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Kernel matrices are popular in machine learning and scientific computing, but they are limited by their quadratic complexity in both construction and storage. It is well-known that as one varies the kernel parameter, e.g., the width parameter in radial basis function kernels, the kernel matrix changes from a smooth low-rank kernel to a diagonally-dominant… (More)

This paper presents an efficient multiscale butterfly algorithm for computing Fourier integral operators (FIOs) of the form (Lf)(x) = R d a(x, ξ)e 2πıΦ(x,ξ) f (ξ)dξ, where Φ(x, ξ) is a phase function, a(x, ξ) is an amplitude function, and f (x) is a given input. The frequency domain is hierarchically decomposed into a union of Cartesian coronas. The… (More)

The hierarchical interpolative factorization (HIF) offers an efficient way for solving or pre-conditioning elliptic partial differential equations. By exploiting locality and low-rank properties of the operators, the HIF achieves quasi-linear complexity for factorizing the discrete elliptic operator and linear complexity for solving the associated linear… (More)

- Yingzhou Li, Haizhao Yang, Ying Lexing
- 2015

This paper introduces the multidimensional butterfly factorization as a data-sparse representation of multidimensional kernel matrices that satisfy the complementary low-rank property. This factorization approximates such a kernel matrix of size N × N with a product of O(log N) sparse matrices, each of which contains O(N) nonzero entries. We also propose… (More)

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