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CHATGPT FOR COMPUTATIONAL TOPOLOGY. 用于计算拓扑学的 chatgpt
IF 1.7
Foundations of data science (Springfield, Mo.) Pub Date : 2024-06-01 DOI: 10.3934/fods.2024009
Jian Liu, Li Shen, Guo-Wei Wei
{"title":"CHATGPT FOR COMPUTATIONAL TOPOLOGY.","authors":"Jian Liu, Li Shen, Guo-Wei Wei","doi":"10.3934/fods.2024009","DOIUrl":"10.3934/fods.2024009","url":null,"abstract":"<p><p>ChatGPT represents a significant milestone in the field of artificial intelligence (AI), finding widespread applications across diverse domains. However, its effectiveness in mathematical contexts has been somewhat constrained by its susceptibility to conceptual errors. Concurrently, topological data analysis (TDA), a relatively new discipline, has garnered substantial interest in recent years. Nonetheless, the advancement of TDA is impeded by the limited understanding of computational algorithms and coding proficiency among theoreticians. This work endeavors to bridge the gap between theoretical topological concepts and their practical implementation in computational topology through the utilization of ChatGPT. We showcase how a pure theoretician, devoid of computational experience and coding skills, can effectively transform mathematical formulations and concepts into functional codes for computational topology with the assistance of ChatGPT. Our strategy outlines a productive process wherein a mathematician trains ChatGPT on pure mathematical concepts, steers ChatGPT towards generating computational topology codes, and subsequently validates the generated codes using established examples. Our specific case studies encompass the computation of Betti numbers, Laplacian matrices, and Dirac matrices for simplicial complexes, as well as the persistence of various homologies and Laplacians. Furthermore, we explore the application of ChatGPT in computing recently developed topological theories for hypergraphs and digraphs, as well as the persistent harmonic space, which has not been computed in the literature, to the best of our knowledge. This work serves as an initial step towards effectively transforming pure mathematical theories into practical computational tools, with the ultimate goal of enabling real applications across diverse fields.</p>","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11463974/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142395644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
PERSISTENT PATH LAPLACIAN. 持久路径拉普拉斯算子。
Foundations of data science (Springfield, Mo.) Pub Date : 2023-03-01 DOI: 10.3934/fods.2022015
Rui Wang, Guo-Wei Wei
{"title":"PERSISTENT PATH LAPLACIAN.","authors":"Rui Wang,&nbsp;Guo-Wei Wei","doi":"10.3934/fods.2022015","DOIUrl":"https://doi.org/10.3934/fods.2022015","url":null,"abstract":"<p><p>Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs and networks. Persistent path homology (PPH) extends the path homology with filtration to deal with asymmetry structures. However, PPH is constrained to purely topological persistence and cannot track the homotopic shape evolution of data during filtration. To overcome the limitation of PPH, persistent path Laplacian (PPL) is introduced to capture the shape evolution of data. PPL's harmonic spectra fully recover PPH's topological persistence and its non-harmonic spectra reveal the homotopic shape evolution of data during filtration.</p>","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10575407/pdf/nihms-1888540.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41241769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
Diagnostic of the Lévy area for geophysical flow models in view of defining high order stochastic discrete-time schemes 基于高阶随机离散时间格式的地球物理流动模型lsamvy区域诊断
Foundations of data science (Springfield, Mo.) Pub Date : 2023-01-01 DOI: 10.3934/fods.2023011
Pierre-Marie Boulvard, Etienne Mémin
