Interpolative Decomposition Butterfly Factorization

This paper introduces a kernel-independent interpolative decomposition butterfly factorization (IDBF) as a data-sparse approximation for matrices that satisfy a complementary low-rank property. The IDBF can be constructed in $O(Nlog N)$ operations for an $Ntimes N$ matrix via hierarchical interpolative decompositions (IDs), if matrix entries can be sampled individually and each sample takes $O(1)$ operations. The resulting factorization is a product of $O(log N)$ sparse matrices, each with $O(N)$ non-zero entries. Hence, it can be applied to a vector rapidly in $O(Nlog N)$ operations. IDBF is a general framework for nearly optimal fast matvec useful in a wide range of applications, e.g., special function transformation, Fourier integral operators, high-frequency wave computation. Numerical results are provided to demonstrate the effectiveness of the butterfly factorization and its construction algorithms.