$\ell_p$ Slack Norm Support Vector Data Description
This work addresses one-class classification for outlier detection, but it is incremental as it modifies an existing method with a generalized norm.
The authors tackled the problem of one-class classification by generalizing the support vector data description (SVDD) to use an ℓp-norm slack penalty instead of the standard ℓ1-norm, enabling a non-linear cost function and sparsity tuning. Experimental results on several datasets showed improved performance compared to alternatives, though no concrete numbers were provided.
The support vector data description (SVDD) approach serves as a de facto standard for one-class classification where the learning task entails inferring the smallest hyper-sphere to enclose target objects while linearly penalising any errors/slacks via an $\ell_1$-norm penalty term. In this study, we generalise this modelling formalism to a general $\ell_p$-norm ($p\geq1$) slack penalty function. By virtue of an $\ell_p$ slack norm, the proposed approach enables formulating a non-linear cost function with respect to slacks. From a dual problem perspective, the proposed method introduces a sparsity-inducing dual norm into the objective function, and thus, possesses a higher capacity to tune into the inherent sparsity of the problem for enhanced descriptive capability. A theoretical analysis based on Rademacher complexities characterises the generalisation performance of the proposed approach in terms of parameter $p$ while the experimental results on several datasets confirm the merits of the proposed method compared to other alternatives.