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Learning with Density Matrices and Random Features

Fabio A. González, Alejandro Gallego, Santiago Toledo-Cortés, Vladimir Vargas-Calderón

TL;DR

This work introduces a quantum-inspired framework that uses density matrices as building blocks for machine learning tasks. By coupling density matrices with random Fourier features, the authors show how arbitrary distributions over $\mathbb{R}^n$ can be represented and learned in a differentiable, GPU-friendly manner, enabling density estimation, classification, and regression. The paper develops four methods (DMKDE, DMKDC, QMC, QMR) with differentiable training strategies, including optimization-based (SGD) approaches and optimization-free estimators, and validates them across density estimation, classification, and ordinal regression benchmarks. Empirical results demonstrate competitive performance against standard KDE and SVM baselines, while offering favorable scalability and integration with deep architectures. The work opens avenues for future exploration of complex-valued density matrices and entanglement-inspired representations in quantum-inspired classical learning frameworks.

Abstract

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 $\mathbb{R}^n$. 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.

Learning with Density Matrices and Random Features

TL;DR

This work introduces a quantum-inspired framework that uses density matrices as building blocks for machine learning tasks. By coupling density matrices with random Fourier features, the authors show how arbitrary distributions over can be represented and learned in a differentiable, GPU-friendly manner, enabling density estimation, classification, and regression. The paper develops four methods (DMKDE, DMKDC, QMC, QMR) with differentiable training strategies, including optimization-based (SGD) approaches and optimization-free estimators, and validates them across density estimation, classification, and ordinal regression benchmarks. Empirical results demonstrate competitive performance against standard KDE and SVM baselines, while offering favorable scalability and integration with deep architectures. The work opens avenues for future exploration of complex-valued density matrices and entanglement-inspired representations in quantum-inspired classical learning frameworks.

Abstract

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 . 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.

Paper Structure

This paper contains 24 sections, 3 theorems, 38 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Kernel density estimation
  4. Random features
  5. Density estimation with density matrices
  6. Density matrices
  7. Methods
  8. Density matrix kernel density estimation (DMKDE)
  9. Density matrix kernel density classification (DMKDC)
  10. Quantum measurement classification (QMC)
  11. Quantum measurement regression (QMR)
  12. Related Work
  13. Experimental Evaluation
  14. Density estimation evaluation
  15. Data sets and experimental setup
  16. ...and 9 more sections

Key Result

Theorem 1

rahimi2007rff Let $\mathcal{M}$ be a compact subset of $\mathbb{R}^d$ with a diameter $\text{diam}(\mathcal{M})$. Then for the mapping $\phi_{\mathrm{rff}}$ defined above, we have where, $\sigma^2_p$ is the second momentum of the Fourier transform of $k$. In particular, for the Gaussian kernel $\sigma^2_p = 2d\gamma$, where $\gamma$ is the spread parameter (see Eq. eq:Parzen-Gaussian).

Figures (10)

  • Figure 1: Density matrix kernel density estimation (DMKDE).
  • Figure 2: Density matrix kernel density classification (DMKDC).
  • Figure 3: Quantum measurement classification (QMC).
  • Figure 4: 1-D synthetic dataset. The gray zone is the area of the true density. The estimated pdf by DMKDE ($\gamma=2$) and KDE ($\gamma=4$) is shown.
  • Figure 5: Accuracy of the density estimation of DMKDE for different number of RFF for the 1-D dataset (top left), 2-D dataset (top right) and MNIST dataset (bottom). For the 1-D dataset both KDE and DMKDE are compared against the true density. For the two other datasets the difference between KDE and DMKDE is calculated. In all the cases the RMSE is calculated. The blue shaded zone represents the 95% confidence interval.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof