Table 1.

An overview of linear methods.

Learning representations	Approaches	Mathematical formulations	Optimization problem	Relevant references
Correlation- based learning	Cross-Correlation CCA	$B_i^f(q)u_i\left[k\right]~\sum _{j=1}^{n_y}{A}_j^b(q)y_j\left[k\right]+e\left[k\right]$	Generalized eigenvalue problem	Biesmans et al., 2017; Dmochowski et al., 2017; de Cheveigné et al., 2018; de Cheveigné et al., 2019
Model-based learning	Forward modeling Supervised case: Encoding	$y_j\left[k\right]=B_i^f(q)u_i\left[k\right]+e_j^f\left[k\right]$	Least- squares	Ding and Simon, 2012a; Di Liberto et al., 2015; Alickovic et al., 2016, in rewiev; Fiedler et al., 2017, 2019; Hjortkjær et al., 2018; Kalashnikova et al., 2018; Lesenfants et al., 2018; Lunner et al., 2018; Verschueren et al., 2018; Wong et al., 2018
	Inverse/backward modeling Supervised case: Decoding	$u_i\left[k\right]=\sum _{j=1}^{n_y}{A}_j^b(q)y_j\left[k\right]+e_i^b\left[k\right]$		Mirkovic et al., 2015; O'Sullivan et al., 2015, 2017; Aroudi et al., 2016; Das et al., 2016, 2018; Presacco et al., 2016; Biesmans et al., 2017; Fuglsang et al., 2017; Van Eyndhoven et al., 2017; Zink et al., 2017; Bednar and Lalor, 2018; Ciccarelli et al., 2018; Etard et al., 2018; Hausfeld et al., 2018; Narayanan and Bertrand, 2018; Schäfer et al., 2018; Vanthornhout et al., 2018; Verschueren et al., 2018; Wong et al., 2018; Akbari et al., 2019; Somers et al., 2019