A recursive butterfly factorization with optimality guarantees

We formalize a recursive format for representing a butterfly matrix. This new format naturally leads to a simple recursive algorithm for computing a quasi-optimal butterfly approximation to an arbitrary $N \times N$ matrix $A$. When the entries of $A$ are explicitly available, we show that the algorithm computes a butterfly matrix $B$ in $O(N^2)$ operations with approximation error $\|A - B\|_F$ at most a $O(\sqrt{\log(N)})$ factor away from that of the best possible approximation by a butterfly matrix. We also develop a matrix-free variant of the method, which uses $\widetilde{O}(\sqrt{N})$ matrix-vector products and $\widetilde{O}(N)$ working memory and, with high probability, returns a butterfly approximation with Frobenius norm error within a $O(N^{1/4})$-factor of the optimal error. We show that the algorithm is a reformulation of the hybrid butterfly factorization approach presented in [Liu et. al.; SISC, 43 (2021)]. Our paper therefore provides the first theoretical quasi-optimality guarantee for that algorithm.