Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic Fidelity
In quantum and quantum-inspired machine learning, a key step is to embed the data in the quantum space known as Hilbert space. Studying quantum kernel function, which defines the distances among the samples in the Hilbert space, belongs to the fundamental topics in this direction. In this work, we p...
| Asıl Yazarlar: | , |
|---|---|
| Materyal Türü: | Makale |
| Dil: | English |
| Baskı/Yayın Bilgisi: |
MDPI AG
2022-03-01
|
| Seri Bilgileri: | Mathematics |
| Konular: | |
| Online Erişim: | https://www.mdpi.com/2227-7390/10/6/940 |
| _version_ | 1827628024945704960 |
|---|---|
| author | Wei-Ming Li Shi-Ju Ran |
| author_facet | Wei-Ming Li Shi-Ju Ran |
| author_sort | Wei-Ming Li |
| collection | DOAJ |
| description | In quantum and quantum-inspired machine learning, a key step is to embed the data in the quantum space known as Hilbert space. Studying quantum kernel function, which defines the distances among the samples in the Hilbert space, belongs to the fundamental topics in this direction. In this work, we propose a tunable quantum-inspired kernel function (QIKF) named rescaled logarithmic fidelity (RLF) and a non-parametric algorithm for the semi-supervised learning in the quantum space. The rescaling takes advantage of the non-linearity of the kernel to tune the mutual distances of samples in the Hilbert space, and meanwhile avoids the exponentially-small fidelities between quantum many-qubit states. Being non-parametric excludes the possible effects from the variational parameters, and evidently demonstrates the properties of the kernel itself. Our results on the hand-written digits (MNIST dataset) and movie reviews (IMDb dataset) support the validity of our method, by comparing with the standard fidelity as the QIKF as well as several well-known non-parametric algorithms (naive Bayes classifiers, <i>k</i>-nearest neighbors, and spectral clustering). High accuracy is demonstrated, particularly for the unsupervised case with no labeled samples and the few-shot cases with small numbers of labeled samples. With the visualizations by <i>t</i>-stochastic neighbor embedding, our results imply that the machine learning in the Hilbert space complies with the principles of maximal coding rate reduction, where the low-dimensional data exhibit within-class compressibility, between-class discrimination, and overall diversity. The proposed QIKF and semi-supervised algorithm can be further combined with the parametric models such as tensor networks, quantum circuits, and quantum neural networks. |
| first_indexed | 2024-03-09T13:25:01Z |
| format | Article |
| id | doaj.art-19db77d7e75a4f83910d30e3b07bf491 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T13:25:01Z |
| publishDate | 2022-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-19db77d7e75a4f83910d30e3b07bf4912023-11-30T21:24:24ZengMDPI AGMathematics2227-73902022-03-0110694010.3390/math10060940Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic FidelityWei-Ming Li0Shi-Ju Ran1Department of Physics, Capital Normal University, Beijing 100048, ChinaDepartment of Physics, Capital Normal University, Beijing 100048, ChinaIn quantum and quantum-inspired machine learning, a key step is to embed the data in the quantum space known as Hilbert space. Studying quantum kernel function, which defines the distances among the samples in the Hilbert space, belongs to the fundamental topics in this direction. In this work, we propose a tunable quantum-inspired kernel function (QIKF) named rescaled logarithmic fidelity (RLF) and a non-parametric algorithm for the semi-supervised learning in the quantum space. The rescaling takes advantage of the non-linearity of the kernel to tune the mutual distances of samples in the Hilbert space, and meanwhile avoids the exponentially-small fidelities between quantum many-qubit states. Being non-parametric excludes the possible effects from the variational parameters, and evidently demonstrates the properties of the kernel itself. Our results on the hand-written digits (MNIST dataset) and movie reviews (IMDb dataset) support the validity of our method, by comparing with the standard fidelity as the QIKF as well as several well-known non-parametric algorithms (naive Bayes classifiers, <i>k</i>-nearest neighbors, and spectral clustering). High accuracy is demonstrated, particularly for the unsupervised case with no labeled samples and the few-shot cases with small numbers of labeled samples. With the visualizations by <i>t</i>-stochastic neighbor embedding, our results imply that the machine learning in the Hilbert space complies with the principles of maximal coding rate reduction, where the low-dimensional data exhibit within-class compressibility, between-class discrimination, and overall diversity. The proposed QIKF and semi-supervised algorithm can be further combined with the parametric models such as tensor networks, quantum circuits, and quantum neural networks.https://www.mdpi.com/2227-7390/10/6/940quantum machine learningsemi-supervised learningnon-parametric learningquantum kernel function |
| spellingShingle | Wei-Ming Li Shi-Ju Ran Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic Fidelity Mathematics quantum machine learning semi-supervised learning non-parametric learning quantum kernel function |
| title | Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic Fidelity |
| title_full | Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic Fidelity |
| title_fullStr | Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic Fidelity |
| title_full_unstemmed | Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic Fidelity |
| title_short | Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic Fidelity |
| title_sort | non parametric semi supervised learning in many body hilbert space with rescaled logarithmic fidelity |
| topic | quantum machine learning semi-supervised learning non-parametric learning quantum kernel function |
| url | https://www.mdpi.com/2227-7390/10/6/940 |
| work_keys_str_mv | AT weimingli nonparametricsemisupervisedlearninginmanybodyhilbertspacewithrescaledlogarithmicfidelity AT shijuran nonparametricsemisupervisedlearninginmanybodyhilbertspacewithrescaledlogarithmicfidelity |