A Kernel Method for the Two-Sample-Problem
Open accessAbstract
We propose a framework for analyzing and comparing distributions, allowing us to design statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS). We present two tests based on large deviation bounds for the test statistic, while a third is based on the asymptotic distribution of this statistic. The test statistic can be computed in quadratic time, although efficient linear time approximations are available. Several classical metrics on distributions are recovered when the function space used to compute the difference in expectations is allowed to be more general (eg. a Banach space). We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.
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After EA, the distances between EEG trials from different subjects become zero because the mean covariance matrices of all subjects are identical, which should generally benefit transfer learning.
“If we view the covariance matrices as the feature embeddings of EEG trials, then, after EA, the distances between EEG trials from different subjects become zero (because the mean covariance matrices of all subjects are identical), which should generally benefit TL.”
From Transfer Learning for Brain–Computer Interfaces: A Euclidean Space Data Alignment Approach (2020) · cites this paper as [10]
MMD represents the distances between different distributions as the distances between their mean embeddings of features; smaller distances indicate distributions are more similar and more suitable for transfer learning.
“MMD represents the distances between different distributions as the distances between their mean embeddings of features. Smaller distances indicate that the distributions are more similar, and hence more suitable for TL.”
From Transfer Learning for Brain–Computer Interfaces: A Euclidean Space Data Alignment Approach (2020) · cites this paper as [10]
The idea of EA can be explained using the concept of maximum mean discrepancy (MMD).
“The idea of EA can also be explained using the concept of maximum mean discrepancy (MMD) [10], [39], widely used in TL.”
From Transfer Learning for Brain–Computer Interfaces: A Euclidean Space Data Alignment Approach (2020) · cites this paper as [10]
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Self-citationTransfer Learning for Brain–Computer Interfaces: A Euclidean Space Data Alignment Approach
He, H., & Wu, D. (2020) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2019.2913914 · Cited by 525 · see citing works
“The idea of EA can also be explained using the concept of maximum mean discrepancy (MMD) [10], [40], widely used in TL.”
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- Authors
- Gretton, A., Borgwardt, K. M., Rasch, M., Schölkopf, B., & Smola, A. J.
- Affiliations
- Max Planck Institute for Biological Cybernetics (DE)
- Ludwig-Maximilians-Universität München (DE)
- Graz University of Technology (AT)
- Australian National University (AU)
- Venue
- Advances in Neural Information Processing Systems 19
- Publisher
- The MIT Press
- Published
- 2007-09-07
- Pages
- 513-520
- OpenAlex
- W2125865219
- Cited by
- 2,167
- References
- 2
- Citations by year
- 2026: 63 · 2025: 181 · 2024: 214 · 2023: 200 · 2022: 174 · 2021: 261 · 2020: 268 · 2019: 185 · 2018: 121 · 2017: 76 · 2016: 85 · 2015: 69 · 2014: 49 · 2013: 55 · 2012: 47
- FWCI
- 90.29
- Type
- book chapter
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- The MIT Press eBooks (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Max Planck Institute for Plasma Physics (submittedVersion)
- UCL Discovery (University College London) (submittedVersion)
- ANU Open Research (Australian National University) (submittedVersion)
- MPG.PuRe (Max Planck Society) (submittedVersion, other-oa)
- MPG.PuRe (Max Planck Society) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Deutsches Krebsforschungszentrum
- Australian Government
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Physical Sciences › Computer Science › Artificial Intelligence › Machine Learning and Algorithms
- Keywords
- two-sample problem, reproducing kernel Hilbert space, kernel methods, large deviation bounds, asymptotic distribution, distribution comparison, statistical tests, Banach spaces, attribute matching
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