A Woodbury Matrix Identity

The likelihood function in Eq. 3.46 is used during the sparsiiication procedure in order to optimise the hyperparameters and the sparse points. At first sight, it seems that the inverse of an A x A matrix has to be calculated, the computational cost of which would scale as N. However, by using the matrix inversion lemma, also known as the Woodbury matrix identity, the computational cost scales only with NM if iV M. If we want to find the inverse of a matrix, which can be written in the form Z + UWV, the Woodbury matrix identity states that 

In our case, Z is an A x A diagonal matrix, hence its inverse is trivial, and W - is M X M. The order of the operations can arranged such that none of them requires more than NM floating point operations  

In our implementation, the determinants are calculated together with the inverses, without any computational overhead. 

We also note that at certain values of the hyperparameters the matrix Cjw is ill conditioned. In the original Gaussian Process, the covariance matrix Q can also be ill conditioned, but by adding the diagonal matrix 7 1 this problem is eliminated, except for very small values of the ffy parameters. Snelson suggested that a small 

Bait6k-Paitay, The Gaussian Approximation Potential, Springer Theses, 

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