{"title":"Diagnostic of the Lévy area for geophysical flow models in view of defining high order stochastic discrete-time schemes","authors":"Pierre-Marie Boulvard, Etienne Mémin","doi":"10.3934/fods.2023011","DOIUrl":"https://doi.org/10.3934/fods.2023011","url":null,"abstract":"In this paper we characterize numerically through two criteria the Lévy area related to unresolved fluctuation velocities associated to a stochastic coarse-scale representation of geophysical fluid flow dynamics. We study in particular whether or not the process associated to the random unresolved velocity components exhibits a Lévy area corresponding to a Wiener process, and if the law of this process can reasonably be approached by a centered Dirac measure. This exploration enables us to answer positively to a conjecture made for the constitution of high-order discrete time evolution schemes for stochastic representation defined from stochastic transport.","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136306282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Hierarchical regularization networks for sparsification based learning on noisy datasets 基于噪声数据集学习的分层正则化网络
Foundations of data science (Springfield, Mo.) Pub Date : 2023-01-01 DOI: 10.3934/fods.2023009
P. Shekhar, M. Babu, Abani Patra
{"title":"Hierarchical regularization networks for sparsification based learning on noisy datasets","authors":"P. Shekhar, M. Babu, Abani Patra","doi":"10.3934/fods.2023009","DOIUrl":"https://doi.org/10.3934/fods.2023009","url":null,"abstract":"","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70248298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Persistent hyperdigraph homology and persistent hyperdigraph Laplacians 持久超向位同调和持久超向位拉普拉斯算子
Foundations of data science (Springfield, Mo.) Pub Date : 2023-01-01 DOI: 10.3934/fods.2023010
Dong Chen, Jian Liu, Jie Wu, Guo-Wei Wei
{"title":"Persistent hyperdigraph homology and persistent hyperdigraph Laplacians","authors":"Dong Chen, Jian Liu, Jie Wu, Guo-Wei Wei","doi":"10.3934/fods.2023010","DOIUrl":"https://doi.org/10.3934/fods.2023010","url":null,"abstract":"Hypergraphs are useful mathematical models for describing complex relationships among members of a structured graph, while hyperdigraphs serve as a generalization that can encode asymmetric relationships in the data. However, obtaining topological information directly from hyperdigraphs remains a challenge. To address this issue, we introduce hyperdigraph homology in this work. We also propose topological hyperdigraph Laplacians, which can extract both harmonic spectra and non-harmonic spectra from directed and internally organized data. Moreover, we introduce persistent hyperdigraph homology and persistent hyperdigraph Laplacians through filtration, enabling the capture of topological persistence and homotopic shape evolution of directed and structured data across multiple scales. The proposed methods offer new multiscale algebraic topology tools for topological data analysis.","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136053367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Unsupervised learning of observation functions in state space models by nonparametric moment methods 非参数矩法在状态空间模型中观测函数的无监督学习
Foundations of data science (Springfield, Mo.) Pub Date : 2023-01-01 DOI: 10.3934/fods.2023002
Qingci An, Yannis Kevrekidis, Fei Lu, Mauro Maggioni
{"title":"Unsupervised learning of observation functions in state space models by nonparametric moment methods","authors":"Qingci An, Yannis Kevrekidis, Fei Lu, Mauro Maggioni","doi":"10.3934/fods.2023002","DOIUrl":"https://doi.org/10.3934/fods.2023002","url":null,"abstract":"We investigate the unsupervised learning of non-invertible observation functions in nonlinear state space models. Assuming abundant data of the observation process along with the distribution of the state process, we introduce a nonparametric generalized moment method to estimate the observation function via constrained regression. The major challenge comes from the non-invertibility of the observation function and the lack of data pairs between the state and observation. We address the fundamental issue of identifiability from quadratic loss functionals and show that the function space of identifiability is the closure of a RKHS that is intrinsic to the state process. Numerical results show that the first two moments and temporal correlations, along with upper and lower bounds, can identify functions ranging from piecewise polynomials to smooth functions, leading to convergent estimators. The limitations of this method, such as non-identifiability due to symmetry and stationarity, are also discussed.","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135534595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Noise calibration for SPDEs: A case study for the rotating shallow water model spde的噪声校正:以旋转浅水模型为例
