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 Cat Food-Wet 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 Mittens with the standard fidelity as the QIKF as well as several well-known non-parametric algorithms (naive Bayes classifiers, k-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 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.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.