Dimension reduction (DR) algorithms have proven to be extremely useful for gaining insight into large-scale high-dimensional datasets, particularly finding clusters in transcriptomic data. The initial phase of these DR methods often involves converting the original high-dimensional data into a graph. In this graph, each edge represents the similarity or dissimilarity between pairs of data points. However, this graph is frequently suboptimal due to unreliable high-dimensional distances and the limited information extracted from the high-dimensional data. This problem is exacerbated as the dataset size increases. If we reduce the size of the dataset by selecting points for a specific sections of the embeddings, the clusters observed through DR are more separable since the extracted subgraphs are more reliable. In this paper, we introduce LocalMAP, a new dimensionality reduction algorithm that dynamically and locally adjusts the graph to address this challenge. By dynamically extracting subgraphs and updating the graph on-the-fly, LocalMAP is capable of identifying and separating real clusters within the data that other DR methods may overlook or combine. We demonstrate the benefits of LocalMAP through a case study on biological datasets, highlighting its utility in helping users more accurately identify clusters for real-world problems.

Parametric dimensionality reduction methods have gained prominence for their ability to generalize to unseen datasets, an advantage that traditional non-parametric approaches typically lack. Despite their growing popularity, there remains a prevalent misconception among practitioners about the equivalence in performance between parametric and non-parametric methods. Here, we show that these methods are not equivalent -- parametric methods retain global structure but lose significant local details. To explain this, we provide evidence that parameterized approaches lack the ability to repulse negative samples, and the choice of loss function also has an impact. Addressing these issues, we developed a new parametric method, ParamRepulsor, that incorporates Hard Negative Mining and a loss function that applies a strong repulsive force. This new method achieves state-of-the-art performance on local structure preservation for parametric methods without sacrificing the fidelity of global structural representation. Our code is available at https://github.com/hyhuang00/ParamRepulsor.

We incorporate group fairness into the algorithmic centroid clustering problem, where $k$ centers are to be located to serve $n$ agents distributed in a metric space. We refine the notion of proportional fairness proposed in [Chen et al., ICML 2019] as {\em core fairness}. A $k$-clustering is in the core if no coalition containing at least $n/k$ agents can strictly decrease their total distance by deviating to a new center together. Our solution concept is motivated by the situation where agents are able to coordinate and utilities are transferable. A string of existence, hardness and approximability results is provided. Particularly, we propose two dimensions to relax core requirements: one is on the degree of distance improvement, and the other is on the size of deviating coalition. For both relaxations and their combination, we study the extent to which relaxed core fairness can be satisfied in metric spaces including line, tree and general metric space, and design approximation algorithms accordingly. We also conduct experiments on synthetic and real-world data to examine the performance of our algorithms.

Dimension reduction (DR) techniques such as t-SNE, UMAP, and TriMap have demonstrated impressive visualization performance on many real-world datasets. One tension that has always faced these methods is the trade-off between preservation of global structure and preservation of local structure: these methods can either handle one or the other, but not both. In this work, our main goal is to understand what aspects of DR methods are important for preserving both local and global structure: it is difficult to design a better method without a true understanding of the choices we make in our algorithms and their empirical impact on the low-dimensional embeddings they produce. Towards the goal of local structure preservation, we provide several useful design principles for DR loss functions based on our new understanding of the mechanisms behind successful DR methods. Towards the goal of global structure preservation, our analysis illuminates that the choice of which components to preserve is important. We leverage these insights to design a new algorithm for DR, called Pairwise Controlled Manifold Approximation Projection (PaCMAP), which preserves both local and global structure. Our work provides several unexpected insights into what design choices both to make and avoid when constructing DR algorithms.

Interferential background, boundary uncertainty, and noises are usually involved in infrared pedestrian imaging, which erect barrier for accurate segmentation. To counter the conundrum rising in these cases, we present a novel intuitionistic fuzzy clustering-based segmentation method, which integrates structural symmetry and local homoplasy information, for precise infrared pedestrian segmentation. Enlightened by the multiapplication and favorable performance of fuzzy clustering methods, intuitionistic fuzzy c-means (IFCM) is applied as the backbone of our segmentation method. The contributions of the proposed method mainly include two parts. First, motivated by potential target characteristics and tendency for clearer contour description, structural symmetry information is utilized, which is an intrinsic shape feature of objects and would be significant especially when the texture and details of the target are lost in infrared images. Further, a map that represents the probability of pixels belonging to the target is constructed in the form of ellipse symmetry region. Combined with the probability map, symmetry information is utilized to establish a novel dissimilarity function in fuzzy clustering. Second, local homoplasy information which is designed based on region similarity is introduced to suppress the intensity inhomogeneity and noises to further improve the performance of the proposed method. Finally, a dataset containing 500 infrared pedestrian images paired with corresponding pixel-wise annotation is constructed to verify segmentation effectiveness. The proposed SR-IFCM is compared with 12 state-of-the-art segmentation methods. The experimental results indicate that the proposed method outperforms the comparison methods and works better for infrared pedestrian segmentation.

In this paper, an intuitionistic center-free fuzzy c-means clustering method (ICFFCM) is proposed for magnetic resonance (MR) brain image segmentation. First, in order to suppress the effect of noise in MR brain images, a pixel-to-pixel similarity with spatial information is defined. Then, for the purpose of handling the vagueness in MR brain images as well as the uncertainty in clustering process, a pixel-to-cluster similarity measure is defined by employing the intuitionistic fuzzy membership function. These two similarities are used to modify the center-free FCM so that the ability of the method for MR brain image segmentation could be improved. Second, on the basis of the improved center-free FCM method, a local information term, which is also intuitionistic and center-free, is appended to the objective function. This generates the final proposed ICFFCM. The consideration of local information further enhances the robustness of ICFFCM to the noise in MR brain images. Experimental results on the simulated and real MR brain image datasets show that ICFFCM is effective and robust. Moreover, ICFFCM could outperform several fuzzy-clustering-based methods and could achieve comparable results to the standard published methods like statistical parametric mapping and FMRIB automated segmentation tool.

Pedestrians are highlighted regions in infrared images. Hence, infrared images could be used for pedestrian detection. Segmentation is an important step for detection and an accurate segmentation would be helpful for infrared pedestrian detection. However, intensity inhomogeneity is a common drawback in infrared images due to occlusion or uneven heat dissipation. This phenomenon would have negative influences on pedestrian segmentation in infrared images. To address this problem, an intensity inhomogeneity suppressed fuzzy C-means method is proposed in this paper for the segmentation of infrared pedestrians with intensity inhomogeneity. Two improvements are made in the proposed method to suppress intensity inhomogeneity: 1) a weight based on membership information and image intensity information of infrared pedestrians is added to the objective function and 2) neighborhood information is considered by adding a regularization term into the objective function to suppress the intensity inhomogeneity in infrared images. Eight famous segmentation methods are utilized as comparison methods in our experiment. The experimental results show that the proposed method could effectively suppress the intensity inhomogeneity in infrared pedestrian images and perform better for segmentation than the comparison methods.

Pedestrian detection in infrared images is always a challenging task. Segmentation is an important step of pedestrian detection. An accurate segmentation could provide more information for further analysis. In this paper, an improved Fuzzy C-Means clustering method, which incorporates geometric symmetry information, is proposed for infrared pedestrian segmentation. In the proposed method, symmetry information is introduced by Markov random field theory. Moreover, a new metric is utilized to handle the weak symmetry of pedestrian. In addition, a whole procedure is proposed to extract infrared pedestrians. The experimental results indicate that our method performs better for infrared pedestrian segmentation and obtains better segmentation results compared with other state-of-the-art methods.