Fast deterministic approximation of symmetric indefinite kernel matrices with high dimensional datasets Article Swipe
Difeng Cai
,
J.I. Nagy
,
Yuanzhe Xi
·
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2102.05215
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2102.05215
Kernel methods are used frequently in various applications of machine learning. For large-scale high dimensional applications, the success of kernel methods hinges on the ability to operate certain large dense kernel matrix K. An enormous amount of literature has been devoted to the study of symmetric positive semi-definite (SPSD) kernels, where Nystrom methods compute a low-rank approximation to the kernel matrix via choosing landmark points. In this paper, we study the Nystrom method for approximating both symmetric indefinite kernel matrices as well SPSD ones. We first develop a theoretical framework for general symmetric kernel matrices, which provides a theoretical guidance for the selection of landmark points. We then leverage discrepancy theory to propose the anchor net method for computing accurate Nystrom approximations with optimal complexity. The anchor net method operates entirely on the dataset without requiring the access to $K$ or its matrix-vector product. Results on various types of kernels (both indefinite and SPSD ones) and machine learning datasets demonstrate that the new method achieves better accuracy and stability with lower computational cost compared to the state-of-the-art Nystrom methods.
Related Topics
Concepts
Kernel (algebra)
Leverage (statistics)
Computer science
Nyström method
Kernel method
Algorithm
Matrix (chemical analysis)
Positive-definite matrix
Applied mathematics
Mathematical optimization
Mathematics
Support vector machine
Artificial intelligence
Integral equation
Mathematical analysis
Combinatorics
Eigenvalues and eigenvectors
Materials science
Quantum mechanics
Composite material
Physics
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2102.05215
- https://arxiv.org/pdf/2102.05215
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4310233492
All OpenAlex metadata
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- OpenAlex ID
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https://openalex.org/W4310233492Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2102.05215Digital Object Identifier
- Title
-
Fast deterministic approximation of symmetric indefinite kernel matrices with high dimensional datasetsWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
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2021Year of publication
- Publication date
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2021-02-10Full publication date if available
- Authors
-
Difeng Cai, J.I. Nagy, Yuanzhe XiList of authors in order
- Landing page
-
https://arxiv.org/abs/2102.05215Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2102.05215Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/2102.05215Direct OA link when available
- Concepts
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Kernel (algebra), Leverage (statistics), Computer science, Nyström method, Kernel method, Algorithm, Matrix (chemical analysis), Positive-definite matrix, Applied mathematics, Mathematical optimization, Mathematics, Support vector machine, Artificial intelligence, Integral equation, Mathematical analysis, Combinatorics, Eigenvalues and eigenvectors, Materials science, Quantum mechanics, Composite material, PhysicsTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
- Related works (count)
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.Nystrom | 51, 71, 120, 177 |
| abstract_inverted_index.Results | 144 |
| abstract_inverted_index.ability | 24 |
| abstract_inverted_index.certain | 27 |
| abstract_inverted_index.compute | 53 |
| abstract_inverted_index.dataset | 133 |
| abstract_inverted_index.develop | 86 |
| abstract_inverted_index.devoted | 40 |
| abstract_inverted_index.general | 91 |
| abstract_inverted_index.kernels | 149 |
| abstract_inverted_index.machine | 9, 156 |
| abstract_inverted_index.methods | 1, 20, 52 |
| abstract_inverted_index.operate | 26 |
| abstract_inverted_index.optimal | 123 |
| abstract_inverted_index.points. | 64, 105 |
| abstract_inverted_index.propose | 112 |
| abstract_inverted_index.success | 17 |
| abstract_inverted_index.various | 6, 146 |
| abstract_inverted_index.without | 134 |
| abstract_inverted_index.accuracy | 166 |
| abstract_inverted_index.accurate | 119 |
| abstract_inverted_index.achieves | 164 |
| abstract_inverted_index.choosing | 62 |
| abstract_inverted_index.compared | 173 |
| abstract_inverted_index.datasets | 158 |
| abstract_inverted_index.enormous | 34 |
| abstract_inverted_index.entirely | 130 |
| abstract_inverted_index.guidance | 99 |
| abstract_inverted_index.kernels, | 49 |
| abstract_inverted_index.landmark | 63, 104 |
| abstract_inverted_index.learning | 157 |
| abstract_inverted_index.leverage | 108 |
| abstract_inverted_index.low-rank | 55 |
| abstract_inverted_index.matrices | 79 |
| abstract_inverted_index.methods. | 178 |
| abstract_inverted_index.operates | 129 |
| abstract_inverted_index.positive | 46 |
| abstract_inverted_index.product. | 143 |
| abstract_inverted_index.provides | 96 |
| abstract_inverted_index.computing | 118 |
| abstract_inverted_index.framework | 89 |
| abstract_inverted_index.learning. | 10 |
| abstract_inverted_index.matrices, | 94 |
| abstract_inverted_index.requiring | 135 |
| abstract_inverted_index.selection | 102 |
| abstract_inverted_index.stability | 168 |
| abstract_inverted_index.symmetric | 45, 76, 92 |
| abstract_inverted_index.frequently | 4 |
| abstract_inverted_index.indefinite | 77, 151 |
| abstract_inverted_index.literature | 37 |
| abstract_inverted_index.complexity. | 124 |
| abstract_inverted_index.demonstrate | 159 |
| abstract_inverted_index.dimensional | 14 |
| abstract_inverted_index.discrepancy | 109 |
| abstract_inverted_index.large-scale | 12 |
| abstract_inverted_index.theoretical | 88, 98 |
| abstract_inverted_index.applications | 7 |
| abstract_inverted_index.applications, | 15 |
| abstract_inverted_index.approximating | 74 |
| abstract_inverted_index.approximation | 56 |
| abstract_inverted_index.computational | 171 |
| abstract_inverted_index.matrix-vector | 142 |
| abstract_inverted_index.semi-definite | 47 |
| abstract_inverted_index.approximations | 121 |
| abstract_inverted_index.state-of-the-art | 176 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 3 |
| citation_normalized_percentile |