Quantum mean embedding of probability distributions
The kernel mean embedding of probability distributions is commonly used in machine learning as an injective mapping from distributions to functions in an infinite-dimensional Hilbert space. It allows us, for example, to define a distance measure between probability distributions, called the maximum...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
American Physical Society
2019-12-01
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| Series: | Physical Review Research |
| Online Access: | http://doi.org/10.1103/PhysRevResearch.1.033159 |
| _version_ | 1827286292506869760 |
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| author | Jonas M. Kübler Krikamol Muandet Bernhard Schölkopf |
| author_facet | Jonas M. Kübler Krikamol Muandet Bernhard Schölkopf |
| author_sort | Jonas M. Kübler |
| collection | DOAJ |
| description | The kernel mean embedding of probability distributions is commonly used in machine learning as an injective mapping from distributions to functions in an infinite-dimensional Hilbert space. It allows us, for example, to define a distance measure between probability distributions, called the maximum mean discrepancy. In this work, we propose to represent probability distributions in a pure quantum state of a system that is described by an infinite-dimensional Hilbert space and prove that the representation is unique if the corresponding kernel function is c_{0} universal. This enables us to work with an explicit representation of the mean embedding, whereas classically one can only work implicitly with an infinite-dimensional Hilbert space through the use of the kernel trick. We show how this explicit representation can speed up methods that rely on inner products of mean embeddings and discuss the theoretical and experimental challenges that need to be solved in order to achieve these speedups. |
| first_indexed | 2024-04-24T10:28:55Z |
| format | Article |
| id | doaj.art-2e2f883598bc4160963daa327e9a7c68 |
| institution | Directory Open Access Journal |
| issn | 2643-1564 |
| language | English |
| last_indexed | 2024-04-24T10:28:55Z |
| publishDate | 2019-12-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | Physical Review Research |
| spelling | doaj.art-2e2f883598bc4160963daa327e9a7c682024-04-12T16:47:31ZengAmerican Physical SocietyPhysical Review Research2643-15642019-12-011303315910.1103/PhysRevResearch.1.033159Quantum mean embedding of probability distributionsJonas M. KüblerKrikamol MuandetBernhard SchölkopfThe kernel mean embedding of probability distributions is commonly used in machine learning as an injective mapping from distributions to functions in an infinite-dimensional Hilbert space. It allows us, for example, to define a distance measure between probability distributions, called the maximum mean discrepancy. In this work, we propose to represent probability distributions in a pure quantum state of a system that is described by an infinite-dimensional Hilbert space and prove that the representation is unique if the corresponding kernel function is c_{0} universal. This enables us to work with an explicit representation of the mean embedding, whereas classically one can only work implicitly with an infinite-dimensional Hilbert space through the use of the kernel trick. We show how this explicit representation can speed up methods that rely on inner products of mean embeddings and discuss the theoretical and experimental challenges that need to be solved in order to achieve these speedups.http://doi.org/10.1103/PhysRevResearch.1.033159 |
| spellingShingle | Jonas M. Kübler Krikamol Muandet Bernhard Schölkopf Quantum mean embedding of probability distributions Physical Review Research |
| title | Quantum mean embedding of probability distributions |
| title_full | Quantum mean embedding of probability distributions |
| title_fullStr | Quantum mean embedding of probability distributions |
| title_full_unstemmed | Quantum mean embedding of probability distributions |
| title_short | Quantum mean embedding of probability distributions |
| title_sort | quantum mean embedding of probability distributions |
| url | http://doi.org/10.1103/PhysRevResearch.1.033159 |
| work_keys_str_mv | AT jonasmkubler quantummeanembeddingofprobabilitydistributions AT krikamolmuandet quantummeanembeddingofprobabilitydistributions AT bernhardscholkopf quantummeanembeddingofprobabilitydistributions |