## 4 Conclusions  
In this paper, a novel dimensionality reduction approach is proposed, named FMDR. The existing manifold learning dimensionality reduction techniques mainly treat data in the original space and estimate the similarity between samples by distance such as absolute distance and Gaussian distance. When samples are corrupted by high-energy or non-sparse interference, such as shadows and occlusions, the distance calculated in the original space cannot accurately estimate the manifold relationships between samples. Although such noise is challenging to handle in the original space, it can be efficiently captured and separated in the frequency domain. Inspired by this idea, FMDR transforms the data into the frequency domain for noise filtering and calculates the manifold relationships between samples via a novel kernel function. Then, the graph structure is constructed based on the correlationship in the frequency domain and can be conveniently utilized to find the accurate low-dimensional representation and solve recognition tasks.  
In order to evaluate the effectiveness of FMDR comprehensively, several experiments are conducted in the public image datasets. Firstly, dimensions of samples from ORL and YaleB are compressed into two dimensions by FMDR and visualized by the feature dimensions. The illustrations indicate that FMDR is able to recognize the major components for distinguishing samples and dimensionality reduction. Even with the interference such as shadows, angles and various expression, samples can be effectively distinguished in the low-dimensional space. Further, FMDR is employed to solve the semi-supervised classification problems in five datasets. According to four indicators, our approach achieves the best classification performances in the most cases compared to the baselines and the state-of-the-art algorithms. At last, by setting up labeled datasets with varying proportions, FMDR demonstrates its superiority and stability of the recognition performance as the proportion of labeled data increases from low to high.  
In conclusion, FMDR leverages the correlationship in the frequency domain of samples to obtain the intrinsic high-dimensional manifold relationship, which enables FMDR to find an optimal low-dimensional representation of the original data. According to its outstanding dimensionality reduction performance and operational efficiency, FMDR can be conveniently applied across multiple domains. It can be utilized for preprocessing tasks such as feature extraction and compressed sensing for high-dimensional data, as well as for recognition and clustering of multi-type data. This facilitates the development of subsequent intelligent algorithms and applications.  
Conceptualization, Z.L.; methodology, Z.L.; software, R.G. and G.T.; validation, G.T. and S.J.; formal analysis, G.T. and R.Z.; investigation, Z.L.; resources, R.Z.; writing--original draft preparation, Z.L.; writing--review and editing, R.Z.; visualization, S.J.; supervision, R.Z.; project administration, R.Z.; funding acquisition, R.Z. All authors have read and agreed to the published version of the manuscript.  
Figure 7: The Variation in algorithm performance with changing proportions of labeled data. (**a**) The accuracy varies with the change in labeled data in ORL. (**b**) The accuracy varies with the change in labeled data in AR. (**c**) The accuracy varies with the change in labeled data in Yale. (**d**) The accuracy varies with the change in labeled data in YaleB.  
**Funding:** This research was funded by Key-Area Research and Development Program of Guangdong Province grant number 2022B0701180001.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.