Pantelis Bouboulis

Pantelis Bouboulis, Symeon Chouvardas, Sergios Theodoridis

We present a novel diffusion scheme for online kernel-based learning over networks. So far, a major drawback of any online learning algorithm, operating in a reproducing kernel Hilbert space (RKHS), is the need for updating a growing number of parameters as time iterations evolve. Besides complexity, this leads to an increased need of communication resources, in a distributed setting. In contrast, the proposed method approximates the solution as a fixed-size vector (of larger dimension than the input space) using Random Fourier Features. This paves the way to use standard linear combine-then-adapt techniques. To the best of our knowledge, this is the first time that a complete protocol for distributed online learning in RKHS is presented. Conditions for asymptotic convergence and boundness of the networkwise regret are also provided. The simulated tests illustrate the performance of the proposed scheme.

Pantelis Bouboulis, Spyridon Pougkakiotis, Sergios Theodoridis

We present a new framework for online Least Squares algorithms for nonlinear modeling in RKH spaces (RKHS). Instead of implicitly mapping the data to a RKHS (e.g., kernel trick), we map the data to a finite dimensional Euclidean space, using random features of the kernel's Fourier transform. The advantage is that, the inner product of the mapped data approximates the kernel function. The resulting "linear" algorithm does not require any form of sparsification, since, in contrast to all existing algorithms, the solution's size remains fixed and does not increase with the iteration steps. As a result, the obtained algorithms are computationally significantly more efficient compared to previously derived variants, while, at the same time, they converge at similar speeds and to similar error floors.

George Papageorgiou, Pantelis Bouboulis, Sergios Theodoridis

We consider the task of robust non-linear regression in the presence of both inlier noise and outliers. Assuming that the unknown non-linear function belongs to a Reproducing Kernel Hilbert Space (RKHS), our goal is to estimate the set of the associated unknown parameters. Due to the presence of outliers, common techniques such as the Kernel Ridge Regression (KRR) or the Support Vector Regression (SVR) turn out to be inadequate. Instead, we employ sparse modeling arguments to explicitly model and estimate the outliers, adopting a greedy approach. The proposed robust scheme, i.e., Kernel Greedy Algorithm for Robust Denoising (KGARD), is inspired by the classical Orthogonal Matching Pursuit (OMP) algorithm. Specifically, the proposed method alternates between a KRR task and an OMP-like selection step. Theoretical results concerning the identification of the outliers are provided. Moreover, KGARD is compared against other cutting edge methods, where its performance is evaluated via a set of experiments with various types of noise. Finally, the proposed robust estimation framework is applied to the task of image denoising, and its enhanced performance in the presence of outliers is demonstrated.

Pantelis Bouboulis, Sergios Theodoridis, Charalampos Mavroforakis et al.

The paper presents a new framework for complex Support Vector Regression as well as Support Vector Machines for quaternary classification. The method exploits the notion of widely linear estimation to model the input-out relation for complex-valued data and considers two cases: a) the complex data are split into their real and imaginary parts and a typical real kernel is employed to map the complex data to a complexified feature space and b) a pure complex kernel is used to directly map the data to the induced complex feature space. The recently developed Wirtinger's calculus on complex reproducing kernel Hilbert spaces (RKHS) is employed in order to compute the Lagrangian and derive the dual optimization problem. As one of our major results, we prove that any complex SVM/SVR task is equivalent with solving two real SVM/SVR tasks exploiting a specific real kernel which is generated by the chosen complex kernel. In particular, the case of pure complex kernels leads to the generation of new kernels, which have not been considered before. In the classification case, the proposed framework inherently splits the complex space into four parts. This leads naturally in solving the four class-task (quaternary classification), instead of the typical two classes of the real SVM. In turn, this rationale can be used in a multiclass problem as a split-class scenario based on four classes, as opposed to the one-versus-all method; this can lead to significant computational savings. Experiments demonstrate the effectiveness of the proposed framework for regression and classification tasks that involve complex data.