# Computing Haar Measures

## TL;DR

This paper proves that Haar integrals on computably compact groups are computable, provides explicit algorithms with error bounds, and analyzes their complexity, linking them to classical integration and complexity classes.

## Contribution

It establishes the computability of Haar integrals on computably compact groups and presents explicit algorithms with rigorous error analysis.

## Key findings

- Haar integrals are computable on all computably compact groups.
- Explicit algorithms with guaranteed convergence and error bounds are provided.
- Haar integrals on SO(3) and SU(2) reduce to Euclidean integration and characterize #P$_1$.

## Abstract

According to Haar's Theorem, every compact group $G$ admits a unique (regular, right and) left-invariant Borel probability measure $\mu_G$. Let the Haar integral (of $G$) denote the functional $\int_G:\mathcal{C}(G)\ni f\mapsto \int f\,d\mu_G$ integrating any continuous function $f:G\to\mathbb{R}$ with respect to $\mu_G$. This generalizes, and recovers for the additive group $G=[0;1)\mod 1$, the usual Riemann integral: computable (cmp. Weihrauch 2000, Theorem 6.4.1), and of computational cost characterizing complexity class #P$_1$ (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 $\mathcal{SO}(3)$ and $\mathcal{SU}(2)$ we reduce the Haar integral to and from Euclidean/Riemann integration. In particular both also characterize #P$_1$. Implementation and empirical evaluation using the iRRAM C++ library for exact real computation confirms the (thus necessary) exponential runtime.

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Source: https://tomesphere.com/paper/1906.12220