Cluster-based Data Reduction for Persistent Homology

Topological Data Analysis (TDA) is a data mining technique to characterize the topological features of data. Persistent Homology (PH) is an important tool of TDA that has been applied to a wide range of applications. However its time and space complexities motivates a need for new methods to compute the PH of high-dimensional data. An important, and memory intensive, element in the computation of PH is the complex constructed from the input data. In general, PH tools use and focus on optimizing simplicial complexes; less frequently cubical complexes are also studied. This paper develops a method to construct polytopal complexes (or complexes constructed of any mix of convex polytopes) in any dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{R}^{n}$</tex> In general, polytopal complexes are significantly smaller than simplicial or cubical complexes. This paper includes an experimental assessment of the impact that polytopal complexes have on memory complexity and output results of a PH computation.

In a variety of disciplines, from biology and physics to computer science and the social sciences, computational topology has become a potent tool for data analysis and visualisation. This study explores computational topology's uses in data analysis and visualisation, showing its potential for revealing hidden structures and patterns in large, complicated datasets.The paper starts out by giving a general introduction of computational topology and outlining its core ideas and methods. The use of it for extracting important information from big, high-dimensional datasets is then explored in relation to data analysis. Computational topology allows for the discovery of topological properties such as holes, voids, and connection patterns by modelling data as topological structures, such as simplicial complexes or persistent homology diagrams.The study also explores the function of computational topology in data visualisation, highlighting its capacity to offer clear and insightful visual representations of challenging datasets. Computational topology enables the construction of simpler and aesthetically pleasing representations while preserving the fundamental topological properties of the data through methods like topological simplification and dimensionality reduction.The usefulness of computational topology in several application domains, such as genomics, image analysis, and network analysis, is demonstrated through a number of case studies. These instances show how computational topology can improve activities like data exploration, clustering, classification, and anomaly detection, resulting in fresh perceptions and learnings.This study emphasises the enormous potential of computational topology for applications in data processing and visualisation. Computational topology provides a new perspective that complements current approaches by utilising the inherent geometric and topological properties of data. This allows researchers and practitioners to better understand complex datasets and make decisions based on the knowledge gleaned from them.

High dimensionality data have become common in neuroimaging fields, especially group‐level functional magnetic resonance imaging (fMRI) datasets. fMRI connectivity analysis is a widely used, powerful technique for studying functional brain networks to probe underlying mechanisms of brain function and neuropsychological disorders. However, data‐driven technique like independent components analysis (ICA), can yield unstable and inconsistent results, confounding the true effects of interest and hindering the understanding of brain functionality and connectivity. A key contributing factor to this instability is the information loss that occurs during fMRI data reduction. Data reduction of high dimensionality fMRI data in the temporal domain to identify the important information within group datasets is necessary for such analyses and is crucial to ensure the accuracy and stability of the outputs. In this study, we describe an fMRI data reduction strategy based on an adapted neighborhood preserving embedding (NPE) algorithm. Both simulated and real data results indicate that, compared with the widely used data reduction method, principal component analysis, the NPE‐based data reduction method (a) shows superior performance on efficient data reduction, while enhancing group‐level information, (b) develops a unique stratagem for selecting components based on an adjacency graph of eigenvectors, (c) generates more reliable and reproducible brain networks under different model orders when the outputs of NPE are used for ICA, (d) is more sensitive to revealing task‐evoked activation for task fMRI, and (e) is extremely attractive and powerful for the increasingly popular fast fMRI and very large datasets.

Computational time in optimization models scales with the number of time steps. To save time, solver time resolution can be reduced and input data can be down-sampled into representative periods such as one or a few representative days per month. However, such data reduction can come at the expense of solution accuracy. In this work, the impact of reduction of input data is systematically isolated considering an optimization which solves an energy system using representative days. A new data reduction method aggregates annual hourly demand data into representative days which preserve demand peaks in the original profiles. The proposed data reduction approach is tested on a real energy system and real annual hourly demand data where the system is optimized to minimize total annual costs. Compared to the full-resolution optimization of the energy system, the total annual energy cost error is found to be equal or less than 0.22% when peaks in customer demand are preserved. Errors are significantly larger for reduction methods that do not preserve peak demand. Solar photovoltaic data reduction effects are also analyzed. This paper demonstrates a need for data reduction methods which consider demand peaks explicitly.

