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Max-convolution is an important problem closely resembling standardconvolution; as such, max-convolution occurs frequently across many fields.Here we extend the method with fastest known worst-case runtime, which can beapplied to nonnegative vectors by numerically approximating the Chebyshev norm$\| \cdot \|_\infty$, and use this approach to derive two numerically stablemethods based on the idea of computing $p$-norms via fast convolution: Thefirst method proposed, with runtime in $O( k \log(k) \log(\log(k)) )$ (which isless than $18 k \log(k)$ for any vectors that can be practically realized),uses the $p$-norm as a direct approximation of the Chebyshev norm. The secondapproach proposed, with runtime in $O( k \log(k) )$ (although in practice bothperform similarly), uses a novel null space projection method, which extractsinformation from a sequence of $p$-norms to estimate the maximum value in thevector (this is equivalent to querying a small number of moments from adistribution of bounded support in order to estimate the maximum). The $p$-normapproaches are compared to one another and are shown to compute anapproximation of the Viterbi path in a hidden Markov model where the transitionmatrix is a Toeplitz matrix; the runtime of approximating the Viterbi path isthus reduced from $O( n k^2 )$ steps to $O( n $k \log(k))$ steps in practice,and is demonstrated by inferring the U.S. unemployment rate from the S&P 500stock index.

A Bounded $p$-norm Approximation of Max-Convolution for Sub-Quadratic Bayesian Inference on Additive Factors