Variable Projection Support Vector Machines and Some Applications Using Adaptive Hermite Expansions

In this paper, we develop the so-called variable projection support vector machine (VP-SVM) algorithm that is a generalization of the classical SVM. In fact, the VP block serves as an automatic feature extractor to the SVM, which are trained simultaneously. We consider the primal form of the arising optimization task and investigate the use of nonlinear kernels. We show that by choosing the so-called adaptive Hermite function system as the basis of the orthogonal projections in our classification scheme, several real-world signal processing problems can be successfully solved. In particular, we test the effectiveness of our method in two case studies corresponding to anomaly detection. First, we consider the detection of abnormal peaks in accelerometer data caused by sensor malfunction. Then, we show that the proposed classification algorithm can be used to detect abnormalities in ECG data. Our experiments show that the proposed method produces comparable results to the state-of-the-art while retaining desired properties of SVM classification such as light weight architecture and interpretability. We implement the proposed method on a microcontroller and demonstrate its ability to be used for real-time applications. To further minimize computational cost, discrete orthogonal adaptive Hermite functions are introduced for the first time.