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2021 Large-sample properties of unsupervised estimation of the linear discriminant using projection pursuit
Una Radojičić, Klaus Nordhausen, Joni Virta
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Electron. J. Statist. 15(2): 6677-6739 (2021). DOI: 10.1214/21-EJS1956

Abstract

We study the estimation of the linear discriminant with projection pursuit, a method that is unsupervised in the sense that it does not use the class labels in the estimation. Our viewpoint is asymptotic and, as our main contribution, we derive central limit theorems for estimators based on three different projection indices, skewness, kurtosis, and their convex combination. The results show that in each case the limiting covariance matrix is proportional to that of linear discriminant analysis (LDA), a supervised estimator of the discriminant. An extensive comparative study between the asymptotic variances reveals that projection pursuit gets arbitrarily close in efficiency to LDA when the distance between the groups is large enough and their proportions are reasonably balanced. Additionally, we show that consistent unsupervised estimation of the linear discriminant can be achieved also in high-dimensional regimes where the dimension grows at a suitable rate to the sample size, for example, pn=o(n13) is sufficient under skewness-based projection pursuit. We conclude with a real data example and a simulation study investigating the validity of the obtained asymptotic formulas for finite samples.

Funding Statement

The work of Joni Virta was supported by the Academy of Finland (Grant 335077).

Acknowledgments

The authors thank the Editors and two referees for helpful and insightful comments and suggestions that greatly improved the manuscript.The work of Joni Virta was supported by the Academy of Finland (Grant 335077).

Citation

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Una Radojičić. Klaus Nordhausen. Joni Virta. "Large-sample properties of unsupervised estimation of the linear discriminant using projection pursuit." Electron. J. Statist. 15 (2) 6677 - 6739, 2021. https://doi.org/10.1214/21-EJS1956

Information

Received: 1 March 2021; Published: 2021
First available in Project Euclid: 30 December 2021

Digital Object Identifier: 10.1214/21-EJS1956

Keywords: clustering , kurtosis , linear discriminant analysis , Projection pursuit , skewness

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Vol.15 • No. 2 • 2021
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