Foundations of data science (Springfield, Mo.) Pub Date : 2023-01-01 DOI: 10.3934/fods.2023012
Dan Crisan, Oana Lang, Alexander Lobbe, Peter-Jan van Leeuwen, Roland Potthast
{"title":"Noise calibration for SPDEs: A case study for the rotating shallow water model","authors":"Dan Crisan, Oana Lang, Alexander Lobbe, Peter-Jan van Leeuwen, Roland Potthast","doi":"10.3934/fods.2023012","DOIUrl":"https://doi.org/10.3934/fods.2023012","url":null,"abstract":"","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134980769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Weight set decomposition for weighted rank and rating aggregation: An interpretable and visual decision support tool 加权排名和评级聚合的权重集分解:一个可解释和可视化的决策支持工具
Foundations of data science (Springfield, Mo.) Pub Date : 2023-01-01 DOI: 10.3934/fods.2023001
Tyler A. Perini, A. Langville, Glenn Kramer, Jeff Shrager, Mark Shapiro
{"title":"Weight set decomposition for weighted rank and rating aggregation: An interpretable and visual decision support tool","authors":"Tyler A. Perini, A. Langville, Glenn Kramer, Jeff Shrager, Mark Shapiro","doi":"10.3934/fods.2023001","DOIUrl":"https://doi.org/10.3934/fods.2023001","url":null,"abstract":"","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70248223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
GEOMETRIC STRUCTURE GUIDED MODEL AND ALGORITHMS FOR COMPLETE DECONVOLUTION OF GENE EXPRESSION DATA. 基因表达数据完全反褶积的几何结构导向模型和算法
Foundations of data science (Springfield, Mo.) Pub Date : 2022-09-01 DOI: 10.3934/fods.2022013
Duan Chen, Shaoyu Li, Xue Wang
{"title":"GEOMETRIC STRUCTURE GUIDED MODEL AND ALGORITHMS FOR COMPLETE DECONVOLUTION OF GENE EXPRESSION DATA.","authors":"Duan Chen, Shaoyu Li, Xue Wang","doi":"10.3934/fods.2022013","DOIUrl":"10.3934/fods.2022013","url":null,"abstract":"<p><p>Complete deconvolution analysis for bulk RNA-seq data is important and helpful to distinguish whether the differences of disease-associated GEPs (gene expression profiles) in tissues of patients and normal controls are due to changes in cellular composition of tissue samples, or due to GEPs changes in specific cells. One of the major techniques to perform complete deconvolution is nonnegative matrix factorization (NMF), which also has a wide-range of applications in the machine learning community. However, the NMF is a well-known strongly ill-posed problem, so a direct application of NMF to RNA-seq data will suffer severe difficulties in the interpretability of solutions. In this paper, we develop an NMF-based mathematical model and corresponding computational algorithms to improve the solution identifiability of deconvoluting bulk RNA-seq data. In our approach, we combine the biological concept of marker genes with the solvability conditions of the NMF theories, and develop a geometric structures guided optimization model. In this strategy, the geometric structure of bulk tissue data is first explored by the spectral clustering technique. Then, the identified information of marker genes is integrated as solvability constraints, while the overall correlation graph is used as manifold regularization. Both synthetic and biological data are used to validate the proposed model and algorithms, from which solution interpretability and accuracy are significantly improved.</p>","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10798655/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42614124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
ASPECTS OF TOPOLOGICAL APPROACHES FOR DATA SCIENCE. 数据科学拓扑方法的各个方面。
Foundations of data science (Springfield, Mo.) Pub Date : 2022-06-01 DOI: 10.3934/fods.2022002
Jelena Grbić, Jie Wu, Kelin Xia, Guo-Wei Wei
{"title":"ASPECTS OF TOPOLOGICAL APPROACHES FOR DATA SCIENCE.","authors":"Jelena Grbić, Jie Wu, Kelin Xia, Guo-Wei Wei","doi":"10.3934/fods.2022002","DOIUrl":"10.3934/fods.2022002","url":null,"abstract":"<p><p>We establish a new theory which unifies various aspects of topological approaches for data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.</p>","PeriodicalId":73054,"journal":{"name":"Foundations of data science (Springfield, Mo.)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9881677/pdf/nihms-1825620.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10592051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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