Quantum algorithms for training Gaussian Processes

TL;DR

This paper introduces a quantum algorithm that significantly accelerates the computation of the log marginal likelihood in Gaussian processes, potentially transforming training efficiency in machine learning models.

Contribution

The paper presents a novel quantum algorithm for efficiently computing the log marginal likelihood of Gaussian processes, leveraging logarithmic time algorithms for determinants and linear systems.

Findings

Quantum algorithm computes LML exponentially faster for sparse matrices.

Logarithmic time algorithms for determinants and linear systems are applied.

Potential for improved GP training efficiency in quantum computing environments.

Abstract

Gaussian processes (GPs) are important models in supervised machine learning. Training in Gaussian processes refers to selecting the covariance functions and the associated parameters in order to improve the outcome of predictions, the core of which amounts to evaluating the logarithm of the marginal likelihood (LML) of a given model. LML gives a concrete measure of the quality of prediction that a GP model is expected to achieve. The classical computation of LML typically carries a polynomial time overhead with respect to the input size. We propose a quantum algorithm that computes the logarithm of the determinant of a Hermitian matrix, which runs in logarithmic time for sparse matrices. This is applied in conjunction with a variant of the quantum linear system algorithm that allows for logarithmic time computation of the form $\mathbf{y}^TA^{-1}\mathbf{y}$, where $\mathbf{y}$ is a dense vector and $A$ is the covariance matrix. We hence show that quantum computing can be used to estimate the LML of a GP with exponentially improved efficiency under certain conditions.