Sketch Kernel Ridge Regression Using Circulant Matrix: Algorithm and Theory
Kernel ridge regression (KRR) is a powerful method for nonparametric regression. The time and space complexity of computing the KRR estimate directly are <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(n^{3}) </tex-math></inline-formula> and <inline-form...
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| Vydáno v: | IEEE transaction on neural networks and learning systems Ročník 31; číslo 9; s. 3512 - 3524 |
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| Jazyk: | angličtina |
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IEEE
01.09.2020
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 2162-237X, 2162-2388, 2162-2388 |
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| Abstract | Kernel ridge regression (KRR) is a powerful method for nonparametric regression. The time and space complexity of computing the KRR estimate directly are <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(n^{3}) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(n^{2}) </tex-math></inline-formula>, respectively, which are prohibitive for large-scale data sets, where <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> is the number of data. In this article, we propose a novel random sketch technique based on the circulant matrix that achieves savings in storage space and accelerates the solution of the KRR approximation. The circulant matrix has the following advantages: It can save time complexity by using the fast Fourier transform (FFT) to compute the product of matrix and vector, its space complexity is linear, and the circulant matrix, whose entries in the first column are independent of each other and obey the Gaussian distribution, is almost as effective as the i.i.d. Gaussian random matrix for approximating KRR. Combining the characteristics of the circulant matrix and our careful design, theoretical analysis and experimental results demonstrate that our proposed sketch method, making the estimate kernel methods scalable and practical for large-scale data problems, outperforms the state-of-the-art KRR estimates in time complexity while retaining similar accuracies. Meanwhile, our sketch method provides the theoretical bound that keeps the optimal convergence rate for approximating KRR. |
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| AbstractList | Kernel ridge regression (KRR) is a powerful method for nonparametric regression. The time and space complexity of computing the KRR estimate directly are [Formula Omitted] and [Formula Omitted], respectively, which are prohibitive for large-scale data sets, where [Formula Omitted] is the number of data. In this article, we propose a novel random sketch technique based on the circulant matrix that achieves savings in storage space and accelerates the solution of the KRR approximation. The circulant matrix has the following advantages: It can save time complexity by using the fast Fourier transform (FFT) to compute the product of matrix and vector, its space complexity is linear, and the circulant matrix, whose entries in the first column are independent of each other and obey the Gaussian distribution, is almost as effective as the i.i.d. Gaussian random matrix for approximating KRR. Combining the characteristics of the circulant matrix and our careful design, theoretical analysis and experimental results demonstrate that our proposed sketch method, making the estimate kernel methods scalable and practical for large-scale data problems, outperforms the state-of-the-art KRR estimates in time complexity while retaining similar accuracies. Meanwhile, our sketch method provides the theoretical bound that keeps the optimal convergence rate for approximating KRR. Kernel ridge regression (KRR) is a powerful method for nonparametric regression. The time and space complexity of computing the KRR estimate directly are O(n3) and O(n2) , respectively, which are prohibitive for large-scale data sets, where n is the number of data. In this article, we propose a novel random sketch technique based on the circulant matrix that achieves savings in storage space and accelerates the solution of the KRR approximation. The circulant matrix has the following advantages: It can save time complexity by using the fast Fourier transform (FFT) to compute the product of matrix and vector, its space complexity is linear, and the circulant matrix, whose entries in the first column are independent of each other and obey the Gaussian distribution, is almost as effective as the i.i.d. Gaussian random matrix for approximating KRR. Combining the characteristics of the circulant matrix and our careful design, theoretical analysis and experimental results demonstrate that our proposed sketch method, making the estimate kernel methods scalable and practical for large-scale data problems, outperforms the state-of-the-art KRR estimates in time complexity while retaining similar accuracies. Meanwhile, our sketch method provides the theoretical bound that keeps the optimal convergence rate for approximating KRR.Kernel ridge regression (KRR) is a powerful method for nonparametric regression. The time and space complexity of computing the KRR estimate directly are O(n3) and O(n2) , respectively, which are prohibitive for large-scale data sets, where n is the number of data. In this article, we propose a novel random sketch technique based on the circulant matrix that achieves savings in storage space and accelerates the solution of the KRR approximation. The circulant matrix has the following advantages: It can save time complexity by using the fast Fourier transform (FFT) to compute the product of matrix and vector, its space complexity is linear, and the circulant matrix, whose entries in the first column are independent of each other and obey the Gaussian distribution, is almost as effective as the i.i.d. Gaussian random matrix for approximating KRR. Combining the characteristics of the circulant matrix and our careful design, theoretical analysis and experimental results demonstrate that our proposed sketch method, making the estimate kernel methods scalable and practical for large-scale data problems, outperforms the state-of-the-art KRR estimates in time complexity while retaining similar accuracies. Meanwhile, our sketch method provides the theoretical bound that keeps the optimal convergence rate for approximating KRR. Kernel ridge regression (KRR) is a powerful method for nonparametric regression. The time and space complexity of computing the KRR estimate directly are <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(n^{3}) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(n^{2}) </tex-math></inline-formula>, respectively, which are prohibitive for large-scale data sets, where <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> is the number of data. In this article, we propose a novel random sketch technique based on the circulant matrix that achieves savings in storage space and accelerates the solution of the KRR approximation. The circulant matrix has the following advantages: It can save time complexity by using the fast Fourier transform (FFT) to compute the product of matrix and vector, its space complexity is linear, and the circulant matrix, whose entries in the first column are independent of each other and obey the Gaussian distribution, is almost as effective as the i.i.d. Gaussian random matrix for approximating KRR. Combining the characteristics of the circulant matrix and our careful design, theoretical analysis and experimental results demonstrate that our proposed sketch method, making the estimate kernel methods scalable and practical for large-scale data problems, outperforms the state-of-the-art KRR estimates in time complexity while retaining similar accuracies. Meanwhile, our sketch method provides the theoretical bound that keeps the optimal convergence rate for approximating KRR. |
| Author | Wang, Weiping Yin, Rong Meng, Dan Liu, Yong |
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| Snippet | Kernel ridge regression (KRR) is a powerful method for nonparametric regression. The time and space complexity of computing the KRR estimate directly are... Kernel ridge regression (KRR) is a powerful method for nonparametric regression. The time and space complexity of computing the KRR estimate directly are O(n3)... |
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| SubjectTerms | Acceleration Algorithms Approximation Approximation algorithms Circulant matrix Complexity Convergence Fast Fourier transformations Fourier analysis Fourier transforms Kernel kernel ridge regression (KRR) Kernels large scale Mathematical analysis Matrix algebra Matrix methods Normal distribution random sketch Regression analysis Statistical analysis Theoretical analysis Time complexity Training |
| Title | Sketch Kernel Ridge Regression Using Circulant Matrix: Algorithm and Theory |
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