Compress Then Test: Powerful Kernel Testing in Near-linear Time
Carles Domingo-Enrich, Raaz Dwivedi, Lester Mackey
Abstract
Kernel two-sample testing provides a powerful framework for distinguishing any pair of distributions based on n sample points. However, existing kernel tests either run in $n^2$ time or sacrifice undue power to improve runtime. To address these shortcomings, we introduce Compress Then Test (CTT), a new framework for high-powered kernel testing based on sample compression. CTT cheaply approximates an expensive test by compressing each n point sample into a small but provably high-fidelity coreset. For standard kernels and subexponential distributions, CTT inherits the statistical behavior of a quadratic-time test—recovering the same optimal detection boundary—while running in near-linear time. We couple these advances with cheaper permutation testing, justified by new power analyses; improved time-vs.-quality guarantees for low-rank approximation; and a fast aggregation procedure for identifying especially discriminating kernels. In our experiments with real and simulated data, CTT and its extensions provide 20–200x speed-ups over state-of-the-art approximate MMD tests with no loss of power.
BibTeX
@InProceedings{pmlr-v206-domingo-enrich23a,
title = {Compress Then Test: Powerful Kernel Testing in Near-linear Time},
author = {Domingo-Enrich, Carles and Dwivedi, Raaz and Mackey, Lester},
booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics},
pages = {1174--1218},
year = {2023},
editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem},
volume = {206},
series = {Proceedings of Machine Learning Research},
month = {25--27 Apr},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v206/domingo-enrich23a/domingo-enrich23a.pdf},
url = {https://proceedings.mlr.press/v206/domingo-enrich23a.html},
abstract = {Kernel two-sample testing provides a powerful framework for distinguishing any pair of distributions based on n sample points. However, existing kernel tests either run in $n^2$ time or sacrifice undue power to improve runtime. To address these shortcomings, we introduce Compress Then Test (CTT), a new framework for high-powered kernel testing based on sample compression. CTT cheaply approximates an expensive test by compressing each n point sample into a small but provably high-fidelity coreset. For standard kernels and subexponential distributions, CTT inherits the statistical behavior of a quadratic-time test—recovering the same optimal detection boundary—while running in near-linear time. We couple these advances with cheaper permutation testing, justified by new power analyses; improved time-vs.-quality guarantees for low-rank approximation; and a fast aggregation procedure for identifying especially discriminating kernels. In our experiments with real and simulated data, CTT and its extensions provide 20–200x speed-ups over state-of-the-art approximate MMD tests with no loss of power.}
}