Experimental mathematics colloquium

Steffen Lauritzen: When does the Score Matching Estimator (SME) of a concentration matrix exist?

The SME of a pxp concentration matrix K restricted to a linear space depends linearly of the Wishart matrix of sums and products of n observations, as the inverse of a random linear map, if this map is invertible. 

The map in question is the Jordan product K -> (KW+WK)/2, where W is the random Wishart matrix.

If K is an element of a d-dimensional subspace L of the pxp symmetric matrices S_p which contains the identity, the map above is not invertible if

d > T_p-T_{p-n}, where T_k= (k+1)k/2.

For some spaces L, the condition d <= T_p-T_{p-n} ensures that the map is invertible with probability one, and for some spaces this condition is not sufficient. 

I have a class of spaces L that I am particularly interested in (corresponding to graphical Gaussian models with symmetry) and would like to know whether the condition d <= T_p-T_{p-n} is sufficient for these models (it might be). If not, I would like to find conditions on L, d, p, and n which are sufficient.

I shall briefly describe the background for the problem, give a number of equivalent or almost equivalent formulations of the problem, and give some classes of spaces L, where I know the condition to be sufficient, and also examples where it isn't.