5 research outputs found
Computing Haar Measures
According to Haar's Theorem, every compact group admits a unique
(regular, right and) left-invariant Borel probability measure . Let the
Haar integral (of ) denote the functional integrating any continuous function with
respect to . This generalizes, and recovers for the additive group
, the usual Riemann integral: computable (cmp. Weihrauch 2000,
Theorem 6.4.1), and of computational cost characterizing complexity class
#P (cmp. Ko 1991, Theorem 5.32). We establish that in fact every computably
compact computable metric group renders the Haar integral computable: once
asserting computability using an elegant synthetic argument, exploiting
uniqueness in a computably compact space of probability measures; and once
presenting and analyzing an explicit, imperative algorithm based on 'maximum
packings' with rigorous error bounds and guaranteed convergence. Regarding
computational complexity, for the groups and
we reduce the Haar integral to and from Euclidean/Riemann
integration. In particular both also characterize #P. Implementation and
empirical evaluation using the iRRAM C++ library for exact real computation
confirms the (thus necessary) exponential runtime