CAST: Canonical Approximate Schur Tree for Approximate Cholesky on Graphs
Graph-data workloads such as diffusion estimation, ranking, semi-supervised learning, and network optimization often solve many Laplacian or symmetric diagonally dominant M-matrix (SDDM) systems with the same coefficient matrix. Approximate Cholesky preconditioners eliminate vertices one at a time and store the resulting sparse approximate factorization, the \emph{factor}, whose construction cost is amortized across these solves. But eliminating a vertex, the \emph{pivot}, creates a dense Schur-complement clique among its $d$ active neighbors. We introduce CAST (Canonical Approximate Schur Tree), which replaces this clique with a weighted random spanning tree sampled directly from it. Every realization is connected and contains exactly d-1 edges, while reweighting each selected edge by the reciprocal of its tree-inclusion probability makes the update unbiased. The distribution is independent of the ordering of the pivot neighbors, and we prove that its leverage-score marginals minimize the largest normalized reweighted-edge contribution among unbiased inverse-marginal one-tree estimators. We also introduce CAST-$ρ$, which replaces each pivot neighbor with $ρ$ copies, each carrying a $1/ρ$ share of that neighbor's incident weight, samples a weighted random spanning tree on the expanded clique, and contracts the copies back to the original neighborhood. The resulting update remains unbiased and connected, can be sampled exactly in $O(ρd)$ time, and satisfies a $1/ρ$ bound on the second moment of the normalized local Schur error. Increasing $ρ$ therefore reduces certified local sampling variability, but may increase construction cost and downstream fill. Empirically, we observe that CAST-1 is the faster default, whereas CAST-2 is preferable when its additional edge contributions remain inexpensive.