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

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Author(s): John E. Lavery

Journal: Algorithms
ISSN 1999-4893

Volume: 5;
Issue: 4;
Start page: 421;
Date: 2012;
Original page

Keywords: average | heavy-tailed distribution | Lp average | lp average | median | mode | polynomial time | radial monotonicity | statistical structure | univariate

ABSTRACT
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.
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