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Fast Approximate Determinants Using Rational Functions
Published 6 May 2024 in cs.DS, cs.NA, and math.NA | (2405.03474v1)
Abstract: We show how rational function approximations to the logarithm, such as $\log z \approx (z2 - 1)/(z2 + 6z + 1)$, can be turned into fast algorithms for approximating the determinant of a very large matrix. We empirically demonstrate that when combined with a good preconditioner, the third order rational function approximation offers a very good trade-off between speed and accuracy when measured on matrices coming from Mat\'ern-$5/2$ and radial basis function Gaussian process kernels. In particular, it is significantly more accurate on those matrices than the state-of-the-art stochastic Lanczos quadrature method for approximating determinants while running at about the same speed.
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