AI 中文总结
本文提出具有最优性保证的递归蝴蝶分解算法，可计算矩阵的准最优蝴蝶近似，其无矩阵变体也有误差保证，为现有混合蝴蝶分解方法提供了首个理论准最优性证明。
AI 中文摘要
我们形式化了一种表示蝴蝶矩阵的递归格式，该格式自然引出了一种简单的递归算法，用于计算任意N×N矩阵A的准最优蝴蝶近似。当A的元素明确可用时，我们证明该算法在O(N²)次运算中计算出蝴蝶矩阵B，其近似误差‖A - B‖_F与蝴蝶矩阵的最佳可能近似的误差之比最多为O(√log(N))因子。我们还开发了该方法的无矩阵变体，它使用Õ(√N)次矩阵-向量乘积和Õ(N)工作内存，且以高概率返回的蝴蝶近似的Frobenius范数误差在最优误差的O(N^(1/4))因子以内。我们证明该算法是[Liu等人；SISC，43（2021）]中提出的混合蝴蝶分解方法的重新表述，因此本文为该算法提供了首个理论准最优性保证。
英文摘要
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.
