  We propose an algebraic combinatorial method for solving large sparse linear
systems of equations locally - that is, a method which can compute single
evaluations of the signal without computing the whole signal. The method scales
only in the sparsity of the system and not in its size, and allows to provide
error estimates for any solution method. At the heart of our approach is the
so-called regression matroid, a combinatorial object associated to sparsity
patterns, which allows to replace inversion of the large matrix with the
inversion of a kernel matrix that is constant size. We show that our method
provides the best linear unbiased estimator (BLUE) for this setting and the
minimum variance unbiased estimator (MVUE) under Gaussian noise assumptions,
and furthermore we show that the size of the kernel matrix which is to be
inverted can be traded off with accuracy.
