Non-parametric Active Learning and Rate Reduction in Many-body Hilbert
Space with Rescaled Logarithmic Fidelity
- URL: http://arxiv.org/abs/2107.00195v1
- Date: Thu, 1 Jul 2021 03:13:16 GMT
- Title: Non-parametric Active Learning and Rate Reduction in Many-body Hilbert
Space with Rescaled Logarithmic Fidelity
- Authors: Wei-Ming Li and Shi-Ju Ran
- Abstract summary: In quantum and quantum-inspired machine learning, the very first step is to embed the data in quantum space known as Hilbert space.
We propose the rescaled logarithmic fidelity (RLF) and a non-parametric active learning in the quantum space, which we name as RLF-NAL.
Our results imply that the machine learning in the Hilbert space complies with the principles of maximal coding rate reduction.
- Score: 4.781805457699204
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: In quantum and quantum-inspired machine learning, the very first step is to
embed the data in quantum space known as Hilbert space. Developing quantum
kernel function (QKF), which defines the distances among the samples in the
Hilbert space, belongs to the fundamental topics for machine learning. In this
work, we propose the rescaled logarithmic fidelity (RLF) and a non-parametric
active learning in the quantum space, which we name as RLF-NAL. 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. We compare RLF-NAL with several
well-known non-parametric algorithms including naive Bayes classifiers,
$k$-nearest neighbors, and spectral clustering. Our method exhibits excellent
accuracy particularly for the unsupervised case with no labeled samples and the
few-shot cases with small numbers of labeled samples. With the visualizations
by t-SNE, 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. Our proposals can be applied to other
quantum and quantum-inspired machine learning, including the methods using the
parametric models such as tensor networks, quantum circuits, and quantum neural
networks.
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