How to compute scipy sparse matrix determinant without turning it to dense?

Question

I am trying to figure out the fastest method to find the determinant of sparse symmetric and real matrices in python. using scipy sparse module but really surprised

Answer 1

Here are some references I provided as part of an answer here. I think they address the actual problem you are trying to solve:

• notes for an implementation in the Shogun library
• Erlend Aune, Daniel P. Simpson: Parameter estimation in high dimensional Gaussian distributions, particularly section 2.1 (arxiv:1105.5256)
• Ilse C.F. Ipsen, Dean J. Lee: Determinant Approximations (arxiv:1105.0437)
• Arnold Reusken: Approximation of the determinant of large sparse symmetric positive definite matrices (arxiv:hep-lat/0008007)

Quoting from the Shogun notes:

The usual technique for computing the log-determinant term in the likelihood expression relies on Cholesky factorization of the matrix, i.e. Σ=LLT, (L is the lower triangular Cholesky factor) and then using the diagonal entries of the factor to compute log(det(Σ))=2∑ni=1log(Lii). However, for sparse matrices, as covariance matrices usually are, the Cholesky factors often suffer from fill-in phenomena - they turn out to be not so sparse themselves. Therefore, for large dimensions this technique becomes infeasible because of a massive memory requirement for storing all these irrelevant non-diagonal co-efficients of the factor. While ordering techniques have been developed to permute the rows and columns beforehand in order to reduce fill-in, e.g. approximate minimum degree (AMD) reordering, these techniques depend largely on the sparsity pattern and therefore not guaranteed to give better result.

Recent research shows that using a number of techniques from complex analysis, numerical linear algebra and greedy graph coloring, we can, however, approximate the log-determinant up to an arbitrary precision [Aune et. al., 2012]. The main trick lies within the observation that we can write log(det(Σ)) as trace(log(Σ)), where log(Σ) is the matrix-logarithm.