## 1 Introduction  
Face Super-Resolution (FSR) is a subfield of image super-resolution that focuses on restoring Low-Resolution (LR) face images to High-Resolution (HR) counterparts using algorithms. It can increase the resolution of an LR face image of low quality and recover the details. Its purpose is to address the limitations posed by image acquisition systems or environmental conditions in many real-world scenarios [1]. By doing so, FSR aims to enhance the quality and improve the visibility of key facial features that are often degraded in LR images. This technique finds significant applications in various domains, including face vision tasks [2], public security, and other relevant fields [3, 4]. Existing FSR methods can be categorized into two main groups: traditional methods and deep learning-based methods. Traditional methods can be further classified into interpolation-based methods, reconstruction-based methods, and learning-based methods. Notably, [PERSON] et al. [5] were among the first to propose FSR methods that employed manual image priors and Gaussian image pyramids for face reconstruction. Since then, numerous significant advancements have been made in this field. During the initial stages of FSR research, researchers primarily focused on designing shallow learning-based methods that leveraged techniques such as local linear embedding [6], eigentransformation [7], and principal component analysis [8]. However, these methods demonstrated limited representation capabilities, making it challenging to generate high-quality HR face images. In recent years, deep learning-based approaches leveraging Convolutional Neural Networks (CNNs) have made remarkable advancements in various computer vision tasks, including FSR. The limited information available in LR images makes the task of recovering high-resolution images unstable and non-unique. As a result, there are infinite possible high-resolution images corresponding to a single low-resolution image, making super-resolution challenging. To obtain stable and meaningful results, it is essential to employ appropriate optimization algorithms. Researchers have explored various methods, including deep learning networks and image priors, to address the ill-posed nature of super-resolution problems and enhance the performance and stability of super-resolution techniques. Neural networks, with their ability to extract deep features from images, are particularly meaningful for super-resolution tasks with insufficient feature information. [PERSON] et al. [9] utilize an extremely deep residual network to learn feature representations of images. The increased depth enhances the network's capability for deep feature extraction, leading to the learning of more powerful and informative feature representations, thus achieving more accurate and detailed super-resolution reconstruction. On the other hand, [PERSON] et al. [10] employ a Laplacian pyramid to decompose LR images into multiple scales. Deep convolutional neural networks are then applied at each scale to extract features, resulting in improved SR performance. To exploit neural networks for recovering finer facial structures, researchers have proposed several studies to incorporate additional face prior information [11], such as facial parsing maps [12], facial heat maps [13], and facial landmarks [14; 15], to guide the network during face reconstruction. Although the inclusion of prior information enhances network performance, it presents significant limitations. Firstly, acquiring and annotating the prior information requires additional effort, and obtaining reliable priors from low-resolution face images is challenging [16]. Secondly, inaccurate face prior information can lead to erroneous reconstruction outcomes. To address these challenges, alternative methods without prior knowledge have been proposed [16; 17; 18; 19].