## 1 Introduction  
Robust estimation refers to learning a model parameter for data that have been contaminated by noise and outliers, which is crucial in applications such as motion segmentation [1], pose graph initialization in Structure-from-Motion algorithms [2, 3, 4], and image mosaicing [5, 6, 7]. Random sample consensus (RANSAC) facilitates the accurate estimation of model parameters from datasets containing outliers. RANSAC is achieved by iteratively selecting random subsets of observations, called minimal sets, to generate model hypotheses. To assess the quality of the estimations, RANSAC evaluates each model by maximizing its consensus with the entire dataset. Recent approaches, such as marginalizing sample consensus (MAGSAC) [8] and marginalizing sample consensus plus plus (MAGSAC++) [9], further refine this process by scoring models based on a marginalized inlier counting. Deep learning methods adhere to the overall framework of RANSAC, taking data points as input. Through the deep learning network, the probability of each sample being selected for sampling is computed. The Gumbel-Softmax technique is employed to make the probabilistic sampling process differentiable, enabling the identification of the minimum sample set. Subsequently, a model is estimated based on this minimum sample set, yielding the final matrix results. This paper focuses on the application of robust model estimation algorithms in computer vision, specifically using correspondences obtained from pre-processing matching algorithms to estimate the essential matrix and fundamental matrix between stereo views. The essential matrix describes the geometric relationship between two cameras when the intrinsic camera parameters are known, thereby reflecting the mapping relationship between the two coordinate systems. The fundamental matrix, on the other hand, is independent of the camera's intrinsic parameters and primarily serves to map a point in one image to the corresponding epipolar line in another image. However, previous methods typically relied on the confidence distribution of data points, using the distance to the ground truth model as a direct optimization objective. However, a shorter distance between the matches in the minimal sets and the ground truth model does not necessarily indicate a higher-quality estimated model [10]. We show an illustration in Figure 1a; the ill-conditioned minimal sets, despite being closer to the ground truth model, lead to worse results than the normal minimal sets in line fitting problems. Meanwhile, Figure 1b also shows the lower inlier ratio of the minimal sets associated with the best model compared with the minimal sets associated with the good model. This suggests that within a certain range of error, the distance from the points to the model should not be the sole criterion for sampling. Despite fine-tuning the neural network with ill-conditioned samples being able to somewhat improve model optimization [10], it only partially addresses the issue from a data perspective, the underlying cause of under-optimization (single objective function on confidence distribution) remaining unresolved. Hence, we visualize some of the ill-conditioned samples as Figure 2 and find that ill-conditioned samples are densely clustered within a specific region in space. Furthermore, we analyze the minimum distance between inliers in the good minimal sets and the best minimal sets. Shown in Figure 3, it is found that the points of the best model are more sparsely distributed in space. Figure 1: Line fitting example and model residual analysis. (**a**) Taking 2D line fitting as an example, the solid black line in the figure represents the ground truth model. Although the points in the ill–conditioned minimal set are closer to the ground truth, the fitted line exhibits greater error compared to the one derived from the normal minimal set. (**b**) Residual analysis of models with varying levels of quality. We analyzed the residual distribution of data for the \(1\) k minimum sample set selected by models of varying quality.