## 2 State-of-the-Art
### Inverse Problem
Moreover, the fact that simple networks can learn the patterns of single time points and predict reasonable inverse solutions is a major point in lowering complexity and consequently computational cost. Additional ESI neural networks frameworks include a denoising AutoEncoder (DST-DAE) consisting of six layers, three encoding blocks and three decoding blocks. This method was able to directly map the EEG and magnetoencephalogram (MEG) signals to the cortical sources, reducing the localization error to less than a millimeter [46]. In this regard, both temporal and spatial information is utilized on synthetic data, in order to extract inverse mapping. The main advantage of this method lies in its resistance and robustness against low Signal-to-Noise ratio (SNR), resulting in efficient source estimation with excellent denoising properties. Added together, the recent advances in Machine Learning ESI estimation indicate the efficacy of the employed procedures in contrast to the traditional model-driven approaches, especially since in all of the data-driven methods, very few or no mathematical priors were used, while the need for optimizing parameters for new data is absent. Apart from Deep Learning designs, recent studies include hierarchical Bayesian analysis methods for source localization. The importance of Bayesian Models relies on the incorporation of statistical a priori information about the sources, eliminating common problems such as ghost sources and uncorrelated activation transition [39]. The main advantage of such solutions is the combination of data-driven learning with sparse priors, minimizing the cost function and maximizing the probability of correlated sources [47]. Most recently, a modified Bayesian approach was introduced, applying the \(\ell_{20}\) mixed norm instead of \(\ell_{21}\) (primarily used with Bayesian Methods) and multivariate Bernoulli Laplacian priors [12], with the main difference between the Bayesian Models being the probability distributions for the correlation of the sources. This method was able to provide sparser solutions, minimizing the underestimation of the intensity of activations, indicating higher localization performance than other conventional Bayesian methods, in both simulated and in real auditory and visual evoked data but at high computational cost, requiring almost 58 times more time than the \(\ell_{21}\) mixed norm approaches. In a related study [48], a computational efficient Expectation-Maximization algorithm with the use of steady-state Kalman Filter (SS-KF) and steady-state Fixed Interval Smoother (SS-FIS) provided a significant performance enhancement compared to other existing methods, while encapsulating the spatial dependencies between the sources. The computational burden mitigation relies on the use of SS-KF and SS-FIS that are computed once throughout the estimation of the sources without loss of model accuracy, subsequently reducing 12-fold the time needed for results output compared to the full KF/FIS method. It should be pointed out that the aforementioned methods present a theoretically zero localization error, outperforming the solutions provided by neural networks. Nevertheless, the advantage of neural networks lies in the small amount of time needed in order to compute the inverse solution. From this standpoint, the ability of a fast calculation of the source estimates is of utmost importance in the aspiration of developing real-time ESI applications in the near future. A comparison of methods presented, illustrating the current state-of-the-art, is shown in Table 1.