TY - GEN

T1 - A Convex Optimization Approach to Improving Suboptimal Hyperparameters of Sliced Normal Distributions

AU - Colbert, Brendon K.

AU - Crespo, Luis G.

AU - Peet, Matthew M.

N1 - Funding Information:
This work was supported by NASA and NSF grants CNS-1739990, and CMMI-1538374. 1Arizona State University, Tempe, AZ, 85287, USA. Brendon Colbert, brendon.colbert@asu.edu, is the corresponding author. 2NASA Langley Research Center, Hampton, VA, 23681, USA
Publisher Copyright:
© 2020 AACC.

PY - 2020/7

Y1 - 2020/7

N2 - 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.

AB - 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.

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U2 - 10.23919/ACC45564.2020.9147403

DO - 10.23919/ACC45564.2020.9147403

M3 - Conference contribution

AN - SCOPUS:85089600476

T3 - Proceedings of the American Control Conference

SP - 4478

EP - 4483

BT - 2020 American Control Conference, ACC 2020

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2020 American Control Conference, ACC 2020

Y2 - 1 July 2020 through 3 July 2020

ER -