## 4 Optimization for ISMLP
However, setting a large \(\mu\) will impose the learned **R** close to the prior metric \(\textbf{I}_{p}\), which prevents ISMLP to learn the correlations and weights of the feature in the reduced space. Figure 4c documents the results on \(p\). Recall that \(p\) is the dimensionality of the reduced space, and setting a small \(p\) will lose a large amount of information of the original data. As a result, we can observe inferior results with small \(p\); however, as \(p\) increases, the performance becomes stable. To compromise between accuracy and computational efficiency, we set \(p\) as 50 in all experiments. We further conduct an experiment to test the sensitivity of ISMLP on the number of proxy vectors and document the result in Figure 4d. It has nearly the same tendency as Figure 4c; when we learn a small number of proxy vectors, instances from the other classes will be mixed up together, which undoubtedly degrades the discriminative ability of the model. Increasing the number of proxy vectors will boost the performance to some extent,

Figure 4: The R@1 accuracy under different parameters. We keep the other parameters fixed when analyzing the current one. (**a**) \(\lambda\); (**b**) \(\mu\); (**c**) Reduced dimensionality; (**d**) Number of proxy vectors. and it can help to discover the latent pattern within a class. Such an idea is also utilized in some cluster-based multi-metric learning methods [30; 50]. However, allocating too many proxy vectors will cost additional computational resources. ## 6 Conclusions

In this paper, we propose an efficient information-theoretic-based semi-supervised metric learning method called ISMLP. By learning a hierarchical form of the Mahalanobis matrix as well as a set of proxy vectors, ISMLP casts the semi-supervised metric learning problem as a probability model. Importantly, the entropy regularization term is adopted to mine the rich unlabeled information. We further prove that ISMLP can be efficiently solved via the alternating direction method. Extensive experiments on five large-scale image datasets reveal that (1) the proposed probability model based on proxy vectors can accurately mine the rich information of unlabeled data, and is thus profitable for semi-supervised learning tasks; (2) ISMLP can be more efficiently trained than the semi-supervised learning methods used in the experiment; and (3) ISMLP is not sensitive to its parameters to some extent. Despite its promising result, the proposed ISMLP assumes linear separability of the data, which is often unrealistic due to the presence of complex data structures. The kernel tricks or the deep neural networks can be utilized to extract the nonlinear features to enhance the performance. In the future, we plan to extend the proposed ISMLP to multi-model settings to deal with the multi-modal input [51; 52; 53]. Conceptualization, P.C. and H.W. ; methodology, H.W. ; software, H.W. ; validation, P.C. and H.W. ; formal analysis, H.W. ; investigation, H.W. ; resources, P.C. ; data curation, H.W. ; writing--original draft preparation, H.W. ; writing--review and editing, P.C. and H.W. ; visualization, P.C. ; supervision, P.C. All authors have read and agreed to the published version of the manuscript. This research was funded by Dalian Science and Technology Innovation Fund 2021 JJ12 GX028. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: All datasets are publicly available. Conflicts of Interest: The authors declare no conflict of interest. ## References

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