Learning with Density Matrices and Random Features

TL;DR

This paper introduces a novel approach using density matrices and random features to create differentiable machine learning models capable of approximating distributions, with applications in density estimation, classification, and regression.

Contribution

It demonstrates how density matrices combined with random Fourier features can approximate any probability distribution and enables differentiable models for various learning tasks.

Findings

Models successfully approximate complex distributions.

Differentiable models integrate with deep learning architectures.

Evaluation shows competitive performance on benchmark tasks.

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.