Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps
Ronald R. Coifman,Stephane Lafon,Ann B. Lee,Mauro Maggioni,Boaz Nadler,Frederick Warner,Steven W. Zucker +6 more
TLDR
The process of iterating or diffusing the Markov matrix is seen as a generalization of some aspects of the Newtonian paradigm, in which local infinitesimal transitions of a system lead to global macroscopic descriptions by integration.Abstract:
We provide a framework for structural multiscale geometric organization of graphs and subsets of R(n). We use diffusion semigroups to generate multiscale geometries in order to organize and represent complex structures. We show that appropriately selected eigenfunctions or scaling functions of Markov matrices, which describe local transitions, lead to macroscopic descriptions at different scales. The process of iterating or diffusing the Markov matrix is seen as a generalization of some aspects of the Newtonian paradigm, in which local infinitesimal transitions of a system lead to global macroscopic descriptions by integration. We provide a unified view of ideas from data analysis, machine learning, and numerical analysis.read more
Citations
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Manifold Regularization: A Geometric Framework for Learning from Labeled and Unlabeled Examples
TL;DR: A semi-supervised framework that incorporates labeled and unlabeled data in a general-purpose learner is proposed and properties of reproducing kernel Hilbert spaces are used to prove new Representer theorems that provide theoretical basis for the algorithms.
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Semi-Supervised Learning
TL;DR: Semi-supervised learning (SSL) as discussed by the authors is the middle ground between supervised learning (in which all training examples are labeled) and unsupervised training (where no label data are given).
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The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains
TL;DR: The field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process high-dimensional data on graphs as discussed by the authors, which are the analogs to the classical frequency domain and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs.
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SCANPY: large-scale single-cell gene expression data analysis
TL;DR: This work presents Scanpy, a scalable toolkit for analyzing single-cell gene expression data that includes methods for preprocessing, visualization, clustering, pseudotime and trajectory inference, differential expression testing, and simulation of gene regulatory networks, and AnnData, a generic class for handling annotated data matrices.
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Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
TL;DR: This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation, and presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions.
References
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Proceedings ArticleDOI
Normalized cuts and image segmentation
Jianbo Shi,Jitendra Malik +1 more
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The Elements of Statistical Learning
TL;DR: Chapter 11 includes more case studies in other areas, ranging from manufacturing to marketing research, and a detailed comparison with other diagnostic tools, such as logistic regression and tree-based methods.
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