Reverse engineering refers to the process of obtaining a CAD model from an existing physical part. Advances in laser scanning technologies have facilitated this process by sampling part surface data with speed and accuracy. With the help of this technology, it is now possible to acquire the geometry of a part having complex and freeform surfaces. However, it creates the burden of large amounts of point data which must be manipulated, therefore, data from a laser scan must be significantly reduced to proceed with the computations and to lower the storage requirement. Many point data reduction methods for image processing have been developed in the past. However, there is little published work on laser-scanned data, and what exists focuses only on 2D point data. This paper presents a data reduction method that reduces the amount of 3D point data using part geometry information. The method reduces the point data, based on normal values of points using 3D grids. The method is applied to two sample models and the results are discussed.

Accurate binding affinity prediction (BAP) is crucial to structure-based drug design. We present PATH+, a novel, generalizable machine learning algorithm for BAP that exploits recent advances in computational topology. Compared to current binding affinity prediction algorithms, PATH+ shows similar or better accuracy and is more generalizable across orthogonal datasets. PATH+ is not only one of the most accurate algorithms for BAP, it is also the first algorithm that is inherently interpretable. Interpretability is a key factor of trust for an algorithm and alongside generalizability, which allows PATH+ to be trusted in critical applications, such as inhibitor design. We visualized the features captured by PATH+ for two clinically relevant protein-ligand complexes and find that PATH+ captures binding-relevant structural mutations that are corroborated by biochemical data. Our work also sheds light on the features captured by current computational topology BAP algorithms that contributed to their high performance, which have been poorly understood. PATH+ also offers an improvement of 𝒪 (m + n)3 in computational complexity and is empirically over 10 times faster than the dominant (uninterpretable) computational topology algorithm for BAP. Based on insights from PATH+, we built PATH-, a scoring function for differentiating between binders and non-binders that has outstanding accuracy against 11 current algorithms for BAP. In summary, we report progress in a novel combination of interpretability, speed, and accuracy that should further empower topological screening of large virtual inhibitor libraries to protein targets, and allow binding affinity predictions to be understood and trusted. The source code for PATH+ and PATH- is released open-source as part of the OSPREY protein design software package.

Data reduction has become greatly significant prior to applying instance-based machine learning algorithms in the Big Data era. Data reduction is used to reduce the size of data sets while retaining representative data. Existing algorithms, however, suffer from heavy computational cost and in having tradeoff in size reduction rate and learning accuracy. In this paper, we propose a fast data reduction approach by using granular computing to label important instances, i.e., instances with higher contributions to the learning task. The original data set is first granulated into $K$ granules by applying $K$ -means to a mapped lower-dimension space. Then, the importance of each instance in every granule is labeled based on its Hausdorff distance. Those instances whose importance values are lower than an experimentally tuned threshold are eliminated. The presented algorithm is applied to $k$ NN classification tasks with eighteen different sizes of data sets from the UCI repository, and its outstanding performance in classification accuracy, size reduction rate, and runtime is illustrated by comparing with seven data reduction methods. The experimental results demonstrate that the proposed algorithm can greatly reduce the computational cost and achieve a higher classification accuracy when the reduction size is the same for all the compared algorithms.

Density peaks clustering algorithm is a simple but effective clustering method, which requires fewer parameters and iteration, and can determine the number of clusters. But this algorithm has high complexity of time and space which makes it is unsuitable to cluster large-scale data. So that this paper proposes a density peaks clustering algorithm based on the divide-and-conquer strategy. Firstly, divides the large-scale data into a series of data blocks consistent with the original data distribution, then performs density peaks clustering on a randomly selected block to get the clustering center of the data block. Since the data block has the same distribution with the original large-scale data, the clustering center can be used as the clustering center of the original data. Finally, allocates the remaining data blocks to the corresponding clustering center to obtain the final clustering result. Experimental results on real and synthetic datasets verify the effectiveness of the proposed algorithm.

