Clustering, a fundamental task in data science and machine learning, groups a set of objects in such a way that objects in the same cluster are closer to each other than to those in other clusters. In this paper, we consider a well-known structure, so-called @math -nets, which rigorously captures the properties of clustering. We devise algorithms that improve the run-time of approximating @math -nets in high-dimensional spaces with @math and @math metrics from @math to @math , where @math . These algorithms are also used to improve a framework that provides approximate solutions to other high dimensional distance problems. Using this framework, several important related problems can also be solved efficiently, e.g., @math -approximate @math th-nearest neighbor distance, @math -approximate Min-Max clustering, @math -approximate @math -center clustering. In addition, we build an algorithm that @math -approximates greedy permutations in time @math where @math is the spread of the input. This algorithm is used to @math -approximate @math -center with the same time complexity.
An approach on hierarchical clustering was presented by @cite_6 . They construct an algorithm based on furthest first traversal, which is essentially building a greedy permutation and then traversing the permutation in order.
@cite_6
2050932686 2049631158 2128859735 2155162820
This paper presents a new approach to hierarchical clustering of point patterns. Two algorithms for hierarchical location- and density-based clustering are developed. Each method groups points such that maximum intracluster similarity and intercluster dissimilarity are achieved for point locations or point separations. Performance of the clustering methods is compared with four other methods. The approach is applied to a two-step texture analysis, where points represent centroid and average color of the regions in image segmentation. Hierarchical clustering is common method used to determine clusters of similar data points in multi-dimensional spaces. @math algorithms, where @math is the number of points to cluster, have long been known for this problem. This paper discusses parallel algorithms to perform hierarchical clustering using various distance metrics. I describe @math time algorithms for clustering using the single link, average link, complete link, centroid, median, and minimum variance metrics on an @math node CRCW PRAM and @math algorithms for these metrics (except average link and complete link) on @math node butterfly networks or trees. Thus, optimal efficiency is achieved for a significant number of processors using these distance metrics. A general algorithm is given that can be used to perform clustering with the complete link and average link metrics on a butterfly. While this algorithm achieves optimal efficiency for the general class of metrics, it is not optimal for the specific cases of complete link and average link clustering. We present a new method for clustering based on compression. The method does not use subject-specific features or background knowledge, and works as follows: First, we determine a parameter-free, universal, similarity distance, the normalized compression distance or NCD, computed from the lengths of compressed data files (singly and in pairwise concatenation). Second, we apply a hierarchical clustering method. The NCD is not restricted to a specific application area, and works across application area boundaries. A theoretical precursor, the normalized information distance, co-developed by one of the authors, is provably optimal. However, the optimality comes at the price of using the noncomputable notion of Kolmogorov complexity. We propose axioms to capture the real-world setting, and show that the NCD approximates optimality. To extract a hierarchy of clusters from the distance matrix, we determine a dendrogram (ternary tree) by a new quartet method and a fast heuristic to implement it. The method is implemented and available as public software, and is robust under choice of different compressors. To substantiate our claims of universality and robustness, we report evidence of successful application in areas as diverse as genomics, virology, languages, literature, music, handwritten digits, astronomy, and combinations of objects from completely different domains, using statistical, dictionary, and block sorting compressors. In genomics, we presented new evidence for major questions in Mammalian evolution, based on whole-mitochondrial genomic analysis: the Eutherian orders and the Marsupionta hypothesis against the Theria hypothesis. We present a novel clustering algorithm for tagging a face dataset (e. g., a personal photo album). The core of the algorithm is a new dissimilarity, called Rank-Order distance, which measures the dissimilarity between two faces using their neighboring information in the dataset. The Rank-Order distance is motivated by an observation that faces of the same person usually share their top neighbors. Specifically, for each face, we generate a ranking order list by sorting all other faces in the dataset by absolute distance (e. g., L1 or L2 distance between extracted face recognition features). Then, the Rank-Order distance of two faces is calculated using their ranking orders. Using the new distance, a Rank-Order distance based clustering algorithm is designed to iteratively group all faces into a small number of clusters for effective tagging. The proposed algorithm outperforms competitive clustering algorithms in term of both precision recall and efficiency.