Learning with Density Matrices and Random Features
This work introduces a novel approach for machine learning practitioners to integrate quantum-inspired linear algebra with probability, though it appears incremental as it builds on existing random features and density matrix concepts.
The paper tackled the problem of approximating arbitrary probability distributions in machine learning by using density matrices and random Fourier features, resulting in differentiable models for density estimation, classification, and regression that were evaluated on benchmark tasks.
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.