I am trying to generate random draws from a multivariate normal distribution with positive semi-definite covariance matrices. The Distributions.jl
MvNormal only handles positive definite covariance matrices and throws an exception otherwise. There was a discussion about adding support for semi-definite covariance matrices a few years ago here but it is still not supported. While
MvNormal does much more than generating random numbers, I am only interested in the random number generation.
My current solution is to do a pivoted cholesky decomposition and multiply by independent standard normals:
mu = [0,0]
Sig = [1 1;1 1]
chol = cholesky(Sig,Val(true),check=false)
random_draw = mu + chol.L[invperm(chol.p),1:chol.rank]*randn(chol.rank)
I check that the eigenvalues of
Sig are non-negative before I generate the draws:
minimum(eigvals(Sig)) >= 0. This has worked properly in a few problems I have tested.
I would appreciate any feedback on my approach. Is there an easier and/or more efficient way to achieve the same results? Even better, is there any package that already generates multivariate normals with positive semi-definite covariance matrices?
See here: Cholesky decomposition of low-rank positive-semidefinite matrix - #3 by stevengj
The problem is that you may get slightly negative pivots due to roundoff errors, in which case LAPACK may terminate the factorization prematurely.
Ideally there would be a way to specify some tolerance so that slightly negative pivots could be treated as zeros, but I don’t know of any API for this?
You could use
sqrt(Hermitian(A)) instead of Cholesky. This is slower than Cholesky, but it should be more robust to small roundoff errors for semidefinite matrices because we added a tolerance check (which can be customized via an
rtol keyword argument) to treat small negative eigenvalues as zero: RFC: treat small negative λ as 0 for sqrt(::Hermitian) by stevengj · Pull Request #35057 · JuliaLang/julia · GitHub
I am using the
check=false option to ensure that the cholesky decomposition does not stop with small negative pivots. But I need to check that the matrix is positive semi-definite before I can use this, otherwise I may get wrong results without any exceptions if I pass a matrix that is not positive semi-definite. I am using the
eigvals(Sig) to check for positive semi-definiteness.
sqrt is doing the eigendecomposition already. Is there any optional argument to make it throw an exception if the eigenvalues are negative beyond the tolerance? By default it returns a complex matrix.
I’m not sure this does what you think. Read the LAPACK docs in the linked thread and LAPACK technical report #161 section 7: my understanding is that the Cholesky factorization will stop early if it encounters a negative pivot.
check=false just means that you get the partially factorized result with no error thrown.
Wait, I just noticed that you are only looking at columns
1:chol.rank of the
L factor — I think you are right, it is okay in this case:
julia> B = randn(3,7); A = B'B; F = cholesky(A,Val(true),check=false);
julia> L = F.L[invperm(F.p),1:F.rank]; L*L' ≈ A
First, if this doesn’t guarantee that the Cholesky factorization won’t encounter a slightly negative pivot due to roundoff errors. Second, you could easily have a matrix that is theoretically semidefinite but in practice has slightly negative eigenvalues, again due to roundoff errors. For example:
julia> B = randn(3,5); eigvals(Hermitian(B'B))