Not yet reviewed by Pith; the record is open.

In their standard form Gaussian processes (GPs) provide a powerful non-parametric framework for regression and classificaton tasks. Their one limiting property is their $\mathcal{O}(N^{3})$ scaling where $N$ is the number of training data points. In this paper we present a framework for GP training with sequential selection of training data points using an intuitive selection metric. The greedy forward selection strategy is devised to target two factors - regions of high predictive uncertainty and underfit. Under this technique the complexity of GP training is reduced to $\mathcal{O}(M^{3})$ where $(M \ll N)$ if $M$ data points (out of $N$) are eventually selected. The sequential nature of the algorithm circumvents the need to invert the covariance matrix of dimension $N \times N$ and enables the use of favourable matrix inverse update identities. We outline the algorithm and sequential updates to the posterior mean and variance. We demonstrate our method on selected one dimensional functions and show that the loss in accuracy due to using a subset of data points is marginal compared to the computational gains.