We re-examine the theory and orthodox methods that underlie the study of persistent homology, particularly in its calculation of homological cycle representatives that are associated to persistence diagrams. A common background to the subject covers several aspects: schemes to process input data (embedding it in a low-dimensional manifold), categorical descriptions of persistence objects, and algorithms by which the barcode summarizing the homology is found. We overview these aspects, focusing on altered simplicial complexes, traditional computation of persistent homology, and the stability theorem for barcodes. By reformulating these notions in the language of category theory, we can speak more plainly on some recurring notions that are relevant to our discussion. This ultimately sets up for vector space filtrations that prove to be suitable tools for codifying the homology of complexes, including the (co)images and (co)kernels arising from morphisms of complexes. The main body of work then presents an alternative approach to persistent homology, based on filtrations of vector spaces. We elaborate with an interesting example whose persistent homology is readily computed as a quotient of appropriate filtrations; in the process, we produce a representative basis of homological cycles, a step that is often overlooked in existing literature. The proposed algorithm is also notable in that it easily handles the calculation of (co)images and (co)kernels for persistent morphisms, supplying us with the same level of detail; while other algorithms do exist for computing the barcodes of these universal objects, such methods are not easily generalizable. Finally, we compute appropriate homological cycles and use a certain algorithmic matching scheme that both implies the usual barcode matching and attempts to better interpret this interesting behavior.

We propose an algorithm for persistence homology computation of orientable 2-dimensional (2D) manifolds with or without boundary (meshes) represented by 2D combinatorial maps. Having as an input a real function h on the vertices of the mesh, we first compute persistent homology of filtrations obtained by adding cells incident to each vertex of the mesh, The cells to add are controlled by both the function h and a parameter \(\delta \). The parameter \(\delta \) is used to control the number of cells added to each level of the filtration. Bigger \(\delta \) produces less levels in the filtration and consequently more cells in each level. We then simplify each level (cluster) by merging faces of the same cluster. Our experiments demonstrate that our method allows fast computation of persistent homology of big meshes and it is persistent-homology aware in the sense that persistent homology does not change in the simplification process when fixing \(\delta \).

Topological Data Analysis is a machine learning method that summarizes the topological features of a space. Persistent Homology (PH) can identify these topological features as they persist within a point cloud; persisting in respect to the connectedness of the point cloud at increasing distances. The utility of PH is apparent in several fields including bioinformatics, network security, and object classification. However, the memory complexity of PH limits the application to relatively small point clouds for low-dimensional topological feature identification. For this reason, numerous approaches to optimize and approximate the PH have been introduced for providing results over large point clouds. One solution, Partitioned Persistent Homology (PPH), has shown favorable approximation on a single node with significant performance improvement. However, the single-node approach is limited by the available system memory, leading to the need for a distributed approach for additional (especially memory) resources. This paper studies a distributed version of PPH for use with large point clouds over a high-performance compute cluster. Experimental results of the distributed algorithm against previous studies is presented along with scalability of the distributed library.

In this paper, we introduce a fast and memory efficient approach to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by Barmak and Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one. Our approach has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for other types of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot of space and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel. We also focus on the problem of computing persistent homology of a flag tower, i.e. a sequence of flag complexes connected by simplicial maps. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. Moreover we can strong collapse a flag complex knowing only its 1-skeleton and the resulting complex is also a flag complex. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the core flag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremely efficient. As a result and as demonstrated by numerous experiments on publicly available data sets, our approach is extremely fast and memory efficient in practice. Finally, we can compromise between precision and time by choosing the number of simplicial complexes of the sequence we strong collapse.

