Matroid Regression
This addresses the computational bottleneck in large-scale sparse linear algebra for researchers and practitioners, offering a novel algebraic combinatorial approach with incremental improvements in efficiency and error control.
The paper tackles the problem of solving large sparse linear systems locally by computing single evaluations without full signal reconstruction, achieving a method that scales with sparsity rather than system size and provides error estimates. It results in the best linear unbiased estimator (BLUE) and minimum variance unbiased estimator (MVUE) under Gaussian noise, with a trade-off between kernel matrix size and accuracy.
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.