Non-Asymptotic Analysis of $\ell _{1}$ -Norm Support Vector Machines

Support vector machines (SVMs) with the ℓ
₁
-penalty became a standard tool in the analysis of highdimensional classification problems with sparsity constraints in many applications, including bioinformatics and signal processing. We give non-asymptotic results on the performance of ℓ
₁
-SVM in identification of sparse classifiers. We show that an N-dimensional s-sparse classification vector can be (with high probability) well approximated from only O(s log(N)) Gaussian trials. We derive similar estimates also in the presence of misclassifications and for the so-called doubly regularized SVM, which combines the ℓ
₁
- and the ℓ
₂
-penalty. Similar bounds were obtained earlier in the analysis of LASSO and 1-Bit compressed sensing.