The recording of raw or streamed EM survey data, as done by CGG during MEGATEM surveys, allows for the reprocessing of the acquired EM data, including square-wave processing. During the latter, the recorded EM response to the actual half-sine waveform is replaced by the EM response to a square-wave, derived via deconvolution/convolution in the frequency domain. This makes the on- and early-time information more accessible for data modelling, including 1D inversions and conductivity-depth transformations. Square-wave EM data can also be corrected for survey height, transmitter-receiver offset and transmitter attitude. That correction allows for the interpretation of early-time EM response grids, which generally offer better spatial resolution than derived conductivity-depth slices. The advantages of square-wave processing are demonstrated on a MEGATEM data set acquired in 2013 in South America. With survey terrain clearance ranging from 100 – 1600 m, due to the rugged topography, early-time grids of elevation-corrected square-wave data outlined the shallow conductivity structure, whereas early-time grids of the original half-sine data mostly reflected the variable system elevation. Further, derived conductivity-depth sections of the square-wave data show more lateral continuity than the sections derived from the original half-sine data. These results show that the early-time information of square-wave is more accessible than in the original data, facilitating interpretation of shallow conductivity structures.

The Internet of Things (IoT) is expected to grow faster than any other category of connected devices. IoT allows any device with an on-and-off switch to connect to the internet—a concept that has the ability to greatly change our lives and work. These modern systems collect inherently complex data streams due to the volume, velocity, value, variety, variability, and veracity of data, which leads to incremental growth in data traffic on networks and in the cloud. To fulfill the requirements of IoT, including geodistribution, low latency, location awareness, and mobility support, a new paradigm is proposed: edge computing. In edge computing, substantial computing and storage resources are placed at the edge of the network in mobile devices or sensors. The term “edge” is taken from network diagrams; normally, the edge of a network diagram represents the point at which data traffic enters or leaves the workable network. Using the concept of edge computing, an organization can shift huge amounts of data into processed data near the data origin, which helps to reduce data traffic in the network’s central repository (called the “cloud”). Edge computing uses a variety of data reduction techniques close to the data source at the network edge, including data pre-processing, local storage, and filtering. This approach can prevent some critical issues, such as I/O bottlenecks, storage and bandwidth limitations, data traffic increments, and high energy costs. A major advantage of edge computing is improvement of the request-response delay to milliseconds. Edge computing also supports security and network challenges. However, two major obstacles exist toward achieving the benefit of network-edge computing. First, the most efficient algorithms for data reduction in time series (one of the most common types of data in IoT) were developed to work posteriori upon big datasets, but they cannot make decisions for each incoming data item. Secondly, the state of the art lacks systems that can apply any of the possible data reduction methods without adding significant delays or major reconfigurations. Edge computing has also inherited some of the challenges of cloud computing, including data abstraction, naming, and programmability. This chapter presents a detailed taxonomic discussion of edge computing, along with its challenges, opportunities, and data reduction methods.

As the number of pre-trained machine learning (ML) models is growing exponentially, data reduction tools are not catching up. Existing data reduction techniques are not specifically designed for pre-trained model (PTM) dataset files. This is largely due to a lack of understanding of the patterns and characteristics of these datasets, especially those relevant to data reduction and compressibility. This paper presents the first, exhaustive analysis to date of PTM datasets on storage compressibility. Our analysis spans different types of data reduction and compression techniques, from hash-based data deduplication, data similarity detection, to dictionary-coding compression. Our analysis explores these techniques at three data granularity levels, from model layers, model chunks, to model parameters. We draw new observations that indicate that modern data reduction tools are not effective when handling PTM datasets. There is a pressing need for new compression methods that take into account PTMs' data characteristics for effective storage reduction. Motivated by our findings, we design Elf, a simple yet effective, error-bounded, lossy floating-point compression method. Elf transforms floating-point parameters in such a way that the common exponent field of the transformed parameters can be completely eliminated to save storage space. We develop Elves, a compression framework that integrates Elf along with several other data reduction methods. Elves uses the most effective method to compress PTMs that exhibit different patterns. Evaluation shows that Elves achieves an overall compression ratio of 1.52×, which is 1.31×, 1.32× and 1.29× higher than a general-purpose compressor (zstd), an error-bounded lossy compressor (SZ3), and the uniform model quantization, respectively, with negligible model accuracy loss.