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A Comparative Study of $$\text {L}_1$$ L 1 and $$\text {L}_2$$ L 2 Norms in Support Vector Data Descriptions

In: Control Charts and Machine Learning for Anomaly Detection in Manufacturing

Author

Listed:
  • Edgard M. Maboudou-Tchao

    (University of Central Florida)

  • Charles W. Harrison

    (University of Central Florida)

Abstract

The Support Vector Data Description ( $$\text {L}_1$$ L 1 SVDD) is a non-parametric one-class classification algorithm that utilizes the $$\text {L}_1$$ L 1 norm in its objective function. An alternative formulation of SVDD, called $$\text {L}_2$$ L 2 SVDD, uses a $$\text {L}_2$$ L 2 norm in its objective function and has not been extensively studied. $$\text {L}_1$$ L 1 SVDD and $$\text {L}_2$$ L 2 SVDD are formulated as distinct quadratic programming (QP) problems and can be solved with a QP-solver. The $$\text {L}_2$$ L 2 SVDD and $$\text {L}_1$$ L 1 SVDD’s ability to detect small and large shifts in data generated from multivariate normal, multivariate t, and multivariate Laplace distributions is evaluated. Similar comparisons are made using real-world datasets taken from various applications including oncology, activity recognition, marine biology, and agriculture. In both the simulated and real-world examples, $$\text {L}_2$$ L 2 SVDD and $$\text {L}_1$$ L 1 SVDD perform similarly, though, in some cases, one outperforms the other. We propose an extension of the SMO algorithm for $$\text {L}_2$$ L 2 SVDD, and we compare the runtimes of four algorithms: $$\text {L}_2$$ L 2 SVDD (SMO), $$\text {L}_2$$ L 2 SVDD (QP), $$\text {L}_1$$ L 1 SVDD (SMO), and $$\text {L}_1$$ L 1 SVDD (QP). The runtimes favor $$\text {L}_1$$ L 1 SVDD (QP) versus $$\text {L}_2$$ L 2 SVDD (QP), sometimes substantially; however using SMO reduces the difference in runtimes considerably, making $$\text {L}_2$$ L 2 SVDD (SMO) feasible for practical applications. We also present gradient descent and stochastic gradient descent algorithms for linear versions of both the $$\text {L}_1$$ L 1 SVDD and $$\text {L}_2$$ L 2 SVDD. Examples using simulated and real-world data show that both methods perform similarly. Finally, we apply the $$\text {L}_1$$ L 1 SVDD and $$\text {L}_2$$ L 2 SVDD to a real-world dataset that involves monitoring machine failures in a manufacturing process.

Suggested Citation

  • Edgard M. Maboudou-Tchao & Charles W. Harrison, 2022. "A Comparative Study of $$\text {L}_1$$ L 1 and $$\text {L}_2$$ L 2 Norms in Support Vector Data Descriptions," Springer Series in Reliability Engineering, in: Kim Phuc Tran (ed.), Control Charts and Machine Learning for Anomaly Detection in Manufacturing, pages 217-241, Springer.
    • Handle: RePEc:spr:ssrchp:978-3-030-83819-5_9
    DOI: 10.1007/978-3-030-83819-5_9
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