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Univariate Lp and ɭ p Averaging, 0 < p < 1, in Polynomial Time by Utilization of Statistical Structure

1
Mathematical Sciences Division and Computing Sciences Division, Army Research Office, Army Research Laboratory, P.O. Box 12211, Research Triangle Park, NC 27709-2211, USA
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Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA
Algorithms 2012, 5(4), 421-432; https://doi.org/10.3390/a5040421
Received: 28 July 2012 / Revised: 6 September 2012 / Accepted: 17 September 2012 / Published: 5 October 2012
We present evidence that one can calculate generically combinatorially expensive Lp and lp averages, 0 < p < 1, in polynomial time by restricting the data to come from a wide class of statistical distributions. Our approach differs from the approaches in the previous literature, which are based on a priori sparsity requirements or on accepting a local minimum as a replacement for a global minimum. The functionals by which Lp averages are calculated are not convex but are radially monotonic and the functionals by which lp averages are calculated are nearly so, which are the keys to solvability in polynomial time. Analytical results for symmetric, radially monotonic univariate distributions are presented. An algorithm for univariate lp averaging is presented. Computational results for a Gaussian distribution, a class of symmetric heavy-tailed distributions and a class of asymmetric heavy-tailed distributions are presented. Many phenomena in human-based areas are increasingly known to be represented by data that have large numbers of outliers and belong to very heavy-tailed distributions. When tails of distributions are so heavy that even medians (L1 and l1 averages) do not exist, one needs to consider using lp minimization principles with 0 < p < 1. View Full-Text
Keywords: average; heavy-tailed distribution; Lp average; lp average; median; mode; polynomial time; radial monotonicity; statistical structure; univariate average; heavy-tailed distribution; Lp average; lp average; median; mode; polynomial time; radial monotonicity; statistical structure; univariate
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Lavery, J.E. Univariate Lp and ɭ p Averaging, 0 < p < 1, in Polynomial Time by Utilization of Statistical Structure. Algorithms 2012, 5, 421-432.

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