## 1 Introduction
The K-ELM method improves the hyperspectral image classification capacity by coupling the principal component analysis, local binary pattern, and gray wolf optimization algorithm with global search capability [24]. Due to the advantages of ELM over backpropagation, the authors of [25] proposed to train autoencoder networks using an extreme learning machine (ELM-AE). A sparse ELM-AE was presented in [26] by adding the \(L_{1}\)-norm in the quadratic term to give a more significant representation of data. Another approach for unlabeled feature extraction consisted of a generalized ELM-AE in which a manifold regularization factor was added to the objective function of the ELM-AE [27]. An unsupervised learning ELM-AE was proposed in [28], inspired by the use of embedding graphs to capture the structure of the input patterns, and was constructed using local Fisher discrimination analysis. A dense connection autoencoder was introduced into multi-layer learning, in which the output features of the previous layers were used by the following layers [29]. A double random hidden layer ELM autoencoder can achieve efficient extraction of features with dimension reduction capability for deep learning [30]. The \(L_{21}\)-norm regularization has been used for the ELM-AE optimization problem, creating a new unsupervised learning framework that included a disperse representation despite the influence of atypical values and noisy data [31]. In addition, the correntropy-based ELM autoencoder presented in [32] was developed to extract features from noisy input patterns. To lower the number of parameters to be tuned and for the efficient calculation of the pseudo-inverse matrix, an ELM-AE with a kernel function was proposed in [33; 34]. From a different perspective, the need for unsupervised batch learning means that an online sequential ELM-AE [35; 36] is an exciting approach to solving large-scale applications. From the works described above, we see that the backpropagation method and the Moore-Penrose inverse are two classical methods of estimating the weights of an autoencoder network. However, using these methods may cause overfitting when the number of unknown variables is larger than the number of training data. In addition, the accuracy is low when the number of hidden neurons is small, and the training time increases significantly when solving higher-dimensional applications. To overcome these drawbacks, researchers have proposed the use of sparse regularization techniques [37], with FISTA being the most widely used algorithm [38]. Although its origin dates back to 2009, this algorithm is still under study, thanks to its convergence speed, accuracy, and straightforward implementation. Several state-of-the-art studies have demonstrated the successful use of this optimization tool in machine learning. For example, in multilayer network design, the minimization problem defined in the autoencoder uses shrinkage-thresholding optimization techniques for dimension reduction and sparse feature representation [26; 32; 39]. For noisy image and video processing, a mixed scheme was presented in [40], in which sparse coding was adopted with deep learning to solve supervised and unsupervised tasks. A visual dictionary classification scheme was addressed in [41] using a simple (deep) convolutional autoencoder, for which the source of inspiration was sparse optimization and the iterative shrinkage-thresholding algorithm [38]. More recent studies [42; 43] have demonstrated this algorithm's development, continuity, and importance for solving convex minimization problems. In particular, they estimated the output weights of ELM and validated their results on classification problems. Due to the importance of sparse regularization in the mathematical modeling of many real situations and the ability of shrinkage-thresholding algorithms to minimize these optimization problems, our work aims to study these optimization techniques in the architecture of an ELM-AE since there are few studies in the literature.