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.

Answer 2

You cannot do this using the given class. The class you are referencing is equotation solver (which indeed uses cholesky decomposition). To decompose your matrix you should rather use Eigen::LLT. Code example from their website:

MatrixXd A(3,3);
A << 4,-1,2, -1,6,0, 2,0,5;
LLT<MatrixXd> lltOfA(A);
MatrixXd L = lltOfA.matrixL(); 
MatrixXd U = lltOfA.matrixU(); 

Answer 3

As reported somewhere else, e.g., it cannot be done easily. I am copying a possible recommendation (answered by Gael Guennebaud himself), even if somewhat old:

If you really need access to the factor to do your own cooking, then better use the built-in SimplicialL{D}LT<> class. Extracting the factors from the supernodal internal represations of Cholmod/Pardiso is indeed not straightforward and very rarely needed. We have to check, but if Cholmod/Pardiso provide routines to manipulate the factors, like applying it to a vector, then we could let matrix{L,U}() return a pseudo expression wrapping these routines.

Developing code for extracting this is likely beyond SO, and probably a topic for a feature request.

Of course, the solution with LLT is at hand (but not the topic of the OP).