# Deep-Unfolded Tikhonov-Regularized Conjugate Gradient Algorithm for MIMO Detection
## 1 Introduction  
Multiple-input multiple-output (MIMO) systems are essential for enhancing spectral efficiency in modern wireless networks. Spatial multiplexing in MIMO systems allows for simultaneous transmission of multiple information streams across different antennas, setting it apart from diversity systems that focus on reliability by transmitting identical information. Achieving higher data rates through spatial multiplexing presents significant challenges at the receiver, particularly in detection complexity and efficiency, which have been the subject of research for over five decades, driving the evolution of MIMO detection methodologies [1, 2]. The core of MIMO detection involves decoding transmitted symbols using known channel characteristics. While maximum likelihood (ML) detection minimizes bit error rate (BER) optimally, it is computationally impractical for physical implementations involving a large number of antennas. Therefore, alternative methods like sphere decoding (SD), zero forcing (ZF), and linear minimum mean squared error (LMMSE) have been developed for near-optimal performance with lower complexity [2]. There are also methods such as Neumann series expansion (NSE), Gauss-Seidel (GS), and conjugate gradient (CG) which utilize iterative matrix-vector multiplication to reduce system complexity [3, 4, 5, 6]. Nonlinear MIMO detectors are useful in reducing interference for subsequent signals, though errors in interference signals can degrade detection efficacy [7]. Advanced approaches, such as the Belief Propagation (BP) algorithm [8], are effective for high number of antennas and low inter-channel correlation, but may introduce delays and degrade performance in-fading channels due to their iterative nature. Therefore, developing a detection strategy that achieves high reliability without requiring excessive amounts of decoding time is one of the major challenges in MIMO systems [9]. In addition to conventional methods discussed above, recent studies have explored both model-driven and data-driven deep learning approaches [10]. Model-driven techniques enhance iterative algorithms like orthogonal approximate message passing (OAMP) [11], alternating direction method of multipliers (ADMM) [12], Viterbi [13], and expectation propagation [14]. Data-driven solutions use deep learning architectures such as autoencoders [15], deep neural networks (DNNs), and convolutional neural networks (CNNs) [16] for high detection accuracy. These deep learning (DL)-based MIMO detection methods outperform traditional detectors under various channel conditions. Although there are studies that discuss model-driven and data-driven approaches either separately or together, the increasing amount of data in new communication systems increasingly favors model-driven methods. Unsupervised deep learning techniques, such as autoencoders, can be used to learn the entire system for MIMO detection, as demonstrated in data-driven MIMO detection [15]. In addition, DetNet uses a model-driven approach to detection using iterative projected gradient descent [17]. Data-driven methods for MIMO detection in fixed-channel scenarios utilize CNNs and DNNs [16]. Another approach uses conventional deep learning network topologies for signal detection in MIMO systems with erroneous channels [18], while another study employs neural networks to identify decision zones for multi-user MIMO systems [19]. Deep unfolding (DU) algorithms, also known as model-driven deep learning methods, constitute a transformative approach that combines classical iterative methods with the adaptive capabilities of neural networks, and are a common solution for MIMO detection [20; 21]. By structuring known iterative algorithms into neural network layers, each iteration treated is treated as a layer [22] that allows parameters to be trained via backpropagation rather than updated demerimetistically in a traditional way. This leads to improved solutions by incorporating additional or modified parameters to capture features that classical methods may miss [23].