Kernel embeddings and the separation of measure phenomenon.
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| Title: | Kernel embeddings and the separation of measure phenomenon. |
|---|---|
| Authors: | Santoro, Leonardo V.1, Waghmare, Kartik G.2, Panaretos, Victor M.1 victor.panaretos@epfl.ch |
| Source: | Proceedings of the National Academy of Sciences of the United States of America. 6/9/2026, Vol. 123 Issue 23, p1-10. 10p. |
| Subjects: | Gaussian measures, Reproducing kernel (Mathematics), Measure theory, Distribution (Probability theory), Hilbert space, Statistical hypothesis testing |
| Abstract: | We prove that kernel covariance embeddings lead to information-theoretically perfect separation of distinct continuous probability distributions. In statistical terms, we establish that testing for the equality of two nonatomic (Borel) probability measures on a locally compact uncountable Polish space is equivalent to testing for the singularity between two centered Gaussian measures on a reproducing kernel Hilbert space. The corresponding Gaussians are defined via the notion of kernel covariance embedding of a probability measure, and the Hilbert space is that generated by the embedding kernel. Distinguishing singular Gaussians is structurally simpler from an informationtheoretic perspective than nonparametric two-sample testing, particularly in complex or high-dimensional domains. This is because singular Gaussians are supported on essentially separate and affine subspaces. Our proof leverages the classical Feldman-Hájek dichotomy, and shows that even a small perturbation of a continuous distribution will be maximally magnified through its Gaussian embedding. This "separation of measure phenomenon" appears to be a blessing of infinite dimensionality, by means of embedding, with the potential to inform the design of efficient inference tools in considerable generality. The elicitation of this phenomenon also appears to crystallize, in a precise and simple mathematical statement, a core mechanism underpinning the empirical effectiveness of kernel methods. [ABSTRACT FROM AUTHOR] |
| Copyright of Proceedings of the National Academy of Sciences of the United States of America is the property of National Academy of Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 194540934 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Kernel embeddings and the separation of measure phenomenon. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Santoro%2C+Leonardo+V%2E%22">Santoro, Leonardo V.</searchLink><relatesTo>1</relatesTo><br /><searchLink fieldCode="AR" term="%22Waghmare%2C+Kartik+G%2E%22">Waghmare, Kartik G.</searchLink><relatesTo>2</relatesTo><br /><searchLink fieldCode="AR" term="%22Panaretos%2C+Victor+M%2E%22">Panaretos, Victor M.</searchLink><relatesTo>1</relatesTo><i> victor.panaretos@epfl.ch</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Proceedings+of+the+National+Academy+of+Sciences+of+the+United+States+of+America%22">Proceedings of the National Academy of Sciences of the United States of America</searchLink>. 6/9/2026, Vol. 123 Issue 23, p1-10. 10p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Gaussian+measures%22">Gaussian measures</searchLink><br /><searchLink fieldCode="DE" term="%22Reproducing+kernel+%28Mathematics%29%22">Reproducing kernel (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Measure+theory%22">Measure theory</searchLink><br /><searchLink fieldCode="DE" term="%22Distribution+%28Probability+theory%29%22">Distribution (Probability theory)</searchLink><br /><searchLink fieldCode="DE" term="%22Hilbert+space%22">Hilbert space</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+hypothesis+testing%22">Statistical hypothesis testing</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We prove that kernel covariance embeddings lead to information-theoretically perfect separation of distinct continuous probability distributions. In statistical terms, we establish that testing for the equality of two nonatomic (Borel) probability measures on a locally compact uncountable Polish space is equivalent to testing for the singularity between two centered Gaussian measures on a reproducing kernel Hilbert space. The corresponding Gaussians are defined via the notion of kernel covariance embedding of a probability measure, and the Hilbert space is that generated by the embedding kernel. Distinguishing singular Gaussians is structurally simpler from an informationtheoretic perspective than nonparametric two-sample testing, particularly in complex or high-dimensional domains. This is because singular Gaussians are supported on essentially separate and affine subspaces. Our proof leverages the classical Feldman-Hájek dichotomy, and shows that even a small perturbation of a continuous distribution will be maximally magnified through its Gaussian embedding. This "separation of measure phenomenon" appears to be a blessing of infinite dimensionality, by means of embedding, with the potential to inform the design of efficient inference tools in considerable generality. The elicitation of this phenomenon also appears to crystallize, in a precise and simple mathematical statement, a core mechanism underpinning the empirical effectiveness of kernel methods. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Proceedings of the National Academy of Sciences of the United States of America is the property of National Academy of Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1073/pnas.2522504123 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 10 StartPage: 1 Subjects: – SubjectFull: Gaussian measures Type: general – SubjectFull: Reproducing kernel (Mathematics) Type: general – SubjectFull: Measure theory Type: general – SubjectFull: Distribution (Probability theory) Type: general – SubjectFull: Hilbert space Type: general – SubjectFull: Statistical hypothesis testing Type: general Titles: – TitleFull: Kernel embeddings and the separation of measure phenomenon. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Santoro, Leonardo V. – PersonEntity: Name: NameFull: Waghmare, Kartik G. – PersonEntity: Name: NameFull: Panaretos, Victor M. IsPartOfRelationships: – BibEntity: Dates: – D: 09 M: 06 Text: 6/9/2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 00278424 Numbering: – Type: volume Value: 123 – Type: issue Value: 23 Titles: – TitleFull: Proceedings of the National Academy of Sciences of the United States of America Type: main |
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