## 1 Introduction  
Cardiovascular diseases (CVDs) are the leading cause of death worldwide. According to the statistics in [1], one person dies from CVDs every 37 s in the United States. The early diagnosis and detection of CVDs can help in treating and avoiding complications that can lead to death [2]. ECG is the gold standard for monitoring vital signs and for diagnosing, controlling, and preventing CVDs [3, 4]. Studies have shown that young people, especially athletes, are more susceptible to sudden cardiac arrest than ever before [5]. Regular ECG monitoring facilitates the early detection of CVDs by measuring the electrical activity of the heart and conveying information about cardiac function [2]. However, most traditional ECG devices limit the user's physical activity. For instance, the Zio Patch is a wearable device worn on the chest. However, prolonged use during multi-day monitoring may elevate the risk of skin sensitization in individuals with sensitive skin [6]. Conversely, the Apple Watch, a popular wrist-based ECG monitoring device, can only measure signals for 30 s and necessitates active user participation, thereby hinders long-term continuous ECG monitoring. Holter monitor is a device used to record the electrical activity of the heart, especially in dynamic electrocardiogram (DCG). However, its recording leads are limited and cannot reflect the condition of the entire heart. Moreover, the patient is in an active state, which will affect the recording quality of the electrocardiogram to some extent. PPG is a noninvasive signal that reflects the amount of blood pulsation in tissues [7]. The waveform shape (i.e., signal morphology), pulse interval, and amplitude characteristics of PPG provide crucial information about the cardiovascular system [8], including heart rate, heart rate variability [9], respiration [10], and blood pressure [11]. PPGs are more accessible and less expensive to set up than ECGs, making them nearly ubiquitous in clinics and hospitals as finger or toe clips and oximeters. These devices do not require constant user intervention and offer a continuous, long-term detection capability without causing skin irritation. Despite the frequent use of PPG for healthcare monitoring [12], ECG remains the standard and an essential measure for medical diagnosis and is supported by a large body of literature and research. Clinicians still rely on ECG rather than PPG in clinical diagnosis. However, the peak-to-peak intervals of PPG are highly correlated with the RR intervals of ECG [8], suggesting the possibility of deriving ECG from PPG. Therefore, attempts can be made to utilize this correlation to reconstruct ECGs directly from PPGs if ECG signals can be successfully reconstructed from PPG signals acquired from today's wearable devices. In this manner, machine learning would enable the clinical diagnosis of cardiovascular diseases to be conducted at any time, in any location, and in real-time. Several studies have proposed using the discrete cosine transform (DCT) method [13], cross-domain joint dictionary learning (XDJDL) method [14], scattering wavelet transform (SWT) method [15], lightweight neural network model [16], and deep learning models based on encoder-decoder structures [17] to reconstruct ECG signals beat by beat. The first three studies use mathematical methods to investigate the correlation between ECG and PPG. In [13], the authors propose a beat-based linear regression model. However, the relationship between ECG and PPG is not linear. In [14], the authors use dictionary learning to learn the features between ECG and PPG. However, the correlation coefficient on the BIDMC dataset [18] is only 0.82. In [15], the authors propose a beat-based nonlinear regression model. The last two studies use deep learning methods to explore the correlation between ECG and PPG. In [16], the authors propose a deep learning model for beat-based autoencoders.