differences. We provide algorithms for both the original minimax formulation as well as its semidefinite relaxation. In addition to deriving some convergence results, we illustrate how the approach may be applied to identify differences between cell populations in the somatosensory cortex and hippocampus as manifested by single cell RNA-seq. Our broader framework extends beyond the specific choice of Wasserstein divergence.