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  1. Abstract Persistent homology is a computationally intensive and yet extremely powerful tool for Topological Data Analysis. Applying the tool on potentially infinite sequence of data objects is a challenging task. For this reason, persistent homology and data stream mining have long been two important but disjoint areas of data science. The first computational model, that was recently introduced to bridge the gap between the two areas, is useful for detecting steady or gradual changes in data streams, such as certain genomic modifications during the evolution of species. However, that model is not suitable for applications that encounter abrupt changes of extremely short duration. This paper presents another model for computing persistent homology on streaming data that addresses the shortcoming of the previous work. The model is validated on the important real‐world application of network anomaly detection. It is shown that in addition to detecting the occurrence of anomalies or attacks in computer networks, the proposed model is able to visually identify several types of traffic. Moreover, the model can accurately detect abrupt changes of extremely short as well as longer duration in the network traffic. These capabilities are not achievable by the previous model or by traditional data mining techniques. 
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    Free, publicly-accessible full text available July 1, 2024
  2. This paper introduces a framework to compute persistent homology, a principal tool in Topological Data Analysis, on potentially unbounded and evolving data streams. The framework is organized into online and offline components. The online element maintains a summary of the data that preserves the topological structure of the stream. The offline component computes the persistence intervals from the data captured by the summary. The framework is applied to the detection of horizontal or reticulate genomic exchanges during the evolution of species that cannot be identified by phylogenetic inference or traditional data mining. The method effectively detects reticulate evolution that occurs through reassortment and recombination in large streams of genomic sequences of Influenza and HIV viruses. 
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  3. Clustering continues to be an important tool for data engineering and analysis. While advances in deep learning tend to be at the forefront of machine learning, it is only useful for the supervised classification of data sets. Clustering is an essential tool for problems where labeling data sets is either too labor intensive or where there is no agreed upon ground truth. The well studied k-means problem partitions groups of similar vectors into k clusters by iteratively updating the cluster assignment such that it minimizes the within cluster sum of squares metric. Unfortunately k-means can become prohibitive for very large high dimensional data sets as iterative methods often rely on random access to, or multiple passes over, the data set — a requirement that is not often possible for large and potentially unbounded data sets. In this work we explore an randomized, approximate method for clustering called Tree-Walk Random Projection Clustering (TWRP) that is a fast, memory efficient method for finding cluster embedding in high dimensional spaces. TWRP combines random projection with a tree based partitioner to achieve a clustering method that forgoes storing the exhaustive representation of all vectors in the data space and instead performs a bounded search over the implied cluster bifurcation tree represented as approximate vector and count values. The TWRP algorithm is described and experimentally evaluated for scalability and accuracy in the presence of noise against several other well-known algorithms. 
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  4. The massive growth in data generation and collection has brought to the forefront the necessity to develop mechanized methods to analyze and extract information from them. Data clustering is one of the fundamental modes to discover new insights from data. However, high dimensional data has its own challenges where many conventional clustering algorithms fails either in accuracy or scalability. To further complicate the issue, distinct subsets of sensitive data may reside in geographically separated locations with the sensitive nature of the data preventing (or inhibiting) its access for mechanized analysis. Thus, methods to discover information from the collective whole of these secured, distributed data sets that also preserves the integrity of the data must be found. In this paper we develop and assess a distributed algorithm that can cluster geographically separated data while simultaneously preserving the strict privacy requirements of non sharing of protected high dimensional data. We implement our algorithm on the distributed map-reduce based platform Spark and demonstrate its performance by comparing it to the standard data clustering algorithms. 
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  5. Persistent homology is used for computing topological features of a space at different spatial resolutions. It is one of the main tools from computational topology that is applied to the problems of data analysis. Despite several attempts to reduce its complexity, persistent homology remains expensive in both time and space. These limits are such that the largest data sets to which the method can be applied have the number of points of the order of thousands in R^3. This paper explores a technique intended to reduce the number of data points while preserving the salient topological features of the data. The proposed technique enables the computation of persistent homology on a reduced version of the original input data without affecting significant components of the output. Since the run time of persistent homology is exponential in the number of data points, the proposed data reduction method facilitates the computation in a fraction of the time required for the original data. Moreover, the data reduction method can be combined with any existing technique that simplifies the computation of persistent homology. The data reduction is performed by creating small groups of \emph{similar} data points, called nano-clusters, and then replacing the points within each nano-cluster with its cluster center. The persistence homology of the reduced data differs from that of the original data by an amount bounded by the radius of the nano-clusters. The theoretical analysis is backed by experimental results showing that persistent homology is preserved by the proposed data reduction technique. 
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  6. Clustering streaming data has gained importance in recent years due to an expanding opportunity to discover knowledge in widely available data streams. As streams are potentially evolving and unbounded sequence of data objects, clustering algorithms capable of performing fast and incremental processing of data points are necessary. This paper presents a method of clustering high-dimensional data streams using approximate methods called streamingRPHash. streamingRPHash combines random projections with locality-sensitivity hashing to construct a high-performance clustering method. streamingRPHash is amenable to distributed processing frameworks such as Map-Reduce, and also has the benefits of constrained overall complexity growth. This paper describes streamingRPHash algorithm and its various configurations. The clustering performance of streamingRPHash is compared to several alternatives. Experimental results show that streamingRPHash has comparable clustering accuracy and substantially lower runtime and memory usage. 
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  7. The size and amount of data captured from numerous sources has created a situation where the large quantity of data challenges our ability to understand the meaning within the data. This has motivated studies for mechanized data analysis and in particular for the clustering, or partitioning, of data into related groups. In fact, the size of the data has grown to the point where it is now often necessary to stream the data through the system for online and high speed analysis. This paper explores the application of approximate methods for the stream clustering of high-dimensional data (feature sizes contains 100+ measures). In particular, the algorithm that has been developed, called streamingRPHash, combines Random Projection with Locality Sensitive Hashing and a count-min sketch to implement a high-performance method for the parallel and distributed clustering of streaming data in a MapReduce framework. streamingRPHash is able to perform clustering at a rate much faster than traditional clustering algorithms such as K-Means. streamingRPHash provides clustering results that are only slightly less accurate than K-Means, but in runtimes that are nearly half that required by K-Means. The performance advantage for streamingRPHash becomes even more significant as the dimensionality of the input data stream increases. Furthermore, the experimental results show that streamingRPHash has a near linear speedup relative to the number of CPU cores. This speedup efficiency is possible because the approximate methods used in streamingRPHash allow independent and largely unsynchronized analyses to be performed on each streamed data vectors. 
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  8. This paper presents a solution to the approximate k-means clustering problem for very large distributed datasets. Distributed data models have gained popularity in recent years following the efforts of commercial, academic and government organizations, to make data more widely accessible. Due to the sheer volume of available data, in-memory single-core computation quickly becomes infeasible, requiring distributed multi-processing. Our solution achieves comparable clustering performance to other popular clustering algorithms, with improved overall complexity growth while being amenable to distributed processing frameworks such as Map-Reduce. Our solution also maintains certain guarantees regarding data privacy deanonimization. 
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