This paper introduces a new class of nonlinear adaptive filters, whose structure is based on Hammerstein model. Such filters derive from the functional link adaptive filter (FLAF) model, defined by a nonlinear input expansion, which enhances the representation of the input signal through a projection in a higher dimensional space, and a subsequent adaptive filtering. In particular, two robust FLAF-based architectures are proposed and designed ad hoc to tackle nonlinearities in acoustic echo cancellation (AEC). The simplest architecture is the split FLAF, which separates the adaptation of linear and nonlinear elements using two different adaptive filters in parallel. In this way, the architecture can accomplish distinctly at best the linear and the nonlinear modeling. Moreover, in order to give robustness against different degrees of nonlinearity, a collaborative FLAF is proposed based on the adaptive combination of filters. Such architecture allows to achieve the best performance regardless of the nonlinearity degree in the echo path. Experimental results show the effectiveness of the proposed FLAF-based architectures in nonlinear AEC scenarios, thus resulting an important solution to the modeling of nonlinear acoustic channels.

The extreme learning machine (ELM) was recently proposed as a unifying framework for different families of learning algorithms. The classical ELM model consists of a linear combination of a fixed number of nonlinear expansions of the input vector. Learning in ELM is hence equivalent to finding the optimal weights that minimize the error on a dataset. The update works in batch mode, either with explicit feature mappings or with implicit mappings defined by kernels. Although an online version has been proposed for the former, no work has been done up to this point for the latter, and whether an efficient learning algorithm for online kernel-based ELM exists remains an open problem. By explicating some connections between nonlinear adaptive filtering and ELM theory, in this brief, we present an algorithm for this task. In particular, we propose a straightforward extension of the well-known kernel recursive least-squares, belonging to the kernel adaptive filtering (KAF) family, to the ELM framework. We call the resulting algorithm the kernel online sequential ELM (KOS-ELM). Moreover, we consider two different criteria used in the KAF field to obtain sparse filters and extend them to our context. We show that KOS-ELM, with their integration, can result in a highly efficient algorithm, both in terms of obtained generalization error and training time. Empirical evaluations demonstrate interesting results on some benchmarking datasets.

Complex-valued neural networks (CVNNs) are a powerful modeling tool for domains where data can be naturally interpreted in terms of complex numbers. However, several analytical properties of the complex domain (such as holomorphicity) make the design of CVNNs a more challenging task than their real counterpart. In this paper, we consider the problem of flexible activation functions (AFs) in the complex domain, i.e., AFs endowed with sufficient degrees of freedom to adapt their shape given the training data. While this problem has received considerable attention in the real case, very limited literature exists for CVNNs, where most activation functions are generally developed in a split fashion (i.e., by considering the real and imaginary parts of the activation separately) or with simple phase-amplitude techniques. Leveraging over the recently proposed kernel activation functions, and related advances in the design of complex-valued kernels, we propose the first fully complex, nonparametric activation function for CVNNs, which is based on a kernel expansion with a fixed dictionary that can be implemented efficiently on vectorized hardware. Several experiments on common use cases, including prediction and channel equalization, validate our proposal when compared to real-valued neural networks and CVNNs with fixed activation functions.