Inference for Support Vector Regression under ℓ1 Regularization

We provide large-sample distribution theory for support vector regression (SVR) with l1-norm along with error bars for the SVR regression coefficients. Although a classical Wald confidence interval obtains from our theory, its implementation inherently depends on the choice of a tuning parameter that scales the variance estimate and thus the width of the error bars. We address this shortcoming by further proposing an alternative large-sample inference method based on the inversion of a novel test statistic that displays competitive power properties and does not depend on the choice of a tuning parameter.

A mixed ℓ1 regularization approach for sparse simultaneous approximation of parameterized PDEs

We present and analyze a novel sparse polynomial technique for the simultaneous approximation of parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our approach treats the numerical solution as a jointly sparse reconstruction problem through the reformulation of the standard basis pursuit denoising, where the set of jointly sparse vectors is infinite. To achieve global reconstruction of sparse solutions to parameterized elliptic PDEs over both physical and parametric domains, we combine the standard measurement scheme developed for compressed sensing in the context of bounded orthonormal systems with a novel mixed-norm based ℓ1 regularization method that exploits both energy and sparsity. In addition, we are able to prove that, with minimal sample complexity, error estimates comparable to the best s-term and quasi-optimal approximations are achievable, while requiring only a priori bounds on polynomial truncation error with respect to the energy norm. Finally, we perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.

Social Sentiment Sensor in Twitter for Predicting Cyber-Attacks Using ℓ1 Regularization

In recent years, online social media information has been subject of study in several data science fields due to its impact on users as a communication and expression channel. Data~gathered from online platforms such as Twitter has the potential to facilitate research over social phenomena based on sentiment analysis, which usually employs Natural Language Processing and Machine Learning techniques to interpret sentimental tendencies related to users opinions and make predictions about real events. Cyber attacks are not isolated from opinion subjectivity on online social networks. Various security attacks are performed by hacker activists motivated by reactions from polemic social events. In this paper, a methodology for tracking social data that can trigger cyber attacks is developed. Our main contribution lies in the monthly prediction of tweets with content related to security attacks and the incidents detected based on ℓ1 regularization.

Injectivity and weak*-to-weak continuity suffice for convergence rates in ℓ1-regularization

We show that the convergence rate of {\ell^{1}}-regularization for linear ill-posed equations is always {{\mathcal{O}}(\delta)} if the exact solution is sparse and if the considered operator is injective and weak*-to-weak continuous. Under the same assumptions convergence rates in case of non-sparse solutions are proven. The results base on the fact that certain source-type conditions used in the literature for proving convergence rates are automatically satisfied.