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 non-parametric semi-supervised learning in the quantum space, which we name as RLF-NSSL. 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 advantages from the space itself. We compare RLF-NSSL with several well-known non-parametric algorithms including naive Bayes classifiers, k-nearest neighbors, and spectral clustering. Our method exhibits better 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-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. 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.
翻译:在量子和量子启蒙的机器学习中,第一步是将数据嵌入量子空间,称为Hilbert空间。开发量子内核功能(QKF),确定Hilbert空间样本之间的距离,属于机器学习的基本主题。在这项工作中,我们提议在量子空间(我们称之为RLF-NSSL)中重新标定对等性和非参数半监督性学习。 重新扩展利用内核的非线性性来调整Hilbert空间样本的距离,同时避免量子空间样本之间的超小忠实性。非参数排除了变异参数的可能效果,并明显展示了空间本身的优势。我们把RLF-NSSL和一些著名的非参数算法(我们称之为RLF-NSSL)与一些广为人所知的非参数算法(包括天真的Bayes分类器、K-最接近的邻居、以及光谱集集集集,我们的方法更精确地展示了非超级模型,没有标签的样本应用, 也避免了数级级数级的直径网络之间的偏差性。