## 1 Introduction  
Multiple-input-multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) technology plays an important role in fifth-generation mobile communication systems, enabling enhanced spectrum efficiency and transmission rates in wireless communications [1; 2]. Compared with single-input-single-output systems, MIMO systems require multiple signals to be transmitted, resulting in a significant increase in the number of channels. Consequently, data detection becomes more challenging and crucial for MIMO-OFDM systems. Extensive research has been conducted on data detection techniques for MIMO-OFDM systems, with a focus on achieving higher performance while reducing complexity. In [3], linear estimation methods are analyzed, including data detection on the basis of the ZF criterion and the MMSE criterion. In Ref. [4], subspace is used for data detection for MIMO-OFDM systems. This subspace identification algorithm can achieve good estimation performance in scenarios where the channel does not change rapidly. However, it fails to work effectively in situations with rapid channel variations. Ref. [5] employs Monte Carlo methods and iterative approaches to jointly estimate the channel and signals for MIMO-OFDM systems. Ref. [6] analyzes MIMO-OFCDM with zero-forcing successive interference cancellation and imperfect channel estimation. This approach takes into account the correlation between errors, making it more accurate than methods that do not consider such correlation. Ref. [7] propose a joint iterative detection for the MIMO-OFCDM system, which enables space and frequency diversity gains to be jointly exploited. As the number of detections increases and the frequency domain expansion factor grows, the performance improvement becomes more significant. Ref. [8] propose an iterative algorithm based on the variational Bayesian inference framework to address the signal detection problem in OFDM-V-MIMO systems under severe communication environments. In recent years, manifold optimization has gained significant attention and has found wide application in various fields, including data detection [9]. The main idea of Riemannian manifold optimization is if the constraints in an optimization problem possess certain geometric properties, these constraints can be used to construct a Riemannian manifold. Then, gradient descent search can be performed directly on this Riemannian manifold, without the need for computations in the standard Euclidean space. Applying the method of Riemannian manifold optimization can transform a constrained optimization problem into an unconstrained optimization problem on the manifold. Subsequently, unconstrained optimization algorithms based on gradient descent can be used to solve the optimization problem on the Riemannian manifold. This optimization method is equivalent to solving the optimization problem directly in the Euclidean space. Ref. [10] applies manifold learning to channel state information reconstruction. Manifold learning can recover low-dimensional manifolds from high-dimensional spaces while preserving the inherent manifold structure of the data. In [11], a manifold optimization-based data detection method for massive multiuser MIMO has been proposed. This method utilizes manifolds and converts the constrained optimization problem into an unconstrained problem on this manifold. It directly employs the gradient descent method for searching. Ref. [12] introduces a manifold-based joint channel estimation and signal detection method for MIMO, which converts the data detection problem into a manifold optimization problem in the real domain. Although the above methods maintain low algorithm complexity, they are only applicable to MIMO systems with block fading channels. In [13], Riemannian manifold optimization is applied to hybrid beamforming to ensure the beamforming required for radar sensing. It has achieved good results in the radar fields, with high computational efficiency and nearly optimal performance. Ref.