## 1 Introduction
In other words, the relationships between high-dimensional data are measured based on the distances in the original domain. As is widely known, there are various types of interference that are difficult to capture and eliminate in the original domain, but that are much easier to address in the frequency domain. For instance, in the case of facial images affected by extensive occlusions, such as shadows and obstructions, the interference is not sparse and possesses high energy. This makes it challenging to separate in the original domain. PCA and RPCA might recognize the occlusions as the principle components due to their high energy. Distance-based algorithms may erroneously classify different objects that are corrupted by similar occlusion into the same category. On the other hand, by transforming the corrupted data into the frequency domain, the key features and interferences can be effectively separated according to their coordinated frequency band, offering valuable guidance for the subsequent feature extraction and dimensionality reduction processes. Inspired by this idea, in this paper, a novel dimensionality reduction algorithm via frequency domain correlationship is proposed, named Frequency domain-based Manifold Dimensionality Reduction (FMDR), which measures the high-dimensional manifold similarities between data points by the frequency domain information. The overall workflow and processing phases are illustrated in Figure 1. Specifically, FMDR includes the following major steps. Firstly, a transformation is proposed, including transforming data into the frequency domain, extracting the discriminative details, reducing the interference and reconstructing datasets. Secondly, a novel kernel function is proposed for measuring the high-dimensional manifold relationship between samples based on frequency domain information. Finally, according to the relationship measured by the new kernel function, a graph structure is constructed by calculating the Laplacian matrix, which can be effectively utilized for finding low-dimensional mapping for the original data. Once the graph structure is obtained, the manifold relationship of the original data can be preserved, which can be conveniently adopted to solve various problems such as classification, clustering and semi-supervised classification. Figure 1: Demonstration of the FMDR. Data points of the same color belong to the same category. The goal of dimensionality reduction is to search for an accurate low-dimensional mapping that reveals the intrinsic relationships of the original data. Given a set of images, Samples (**a**,**b**), (**c**,**d**) belong to two different categories, respectively. Due to the high energy of the occlusions (such as scarts), samples could be closed to each other in the high-dimensional manifold space, such as (**b**,**c**). FMDR consists of several key steps. (1) Frequency domain transformation and feature extraction. Firstly, the samples are transformed into the frequency domain. According to the frequency bands of noise distribution, interference is filtered out and discriminative details are emphasized. (2) Assisted by a novel kernel function, the similarities between samples are measured based on the correlationship of the frequency domain features. Due to effective extraction and the kernel function, samples of the same type have higher correlations, while the differences between samples of different types are also greater. (3) Graph structure construction. Constructing a manifold topological graph by treating sample points as vertexes and linking pairs of points with edge lengths based on their similarity. Finally, the graph structure can be conveniently applied for dimensionality reduction and recognition tasks. Integrating frequency domain information to design machine learning models has become a new trend. FreMixer [24] is a lightweight machine learning architecture for short-term waves forecasting, which utilizes frequency domain information for data training. [PERSON], [PERSON] et al.