Sparse Gaussian Processes (SGPs) are a class of Gaussian Process (GP) models that aim to address the scalability issues of standard Gaussian Processes. GPs are a powerful and flexible non-parametric Bayesian method for regression and classification tasks, but their computational complexity grows cubically with the number of data points, making them impractical for large datasets. Sparse Gaussian Processes use a smaller set of inducing points to approximate the full GP, resulting in a more scalable and computationally efficient model.

A Gaussian Process is a collection of random variables, any finite number of which have a joint Gaussian distribution. In the context of machine learning, GPs can be used as a prior over functions, which can then be combined with observed data to make predictions. The key idea behind GPs is that they define a distribution over functions, rather than a fixed function as in parametric models.

Given a set of input-output pairs $\{(x_i, y_i)\}_{i=1}^N$, a GP models the relationship between the inputs $x_i$ and outputs $y_i$ as:

where $f(x_i)$ is a latent function and $\epsilon_i$ is Gaussian noise with variance $\sigma^2$. The GP prior over the latent function $f$ is specified by a mean function $m(x)$ and a covariance function (or kernel) $k(x, x')$:

The kernel function encodes the similarity between input points and determines the smoothness and other properties of the functions sampled from the GP.

The main computational bottleneck in Gaussian Processes is the inversion of the covariance matrix, which has a complexity of $\mathcal{O}(N^3)$, where $N$ is the number of data points. This makes GPs infeasible for large datasets. Additionally, the memory requirements for storing the covariance matrix grow quadratically with the number of data points, further limiting the applicability of GPs to large-scale problems.

Sparse Gaussian Processes address the scalability issues of standard GPs by using a smaller set of inducing points to approximate the full GP. The inducing points are a set of $M$ representative input points, where $M \ll N$. The idea is to approximate the full GP by conditioning it on the function values at the inducing points. This results in a more computationally efficient model, as the complexity of the SGP scales as $\mathcal{O}(NM^2)$, which is significantly lower than the complexity of the full GP.

There are several approaches to sparse approximations, including the Sparse Pseudo-input Gaussian Process (SPGP), the Fully Independent Training Conditional (FITC) approximation, and the Variational Free Energy (VFE) approximation. These methods differ in how they select the inducing points and how they approximate the full GP.

The SPGP method selects the inducing points by optimizing a lower bound on the marginal likelihood of the data. The inducing points are treated as additional parameters of the model and are optimized jointly with the kernel hyperparameters. The SPGP approximation can be seen as a low-rank approximation of the full GP covariance matrix.

The FITC approximation assumes that the function values at the training points are conditionally independent given the function values at the inducing points. This results in a sparse approximation of the full GP, where the covariance matrix is block-diagonal. The FITC approximation can be seen as a combination of the SPGP and the full GP, as it includes both the low-rank approximation of the covariance matrix and the diagonal noise term.

The VFE approximation is a variational approach to sparse GPs, where the goal is to find the best approximation to the true posterior distribution over the function values given the inducing points. The VFE method minimizes the Kullback-Leibler (KL) divergence between the approximate and true posterior distributions, resulting in a lower bound on the marginal likelihood of the data. The inducing points and kernel hyperparameters are optimized jointly by maximizing the lower bound.

Sparse Gaussian Processes have been successfully applied to a wide range of regression and classification tasks, including time series prediction, spatial data modeling, and computer vision. They have also been extended to more complex models, such as Deep Gaussian Processes, which combine the flexibility of deep learning with the uncertainty quantification and interpretability of Gaussian Processes.

In summary, Sparse Gaussian Processes are a powerful and scalable alternative to standard Gaussian Processes, enabling the application of GP models to large-scale machine learning problems. By using a smaller set of inducing points to approximate the full GP, SGPs achieve significant computational and memory savings while maintaining the flexibility and expressiveness of the GP framework.