Feat/gp map

[madsendennis]

MR related to https://github.com/unibas-gravis/scalismo/issues/421

Provides handle to directly calculate the .posterior.mean with .MAP without calculating the posterior kernel/covariance matrix.

[madsendennis]

I noted that in the regression function in GaussianProcess.scala, the inverse calculation is what takes up the majority of the time: val invKernel: DenseMatrix[Double] = breeze.linalg.inv(K). Does anyone have an idea how we can calculate this faster?

[madsendennis]

I added an attempt to remove the inv from the mean and kernel calculation with the Cholesky decomposition. The tests pass, and the mean values seem to be the same in the instances I tested, but there seems to be some difference in the kernel calculation. Maybe @JonathanAellen can help have a look, especially on the last commit?

[madsendennis]

I did some more test to check the difference for the inverse vs cholesky. They seem to provide the same output while cholesky is a lot faster to compute.

I see that that the MultivariateNormalDistribution case class has a private regularizedCholesky function. Should we move this out to a separate file and use instead in the GP file?

[Andreas-Forster]

I noted that in the regression function in GaussianProcess.scala, the inverse calculation is what takes up the majority of the time: val invKernel: DenseMatrix[Double] = breeze.linalg.inv(K). Does anyone have an idea how we can calculate this faster?

Maybe you can have a look at using native libraries like blas under the hood (see https://github.com/luhenry/netlib)

[Andreas-Forster]

For speed, there is a paper about cholesky and matrix inversion here: https://arxiv.org/abs/1111.4144

[marcelluethi]

@Andreas-Forster How about calling it posteriorMean instead of Map? This makes it clear that it is just a point estimate and also addresses the possible confusion with Scala's map function.