LEADER 04190nam 22007215 450 001 9910736025803321 005 20240522143355.0 010 $a3-031-33440-X 024 7 $a10.1007/978-3-031-33440-5 035 $a(MiAaPQ)EBC30669135 035 $a(Au-PeEL)EBL30669135 035 $a(DE-He213)978-3-031-33440-5 035 $a(PPN)272254509 035 $a(CKB)27878834500041 035 $a(EXLCZ)9927878834500041 100 $a20230729d2023 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aMathematical Principles of Topological and Geometric Data Analysis /$fby Parvaneh Joharinad, Jürgen Jost 205 $a1st ed. 2023. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2023. 215 $a1 online resource (287 pages) 225 1 $aMathematics of Data,$x2731-4111 ;$v2 311 08$aPrint version: Joharinad, Parvaneh Mathematical Principles of Topological and Geometric Data Analysis Cham : Springer International Publishing AG,c2023 9783031334399 327 $aIntroduction -- Topological foundations, hypercomplexes and homology -- Weighted complexes, cohomology and Laplace operators -- The Laplace operator and the geometry of graphs -- Metric spaces and manifolds -- Linear methods: Kernels, variations, and averaging -- Nonlinear schemes: Clustering, feature extraction and dimension reduction -- Manifold learning, the scheme of Laplacian eigenmaps -- Metrics and curvature. 330 $aThis book explores and demonstrates how geometric tools can be used in data analysis. Beginning with a systematic exposition of the mathematical prerequisites, covering topics ranging from category theory to algebraic topology, Riemannian geometry, operator theory and network analysis, it goes on to describe and analyze some of the most important machine learning techniques for dimension reduction, including the different types of manifold learning and kernel methods. It also develops a new notion of curvature of generalized metric spaces, based on the notion of hyperconvexity, which can be used for the topological representation of geometric information. In recent years there has been a fascinating development: concepts and methods originally created in the context of research in pure mathematics, and in particular in geometry, have become powerful tools in machine learning for the analysis of data. The underlying reason for this is that data are typically equipped with some kind of notion of distance, quantifying the differences between data points. Of course, to be successfully applied, the geometric tools usually need to be redefined, generalized, or extended appropriately. Primarily aimed at mathematicians seeking an overview of the geometric concepts and methods that are useful for data analysis, the book will also be of interest to researchers in machine learning and data analysis who want to see a systematic mathematical foundation of the methods that they use. 410 0$aMathematics of Data,$x2731-4111 ;$v2 606 $aMathematics 606 $aMachine learning 606 $aComputer science 606 $aGeometry 606 $aTopology 606 $aApplications of Mathematics 606 $aMachine Learning 606 $aComputational Geometry 606 $aTopology 606 $aTopologia$2thub 606 $aAnàlisi matemàtica$2thub 608 $aLlibres electrònics$2thub 615 0$aMathematics. 615 0$aMachine learning. 615 0$aComputer science. 615 0$aGeometry. 615 0$aTopology. 615 14$aApplications of Mathematics. 615 24$aMachine Learning. 615 24$aComputational Geometry. 615 24$aTopology. 615 7$aTopologia 615 7$aAnàlisi matemàtica 676 $a514 700 $aJoharinad$b Parvaneh$01380318 701 $aJost$b Jürgen$054734 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910736025803321 996 $aMathematical Principles of Topological and Geometric Data Analysis$93421585 997 $aUNINA