arXiv:2611.13134 · v1

A Fast Algorithm for Subspace Convolution

首次提交 2026-11-10 03:18

版本更新 2026-11-10 03:18

雷达观测 2026-11-11 11:04

摘要

A cornerstone result in Boolean lattice theory is the subset-zeta transform, which computes the $f$-weighted subset summation problem $(Z f)(B)=\sum_{A\subseteq B}f(A)$ for all $B\subseteq[n]$ in $O(N \log N)$ additions instead of the naïve $O(3^n) = O(N^{\log_2 3})$, where $N=2^n$. This paper studies the algebraic analogue of this problem--the subspace summation problem--where subsets are replaced by linear subspaces in $\mathbb{F}_q^n$. Surprisingly, we show that this problem admits a similar $Θ_q(N_q \log N_q)$-time algorithm, where $N_q$ is the number of subspaces. This improves substantially (in both theory and practice) over lattice-based algorithms ([BHK+16, KKW16]), which are known to be optimal for addition-only algorithms over $\mathbb R$. We prove similar runtimes for inversion, rank-constrained sums and span convolution. We describe two main applications: Fast linear algebra on Grassmann graphs: Matrix--vector multiplication for arbitrary distance-invariant kernels on $N$-vertex Grassmann graphs in $O_q(N\log N)$ arithmetic operations, together with fast spectral projections and exact solutions of the corresponding linear systems. Graph coloring of low binary cut-rank graphs: An $n^{2^{r^2/4+O(1)}}$-time algorithm for computing the chromatic number and chromatic polynomial of an $n$-vertex graph, given a rank-decomposition of width $r$. We also give quantitative reductions from satisfiability and monotone constraint problems. Experiments show speedups in measured running time of up to $18$ over the covering-pair algorithm for the zeta transform, $190$ over Poisson summation for rank-constrained sums, and $22$ over the distance-matrix recurrence for Grassmann kernel multiplication. Lean 4 formalizations of the transform identities, arithmetic operation bounds, and graph-coloring constructions accompany the paper [PW26].

作者备注：35 pages, 2 figures. Code, experiments, and Lean 4 formalizations: https://github.com/shlomi-perles/subspace-convolution