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info.json
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info.json
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{
"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.",
"authors": [
"Julianus Pfeuffer",
"Oliver Serang"
],
"id": "15-319",
"issue": 36,
"pages": [
1,
39
],
"title": "A Bounded p-norm Approximation of Max-Convolution for Sub-Quadratic Bayesian Inference on Additive Factors",
"volume": 17,
"year": 2016
}