# On Fast Converging Data-Selective Adaptive Filtering
## 1 Introduction  
In many practical applications, the number of data-acquisition devices is growing at an exponential rate, such as in distributed networks, massive multiple-input multiple-output (MIMO) antennas, and for social networks. This trend calls for the parsimonious use of the acquired data when considering the overwhelming resources required such as storage capacity, device-to-device communications, power consumption, among others. In the era of Big Data, we face the challenge of efficiently utilizing a large amount of data to extract the critical information. In the context of adaptive filtering, the recently proposed strategy to perform data-selection approximating a prescribed update rate appears to be promising [1].  
This new data-selection method prescribes a probability of updating utilizing a threshold based on the mean squared error (MSE) which determines if the acquired data sample contains enough information to justify a change in the parameter estimate. Utilizing a statistical model for the MSE, it is possible to prescribe the probability of updating inherent to the learning algorithm. Also, an additional threshold value can be utilized to verify if the data represents an outlier, i.e., abnormal data.  
Previous work [1] addressed the classical Least Mean Square (LMS), the Affine Projection (AP), and the Recursive Least Squares (RLS) algorithms. Among these algorithms, the RLS has the fastest convergence in stationary environments, and the highest computational complexity, while facing numerical stability issues. In this work, we consider fast converging adaptive filtering algorithms with data selection and apply them to process real data in order to verify their effectiveness in addressing practical problems. The proposed algorithms are the LMS Newton (LMSN) [2], the LMS Quasi-Newton(LMSQN) [3] and the online conjugate gradient (CG) [4; 5; 6; 7; 8; 9]. Like the RLS algorithm, the LMSN has fast convergence even when the input signal is highly correlated. By employing an estimate of the Hessian matrix, the LMSQN methods [10; 11; 12; 13] have a similar computational complexity to the RLS and LMSN algorithms without sacrificing the performance. In fact, one version of the LMSQN algorithm exhibits improved robustness to quantization errors [3]. A low complexity algorithm originates from the Conjugate Gradient (CG) method [4] which does not require the computation of the Hessian matrix inverse as the LMSN, LMSQN, and RLS algorithms. The CG algorithm is attractive since it updates the adaptive filter coefficients based on conjugate directions leading to faster convergence than the gradient-based algorithms such as the LMS.  
We explore the data-selective version of the LMSN, LMSQN and CG algorithms, showing their cost function in different applications, such as equalization and signal enhancement, which were not previously considered in [1; 14; 15]. For each configuration, we describe how to estimate the MSE to allow the prescription of the update rate.  
The performance of the data selective algorithms is evaluated via simulations utilizing synthetic and real data in different adaptive filtering applications. The results indicate that the data selection strategy can indeed be applied to any type of applications of adaptive filters.  
This paper is organized as follows. In Section 2, the data-selection strategy is discussed along with the calculation of the threshold parameter required to achieve a target probability of update. Section 3 describes the data-selective LMSN (DS-LMSN), LMSQN (DS-LMQSN), and the CG (DS-CG) algorithms. Section 4 compares the proposed algorithms. Some concluding remarks are provided in Section 5.