# Non-parametric Semi-Supervised Learning in Many-body Hilbert Space with Rescaled Logarithmic Fidelity

arXiv: https://arxiv.org/abs/2107.00195 · HTML: https://mlsanity.com/paper/2107.00195/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 75 references that are in the map.
- First posted on arXiv 1 Jul 2021.

## Builds on (most cited first, 25 of 25)

- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Variational Quantum Circuits for Deep Reinforcement Learning](https://mlsanity.com/paper/1907.00397/) (2019)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [Learning Diverse and Discriminative Representations via the Principle of Maximal Coding Rate Reduction](https://mlsanity.com/paper/2006.08558/) (2020)
- [Transfer learning in hybrid classical-quantum neural networks](https://mlsanity.com/paper/1912.08278/) (2019)
- [Robust data encodings for quantum classifiers](https://mlsanity.com/paper/2003.01695/) (2020)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Towards Quantum Machine Learning with Tensor Networks](https://mlsanity.com/paper/1803.11537/) (2018)
- [Continuous-variable quantum neural networks](https://mlsanity.com/paper/1806.06871/) (2018)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [q-means: A quantum algorithm for unsupervised machine learning](https://mlsanity.com/paper/1812.03584/) (2018)
- [Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure](https://mlsanity.com/paper/1710.04833/) (2017)
- [Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines](https://mlsanity.com/paper/1712.04144/) (2017)
- [Supervised Learning with Projected Entangled Pair States](https://mlsanity.com/paper/2009.09932/) (2020)
- [TensorNetwork for Machine Learning](https://mlsanity.com/paper/1906.06329/) (2019)
- [Equivalence of restricted Boltzmann machines and tensor network states](https://mlsanity.com/paper/1701.04831/) (2017)
- [Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models](https://mlsanity.com/paper/1609.02542/) (2016)
- [Generative Tensor Network Classification Model for Supervised Machine Learning](https://mlsanity.com/paper/1903.10742/) (2019)
- [Information Scrambling in Quantum Neural Networks](https://mlsanity.com/paper/1909.11887/) (2019)
- [Tangent-Space Gradient Optimization of Tensor Network for Machine Learning](https://mlsanity.com/paper/2001.04029/) (2020)

## Built on by (most cited first, 2 of 2)

- [Tensor networks for interpretable and efficient quantum-inspired machine learning](https://mlsanity.com/paper/2311.11258/) (2023)
- [Intelligent diagnostic scheme for lung cancer screening with Raman spectra data by tensor network machine learning](https://mlsanity.com/paper/2303.06340/) (2023)

## Abstract (arXiv)

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.