How to extract matrixL() and matrixU() when using Eigen::CholmodSupernodalLLT?

Question

I'm trying to use Eigen::CholmodSupernodalLLT for Cholesky decomposition, however, it seems that I could not get matrixL() and matrixU(). How can I extract matrixL() and matrixU() from Eigen::CholmodSupernodalLLT for future use?

Answer 1

A partial answer to integrate what others have said.

Consider Y ~ MultivariateNormal(0, A). One may want to (1) evaluate the (log-)likelihood (a multivariate normal density), (2) sample from such density.

For (1), it is necessary to solve Ax = b where A is symmetric positive-definite, and compute its log-determinant. (2) requires L such that A = L * L.transpose() since Y ~ MultivariateNormal(0, A) can be found as Y = L u where u ~ MultivariateNormal(0, I).

A Cholesky LLT or LDLT decomposition is useful because chol(A) can be used for both purposes. Solving Ax=b is easy given the decomposition, andthe (log)determinant can be easily derived from the (sum)product of the (log-)components of D or the diagonal of L. By definition L can then be used for sampling.

So, in Eigen one can use:

• Eigen::SimplicialLDLT solver(A) (or Eigen::SimplicialLLT), when solver.solve(b) and calculate the determinant using solver.vectorD().diag(). Useful because if A is a covariance matrix, then solver can be used for likelihood evaluations, and matrixL() for sampling.

• Eigen::CholmodDecomposition does not give access to matrixL() or vectorD() but exposes .logDeterminant() to achieve the (1) goal but not (2).

• Eigen::PardisoLDLT does not give access to matrixL() or vectorD() and does not expose a way to get the determinant.

In some applications, step (2) - sampling - can be done at a later stage so Eigen::CholmodDecomposition is enough. At least in my configuration, Eigen::CholmodDecomposition works 2 to 5 times faster than Eigen::SimplicialLDLT (I guess because of the permutations done under the hood to facilitate parallelization)

Example: in Bayesian spatial Gaussian process regression, the spatial random effects can be integrated out and do not need to be sampled. So MCMC can proceed swiftly with Eigen::CholmodDecomposition to achieve convergence for the uknown parameters. The spatial random effects can then be recovered in parallel using Eigen::SimplicialLDLT. Typically this is only a small part of the computations but having matrixL() directly from CholmodDecomposition would simplify them a bit.