TITLE: does the "convolution theorem" apply to weaker algebraic structures?
QUESTION [23 upvotes]: The Convolution Theorem is often exploited to compute the convolution of two sequences efficiently: take the (discrete) Fourier transform of each sequence, multiply them, and then perform the inverse transform on the result.
The same thing can be done for convolutions in the quotient ring Z/pZ via the analogous Number Theoretic Transform.
Does this procedure generalize to other algebraic structures?  Arbitrary rings?  Semirings?  Fast convolutions over the semiring (min,+) would be particularly useful.
I'm led to suspect this because both sorting and FFT can be computed using the same butterfly-like network with simple operations like "min", "max", "+", and "*" at the nodes of the network, and sorting can be thought of as a kind of convolution.

REPLY [3 votes]: I stumbled onto this question after encountering the problem in Bayesian inference (specifically, performing max-product inference on random variables where $M = L + R$ where $L$ and $R$ (or, an an alternate form of the problem, $M$ and $L$) have known discrete distributions and the distribution on $M$ (or $R$, in the alternative problem) is sought. The probability mass function (PMF) on $M$ that is achieved by maxing out $L$ and $R$ will be the result of the max-convolution between the PMFs of $L$ and $R$.
After playing around with it a bit, I noticed that you can rewrite the problem by generating $u^{(m)}$ for each index $m$ of the result (where $u^{(m)}[\ell] =
L[\ell] R[{m-\ell}]$). When the vectors are nonnegative (in my case, this was true because they are PMFs), then you can perform the $\max_\ell u^{(m)}_\ell$ with the Chebyshev norm:
$$
M[m] = \max_\ell u^{(m)}_\ell \\
= \lim_{p \to \infty} \| u^{(m)} \|_p \\
 = \lim_{p \to \infty} {\left( \sum_\ell {\left( u^{(m)}[\ell] \right)}^{p} \right)}^{\frac{1}{p}}
$$
This in turn can be approximated by substituting the $p^*$-norm, where $p^*$ is a sufficiently large constant (essentially, this exploits the fact that the $p^*$-norm does have inverse operations, whereas the $\max$ does not, but the $p^*$-norm can still approximate the $\max$):
$$
  \approx {\left( \sum_\ell {\left( u^{(m)}[\ell] \right)}^{p^*} \right)}^{\frac{1}{p^*}}\\
  = {\left( \sum_\ell {L[\ell]}^{p^*} ~ {R[{m-\ell}]}^{p^*} \right)}^{\frac{1}{p^*}}\\
  = {\left( \sum_\ell {\left(L^{p^*}\right)}[\ell] ~ {\left(R^{p^*}\right)}[{m-\ell}] \right)}^{\frac{1}{p^*}}\\
  = {\left( L^{p^*} ~*~ R^{p^*} \right)}^{\frac{1}{p^*}}[m]
$$
where $*$ is standard convolution, which can be performed in $n \log(n)$ via FFT. A short paper illustrating the approximation for large-scale probabilistic inference problems is in press at the Journal of Computational Biology (Serang 2015 arXiv preprint). 
Afterward a Ph.D. student, Julianus Pfeuffer, and I hacked out a preliminary bound on the absolute error ($p^*$-norm approximations of the Chebyshev norm are poor when $p^*$ is small, but on indices where the normalized result $\frac{M[m]}{\max_{m'} M[m']}$ is very small, large $p^*$ can be numerically unstable). Julianus and I worked out a modified method for cases when the dynamic range of the result $M$ is very large (if not, then the simple method from the first paper works fine). The modified method operates piecewise over $\log(\log(n))$ different $p^*$ values (including runtime constants, this amounts to $\leq 18$ FFTs even when $n$ is on the scale of particles in the observable universe, and those 18 FFTs can be done in parallel). (Pfeuffer and Serang arXiv link which has a link to a Python demonstration of the approach) The approach (and the Python demo code) generalize to tensors (i.e., numpy.arrays) by essentially combining the element-wise Frobenius norm with multidimensional FFT. 
Oliver Serang