A Convex Optimization Approach to Improving Suboptimal Hyperparameters of Sliced Normal Distributions

Brendon K. Colbert, Luis G. Crespo, Matthew M. Peet

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Sliced Normal (SN) distributions are a generalization of Gaussian distributions where the quadratic argument of the exponential is replaced with a sum of squares polynomial. SNs may be used to represent the distribution of a diverse set of random variables including multi-modal, non-symmetric, and skewed distributions. Unfortunately, the likelihood function of a SN includes a normalization constant and the inclusion of this normalization constant makes the likelihood a non-convex function of the hyperparameters which define the SN. In previous work, suboptimal fitting of the hyperparameters was performed by transforming the given data into a higher dimensional monomial basis and selecting the optimal hyperparameters of a Gaussian fit in this space. However, this approach did not account for the effect of lifting on the normalization constant. Indeed, it was observed that as the number of monomials is increased the likelihood of the Sliced Normal can decrease. In this paper, we increase the likelihood of Sliced Normals found using the previous method by developing a convex formulation which scales the covariance matrix of the Gaussian fit such that the likelihood of the Sliced Normal is maximized. The result is significant improvements of the log likelihood of fitted SN distributions, including a significant increase, especially for problems with 500+ monomials.

Original languageEnglish (US)
Title of host publication2020 American Control Conference, ACC 2020
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages4478-4483
Number of pages6
ISBN (Electronic)9781538682661
DOIs
StatePublished - Jul 2020
Event2020 American Control Conference, ACC 2020 - Denver, United States
Duration: Jul 1 2020Jul 3 2020

Publication series

NameProceedings of the American Control Conference
Volume2020-July
ISSN (Print)0743-1619

Conference

Conference2020 American Control Conference, ACC 2020
CountryUnited States
CityDenver
Period7/1/207/3/20

ASJC Scopus subject areas

  • Electrical and Electronic Engineering

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    Colbert, B. K., Crespo, L. G., & Peet, M. M. (2020). A Convex Optimization Approach to Improving Suboptimal Hyperparameters of Sliced Normal Distributions. In 2020 American Control Conference, ACC 2020 (pp. 4478-4483). [9147403] (Proceedings of the American Control Conference; Vol. 2020-July). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.23919/ACC45564.2020.9147403