[Paper Review] Noise Estimation in Gaussian Process Regression

Siavash Ameli, Shawn C. Shadden|arXiv (Cornell University)|Jun 20, 2022

Gaussian Processes and Bayesian Inference5 citations

TL;DR

This paper proposes a computationally efficient method for estimating noise and correlated error variances in Gaussian process regression by reducing the hyperparameter estimation problem to a univariate root-finding task. By deriving analytical bounds and asymptotes of the marginal likelihood, the approach enables robust and scalable parameter estimation without relying on iterative optimization over high-dimensional spaces.

ABSTRACT

We develop a computational procedure to estimate the covariance hyperparameters for semiparametric Gaussian process regression models with additive noise. Namely, the presented method can be used to efficiently estimate the variance of the correlated error, and the variance of the noise based on maximizing a marginal likelihood function. Our method involves suitably reducing the dimensionality of the hyperparameter space to simplify the estimation procedure to a univariate root-finding problem. Moreover, we derive bounds and asymptotes of the marginal likelihood function and its derivatives, which are useful to narrowing the initial range of the hyperparameter search. Using numerical examples, we demonstrate the computational advantages and robustness of the presented approach compared to traditional parameter optimization.

Motivation & Objective

• To address the computational challenge of high-dimensional hyperparameter optimization in Gaussian process regression with additive noise.
• To enable robust and efficient estimation of both correlated error variance and i.i.d. noise variance in semiparametric models.
• To reduce the multivariate optimization problem to a univariate root-finding problem using profiling techniques.
• To derive analytical bounds and asymptotes of the marginal likelihood and its derivatives for narrowing search intervals.
• To improve convergence and computational efficiency compared to traditional gradient-based or global optimization methods.

Proposed method

• The method reduces the hyperparameter space by profiling out the noise and correlated error variances, transforming the problem into a univariate root-finding task.
• It leverages the marginal likelihood function of the Gaussian process model with mixed covariance structure: $\Sigma = \sigma^2 K(\mathbf{x},\mathbf{x}'|\boldsymbol{\theta}) + \sigma_0^2 I(\mathbf{x},\mathbf{x}')$.
• Asymptotic expansions of the marginal likelihood and its derivatives are derived in the limit of large $\eta = \sigma^2 / \sigma_0^2$, enabling analytical bounds on the search space.
• Neumann series expansions are used to approximate inverse covariance matrices and projection operators in the asymptotic regime.
• The method uses the projection matrix $\mathbf{Q} = \mathbf{X}(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top$ and its orthogonal complement $\mathbf{Q}_\perp = \mathbf{I} - \mathbf{Q}$ to decompose the problem.
• Key components include the asymptotic expansion of $\mathbf{P}_\eta = \mathbf{K}_\eta^{-1}\mathbf{X}(\mathbf{X}^\top\mathbf{K}_\eta^{-1}\mathbf{X})^{-1}\mathbf{X}^\top\mathbf{K}_\eta^{-1}$ and $\mathbf{M}_{1,\eta} = \mathbf{K}_\eta^{-1}\mathbf{P}_\eta$, leading to a simplified form in powers of $\eta^{-1}$.

Experimental results

Research questions

• RQ1 Can the joint estimation of noise and correlated error variances in Gaussian process regression be reduced to a univariate root-finding problem?
• RQ2 What are the asymptotic behaviors of the marginal likelihood and its derivatives in the limit of large signal-to-noise ratio?
• RQ3 How can analytical bounds on the marginal likelihood be used to improve the robustness of hyperparameter optimization?
• RQ4 Does the proposed method outperform traditional optimization techniques in terms of convergence and computational cost?
• RQ5 To what extent does the method remain stable and accurate under ill-conditioned or high-dimensional design matrices?

Key findings

• The method reduces the original multivariate optimization problem to a univariate root-finding problem, significantly simplifying computation.
• Asymptotic expansions of the marginal likelihood and its derivatives are derived up to $\mathcal{O}(\eta^{-3})$, enabling precise bounds on the search space.
• The projection matrix $\mathbf{P}_\eta$ is asymptotically expanded as $\mathbf{P}_\eta = \mathbf{Q} + \frac{1}{\eta}\mathbf{Q}_\perp\mathbf{K}\mathbf{Q} - \frac{1}{\eta^2}\mathbf{Q}_\perp(\mathbf{K}\mathbf{Q})^2 + \mathcal{O}(\eta^{-3})$.
• The matrix $\mathbf{M}_{1,\eta} = \mathbf{K}_\eta^{-1}\mathbf{P}_\eta$ is shown to simplify to $\frac{1}{\eta}\mathbf{Q}(\mathbf{I} - \frac{1}{\eta}\mathbf{K}\mathbf{Q} + \frac{1}{\eta^2}(\mathbf{K}\mathbf{Q})^2) + \mathcal{O}(\eta^{-4})$.
• Numerical examples demonstrate superior computational efficiency and robustness compared to standard optimization methods, especially in high-dimensional or ill-conditioned settings.