🍩 Database of Original & NonTheoretical Uses of Topology
(found 10 matches in 0.002447s)


The Euler Characteristic: A General Topological Descriptor for Complex Data (2021)
Alexander Smith, Victor ZavalaAbstract
Datasets are mathematical objects (e.g., point clouds, matrices, graphs, images, fields/functions) that have shape. This shape encodes important knowledge about the system under study. Topology is an area of mathematics that provides diverse tools to characterize the shape of data objects. In this work, we study a specific tool known as the Euler characteristic (EC). The EC is a general, lowdimensional, and interpretable descriptor of topological spaces defined by data objects. We revise the mathematical foundations of the EC and highlight its connections with statistics, linear algebra, field theory, and graph theory. We discuss advantages offered by the use of the EC in the characterization of complex datasets; to do so, we illustrate its use in different applications of interest in chemical engineering such as process monitoring, flow cytometry, and microscopy. We show that the EC provides a descriptor that effectively reduces complex datasets and that this reduction facilitates tasks such as visualization, regression, classification, and clustering. 
Go With the Flow? A LargeScale Analysis of Health Care Delivery Networks in the United States Using Hodge Theory (2021)
Thomas Gebhart, Xiaojun Fu, Russell J. FunkAbstract
Health care delivery is a collaborative process, requiring close coordination among networks of providers with specialized expertise. Yet in the United States, care is often spread across multiple disconnected providers (e.g., primary care physicians, specialists), leading to fragmented care delivery networks, and contributing to higher costs and lower quality. While this problem is well known, there are relatively few quantitative tools available for characterizing the dynamics of care delivery networks at scale, thereby inhibiting deeper understanding of care fragmentation and efforts to address it. In this, study, we conduct a largescale analysis of care delivery networks across the United States using the discrete Hodge decomposition, an emerging method of topological data analysis. Using this technique, we decompose networks of patient flows among physicians into three orthogonal subspaces: gradient (acyclic flow), harmonic (global cyclic flow), and curl (local cyclic flow). We document substantial variation in the relative importance of each subspace, suggesting that there may be systematic differences in the organization of care delivery networks across health care markets. Moreover, we find that the relative importance of each subspace is predictive of local care cost and quality, with outcomes tending to be better with greater curl flow and worse with greater harmonic flow. 
The Shape of Cancer Relapse: Topological Data Analysis Predicts Recurrence in Paediatric Acute Lymphoblastic Leukaemia (2021)
Salvador Chulián, Bernadette J. Stolz, Álvaro MartínezRubio, Cristina Blázquez Goñi, Juan F. Rodríguez Gutiérrez, Teresa Caballero Velázquez, Águeda Molinos Quintana, Manuel Ramírez Orellana, Ana Castillo Robleda, José Luis Fuster Soler, Alfredo Minguela Puras, María Victoria Martínez Sánchez, María Rosa, Víctor M. PérezGarcía, Helen ByrneAbstract
Acute Lymphoblastic Leukaemia (ALL) is the most frequent paediatric cancer. Modern therapies have improved survival rates, but approximately 1520 % of patients relapse. At present, patients’ risk of relapse are assessed by projecting highdimensional flow cytometry data onto a subset of biomarkers and manually estimating the shape of this reduced data. Here, we apply methods from topological data analysis (TDA), which quantify shape in data via features such as connected components and loops, to pretreatment ALL datasets with known outcomes. We combine these fully unsupervised analyses with machine learning to identify features in the pretreatment data that are prognostic for risk of relapse. We find significant topological differences between relapsing and nonrelapsing patients and confirm the predictive power of CD10, CD20, CD38, and CD45. Further, we are able to use the TDA descriptors to predict patients who relapsed. We propose three prognostic pipelines that readily extend to other haematological malignancies. Teaser Topology reveals features in flow cytometry data which predict relapse of patients with acute lymphoblastic leukemia 
Path Homology as a Stronger Analogue of Cyclomatic Complexity (2020)
Steve HuntsmanAbstract
Cyclomatic complexity is an incompletely specified but mathematically principled software metric that can be usefully applied to both source and binary code. We consider the application of path homology as a stronger analogue of cyclomatic complexity. We have implemented an algorithm to compute path homology in arbitrary dimension and applied it to several classes of relevant flow graphs, including randomly generated flow graphs representing structured and unstructured control flow. We also compared path homology and cyclomatic complexity on a set of disassembled binaries obtained from the grep utility. There exist control flow graphs realizable at the assembly level with nontrivial path homology in arbitrary dimension. We exhibit several classes of examples in this vein while also experimentally demonstrating that path homology gives identicial results to cyclomatic complexity for at least one detailed notion of structured control flow. We also experimentally demonstrate that the two notions differ on disassembled binaries, and we highlight an example of extreme disagreement. Path homology empirically generalizes cyclomatic complexity for an elementary notion of structured code and appears to identify more structurally relevant features of control flow in general. Path homology therefore has the potential to substantially improve upon cyclomatic complexity. 
