## 1 Introduction  
Distance Metric Learning (DML), usually referring to learning a Mahalanobis matrix from the given side information, has been an active studying field for the last two decades [1, 2, 3, 4]. Compared to those off-the-shelf distance functions, e.g., Euclidean distance, DML takes the correlations and weights of the features into distance consideration, thus being more appropriate for various downstream tasks. Its efficiency has been validated by a large spectrum of applications [5, 6, 7], for example, few-shot learning [8, 9], face recognition [9, 10], and fault detection [11, 12]. Despite the success of existing DML methods, they rely on massive side information constructed by labeled information [13]. However, manually labeling the data is a labor-consuming task [14], and sometimes it needs domain knowledge [15, 16] to provide meaningful labeling information, e.g., labeling the checkup samples of patients. To solve this issue, researchers have devoted themselves to Semi-Supervised Distance Metric Learning (SSDML). SSDML intends to learn a Mahalanobis matrix from limited labeled data as well as a large amount of unlabeled data, such that under this metric, similar instances are brought closer together whereas dissimilar ones are pushed farther away. Inspired by the unsupervised dimensionality reduction methods, which aim to preserve some properties in the original data space, a lot of SSDML approaches based on manifold-based regularization terms have been proposed in the last decades [1; 17; 18; 19; 20; 21; 22]. For example, [PERSON] [1] proposed to project the data into a new space, where labeled data has the maximum margin constraint, and the unlabeled data has the maximum variance. Similarly, [PERSON] and [PERSON] [18] constructed a novel SDML method by retaining locally linear relationships between close data points in the transformed space and proposed a regularization term based on Locally Linear Embedding (LLE) [23]. The above regularization terms cannot boost the discriminative ability of the model. There are also some SSDML methods based on the Laplacian graph [19; 21; 22; 24; 25; 26; 27], i.e., Laplacian Regularized Metric Learning (LRML) [17] utilized the graph Laplacian to preserve the neighbor relationship of the original space. However, the above graph Laplacian construction process does not take the labeled information into consideration. To mitigate this issue, [PERSON] and [PERSON] [21] proposed to utilize Markov random walk technology to transform the strong limited labeled information into a Laplacian matrix. [PERSON] et al. [20] also took the density information of each instance into the Laplacian graph construction process. However, these methods rely on a default metric to determine the affinities among the samples, which contradicts the goal of metric learning. If the default metric is an appropriate metric, why should we still strive to search for another metric? There are also a few works that do not depend on manifold-based regularization. For example, Semi-Supervised Metric Learning Paradigm with Hyper Sparsity (SERAPH) [28] and Semi-Supervised Regularized Large Margin Distance Metric Learning (S-RLMM) [29]. SERAPH is an information-theoretic metric learning method, and it maximizes the entropy on the labeled data while minimizing the entropy on the unlabeled data. However, the time complexity of these methods is at least quadratically dependent on the number of training instances (Table 4 provides a brief time complexity analysis of some representative methods), which means these methods can hardly scale to large-scale datasets. Moreover, these methods rely on a fixed metric to mine the information similarities between samples, which contradicts the goal of learning a metric from data.