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

Abstract: Max-convolution is an important problem closely resembling standard convolution; as such, max-convolution occurs frequently across many fields. Here we extend the method with fastest known worst-case runtime, which can be applied to nonnegative vectors by numerically approximating the Chebyshev norm $| \cdot |_\infty$, and use this approach to derive two numerically stable methods based on the idea of computing $p$-norms via fast convolution: The first method proposed, with runtime in $O( k \log(k) \log(\log(k)) )$ (which is less 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 second approach proposed, with runtime in $O( k \log(k) )$ (although in practice both perform similarly), uses a novel null space projection method, which extracts information from a sequence of $p$-norms to estimate the maximum value in the vector (this is equivalent to querying a small number of moments from a distribution of bounded support in order to estimate the maximum). The $p$-norm approaches are compared to one another and are shown to compute an approximation of the Viterbi path in a hidden Markov model where the transition matrix is a Toeplitz matrix; the runtime of approximating the Viterbi path is thus 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 500 stock index.