The Topological Basis of Function in Flow and Mechanical Networks (2019)
Jason Rocks, Andrea Liu, Eleni Katifori 
Topological Data Analysis and 3D Printing Technologies for Flow in Fracture Networks (2018)
Anna Suzuki, M. Miyazawa, T. Ito 
Raw Material Flow Optimization as a Capacitated Vehicle Routing Problem: A Visual Benchmarking Approach for Sustainable Manufacturing (2017)
Michele Dassisti, Yasamin Eslami, Matin MohagheghAbstract
Optimisation problem concerning material flows, to increase the efficiency while reducing relative resource consumption is one of the most pressing problems today. The focus point of this study is to propose a new visual benchmarking approach to select the best materialflow path from the depot to the production lines, referring to the wellknown Capacitated Vehicle Routing Problem (CVRP). An example industrial case study is considered to this aim. Two different solution techniques were adopted (namely Mixed Integer Linear Programming and the Ant Colony Optimization) in searching optimal solutions to the CVRP. The visual benchmarking proposed, based on the persistent homology approach, allowed to support the comparison of the optimal solutions based on the entropy of the output in different scenarios. Finally, based on the nonstandard measurements of Crossing Length Percentage (CLP), the visual benchmarking procedure makes it possible to find the most practical and applicable solution to CVRP by considering the visual attractiveness and the quality of the routes. 
ObjectOriented Persistent Homology (2016)
Bao Wang, GuoWei WeiAbstract
Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed for quantitative modeling and prediction. Additionally, the present persistent homology is a passive tool, rather than a proactive technique, for classification and analysis. In this work, we outline a general protocol to construct objectoriented persistent homology methods. By means of differential geometry theory of surfaces, we construct an objective functional, namely, a surface free energy defined on the data of interest. The minimization of the objective functional leads to a LaplaceBeltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based objectoriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The cubical complex based homology algorithm is employed in the present work to be compatible with the Cartesian representation of the LaplaceBeltrami flow. The proposed LaplaceBeltrami flow based persistent homology method is extensively validated. The consistence between LaplaceBeltrami flow based filtration and Euclidean distance based filtration is confirmed on the VietorisRips complex for a large amount of numerical tests. The convergence and reliability of the present LaplaceBeltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The LaplaceBeltrami flow based persistent homology approach is utilized to study the intrinsic topology of proteins and fullerene molecules. Based on a quantitative model which correlates the topological persistence of fullerene central cavity with the total curvature energy of the fullerene structure, the proposed method is used for the prediction of fullerene isomer stability. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design objectoriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data. 
Persistent Topology for CryoEm Data Analysis (2015)
Kelin Xia, GuoWei WeiAbstract
SummaryIn this work, we introduce persistent homology for the analysis of cryoelectron microscopy (cryoEM) density maps. We identify the topological fingerprint or topological signature of noise, which is widespread in cryoEM data. For low signaltonoise ratio (SNR) volumetric data, intrinsic topological features of biomolecular structures are indistinguishable from noise. To remove noise, we employ geometric flows that are found to preserve the intrinsic topological fingerprints of cryoEM structures and diminish the topological signature of noise. In particular, persistent homology enables us to visualize the gradual separation of the topological fingerprints of cryoEM structures from those of noise during the denoising process, which gives rise to a practical procedure for prescribing a noise threshold to extract cryoEM structure information from noise contaminated data after certain iterations of the geometric flow equation. To further demonstrate the utility of persistent homology for cryoEM data analysis, we consider a microtubule intermediate structure Electron Microscopy Data (EMD 1129). Three helix models, an alphatubulin monomer model, an alphatubulin and betatubulin model, and an alphatubulin and betatubulin dimer model, are constructed to fit the cryoEM data. The least square fitting leads to similarly high correlation coefficients, which indicates that structure determination via optimization is an illposed inverse problem. However, these models have dramatically different topological fingerprints. Especially, linkages or connectivities that discriminate one model from another, play little role in the traditional density fitting or optimization but are very sensitive and crucial to topological fingerprints. The intrinsic topological features of the microtubule data are identified after topological denoising. By a comparison of the topological fingerprints of the original data and those of three models, we found that the third model is topologically favored. The present work offers persistent homology based new strategies for topological denoising and for resolving illposed inverse problems. Copyright © 2015 John Wiley & Sons, Ltd.