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

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 | · |_∞, 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 k2 ) 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.