# Learning with Density Matrices and Random Features

arXiv: https://arxiv.org/abs/2102.04394 · HTML: https://mlsanity.com/paper/2102.04394/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 52 references that are in the map.
- First posted on arXiv 8 Feb 2021.

## Builds on (most cited first, 20 of 20)

- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Polylogarithmic width suffices for gradient descent to achieve arbitrarily small test error with shallow ReLU networks](https://mlsanity.com/paper/1909.12292/) (2019)
- [Orthogonal Random Features](https://mlsanity.com/paper/1610.09072/) (2016)
- [Fastfood: Approximate Kernel Expansions in Loglinear Time](https://mlsanity.com/paper/1408.3060/) (2014)
- [Random Fourier Features for Kernel Ridge Regression: Approximation Bounds and Statistical Guarantees](https://mlsanity.com/paper/1804.09893/) (2018)
- [A quantum-inspired classical algorithm for recommendation systems](https://mlsanity.com/paper/1807.04271/) (2018)
- [Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels](https://mlsanity.com/paper/1412.8293/) (2014)
- [Hashing-Based-Estimators for Kernel Density in High Dimensions](https://mlsanity.com/paper/1808.10530/) (2018)
- [But How Does It Work in Theory? Linear SVM with Random Features](https://mlsanity.com/paper/1809.04481/) (2018)
- [Robust Topological Inference: Distance To a Measure and Kernel Distance](https://mlsanity.com/paper/1412.7197/) (2014)
- [Gaussian Quadrature for Kernel Features](https://mlsanity.com/paper/1709.02605/) (2017)
- [Implicit Kernel Learning](https://mlsanity.com/paper/1902.10214/) (2019)
- [ASKIT: Approximate Skeletonization Kernel-Independent Treecode in High Dimensions](https://mlsanity.com/paper/1410.0260/) (2014)
- [A Comprehensive Approach to Mode Clustering](https://mlsanity.com/paper/1406.1780/) (2014)
- [Robust Deep Ordinal Regression Under Label Noise](https://mlsanity.com/paper/1912.03488/) (2019)
- [Supervised Learning with Quantum Measurements](https://mlsanity.com/paper/2004.01227/) (2020)
- [Scale Up Nonlinear Component Analysis with Doubly Stochastic Gradients](https://mlsanity.com/paper/1504.03655/) (2015)
- [Pattern recognition on the quantum Bloch sphere](https://mlsanity.com/paper/1603.00173/) (2016)
- [Cluster Trees on Manifolds](https://mlsanity.com/paper/1307.6515/) (2013)
- [Kernel Method based on Non-Linear Coherent State](https://mlsanity.com/paper/2007.07887/) (2020)

## Built on by (most cited first, 17 of 17)

- [Non-IID Quantum Federated Learning with One-shot Communication Complexity](https://mlsanity.com/paper/2209.00768/) (2022)
- [The Representation Jensen-Shannon Divergence](https://mlsanity.com/paper/2305.16446/) (2023)
- [Quantum Adaptive Fourier Features for Neural Density Estimation](https://mlsanity.com/paper/2208.00564/) (2022)
- [AD-DMKDE: Anomaly Detection through Density Matrices and Fourier Features](https://mlsanity.com/paper/2210.14796/) (2022)
- [Fast Kernel Density Estimation with Density Matrices and Random Fourier Features](https://mlsanity.com/paper/2208.01206/) (2022)
- [InQMAD: Incremental Quantum Measurement Anomaly Detection](https://mlsanity.com/paper/2210.05061/) (2022)
- [Quantum Channel Learning](https://mlsanity.com/paper/2407.04406/) (2024)
- [Quantum Measurement Classification with Qudits](https://mlsanity.com/paper/2107.09781/) (2021)
- [Optimisation-free Classification and Density Estimation with Quantum Circuits](https://mlsanity.com/paper/2203.14452/) (2022)
- [LEAN-DMKDE: Quantum Latent Density Estimation for Anomaly Detection](https://mlsanity.com/paper/2211.08525/) (2022)
- [Kernel Density Matrices for Probabilistic Deep Learning](https://mlsanity.com/paper/2305.18204/) (2023)
- [On-orbit model training for satellite imagery with label proportions](https://mlsanity.com/paper/2306.12461/) (2023)
- [MEMO-QCD: Quantum Density Estimation through Memetic Optimisation for Quantum Circuit Design](https://mlsanity.com/paper/2406.08591/) (2024)
- [Defining Operational Conditions for Safety-Critical AI-Based Systems from Data](https://mlsanity.com/paper/2601.22118/) (2026)
- [Behavioral Heterogeneity as Quantum-Inspired Representation](https://mlsanity.com/paper/2603.22729/) (2026)
- [Prostate Tissue Grading with Deep Quantum Measurement Ordinal Regression](https://mlsanity.com/paper/2103.03188/) (2021)
- [Latent Anomaly Detection Through Density Matrices](https://mlsanity.com/paper/2408.07623/) (2024)

## Abstract (arXiv)

A density matrix describes the statistical state of a quantum system. It is a powerful formalism to represent both the quantum and classical uncertainty of quantum systems and to express different statistical operations such as measurement, system combination and expectations as linear algebra operations. This paper explores how density matrices can be used as a building block for machine learning models exploiting their ability to straightforwardly combine linear algebra and probability. One of the main results of the paper is to show that density matrices coupled with random Fourier features could approximate arbitrary probability distributions over ℝn. Based on this finding the paper builds different models for density estimation, classification and regression. These models are differentiable, so it is possible to integrate them with other differentiable components, such as deep learning architectures and to learn their parameters using gradient-based optimization. In addition, the paper presents optimization-less training strategies based on estimation and model averaging. The models are evaluated in benchmark tasks and the results are reported and discussed.