Finding the Homology of Submanifolds with High Confidence from Random Samples

By Partha Niyogi, Steve Smale, Shmuel Weinberger. Discrete & Computational Geometry, 2008

746 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/345d254795b1aa41c9cb38bae12a580ebec7e8e1

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Citations in the corpus

Stability of Persistence Diagrams (2005, background): “In the smooth but possibly higher-dimensional setting, [18] shows how to build a homotopy equivalent complex, which suffices for homology estimation.”
Nonasymptotic rates for manifold, tangent space and curvature estimation (2017, methodology): “[1, 14] and in topological inference [22].”
To Trust Or Not To Trust A Classifier (2018, background): “As common to related work in the manifold setting, we make the following regularity assumptions which are standard among works in manifold learning (e.g. [41, 21, 2]).”
Topological Data Analysis (2016, methodology): “d-dimensional but it does converge to S in the following sense (Cuevas, 2009; Fasy et al., 2014b; Niyogi et al., 2008; Cuevas et al., 2001; Chazal et al., 2014b).”
Random Projections of Smooth Manifolds (2009, methodology): “1 [31] If M is a submanifold of RN with condition number 1/τ , then the norm of the second fundamental form is bounded by 1/τ in all directions.”
Persistent homology for random fields and complexes (2010, background): “One contributions in the spirit of this paper is [29] by Niyogi, Smale, and Weinberger, who studied the problem of estimating smooth manifolds from finite samples.”
Distance Preserving Embeddings for General n-Dimensional Manifolds (2012, methodology): “Lemma 16 (relating closeby tangent vectors – implicit in the proof of Proposition 6.2 Niyogi et al. (2006)) Pick any two (path-connected) points p, q ∈ M .”
Limit theorems for Betti numbers of random simplicial complexes (2010, result): “This may be of interest in topological data analysis, as in earlier work of Niyogi, Smale, and Weinberger [14].”
A statistical approach to persistent homology (2006, background): “Given a finite set, sometimes called point cloud data (PCD), that is randomly sampled from a subspace X of some metric space, one hopes to recover geometric and topological properties of X. Using random samples, P. Niyogi, S. Smale and S. Weinberger [ NSW05 ] show how to recover the homology of cert”
Random geometric complexes in the thermodynamic regime (2014, background): “[26, 27]) there has been considerable recent interest in the topological properties of CB(Φ, r) and C(Φ, r) that go beyond mere connectivity or the volumetric measures provided by integral geometry.”
The smashed filter for compressive classification and target recognition (2007, background): “The cost in terms of measurements is extremely modest; assuming similarly conditioned manifolds, the difference between (11) and (10) is approximately O(K log(P )) additional measurements.”
Topology and data (2009, background): “Further, even if one could assume that the data lies along a manifold, one is usually not in a position to verify that the stringent hypotheses of [52] are satisfied.”
Testing the Manifold Hypothesis (2013, background): “We refer the interested reader to a limited set of papers associated with this field; see [3, 8, 9, 11, 16, 20, 26, 31, 32, 41, 43, 47, 50, 52, 55] and the references therein.”
Finding and Leveraging Structure in Learning Problems (2012, methodology): “Building on the work of [Niyogi et al., 2008, 2011], we derive efficient estimators under a variety of noise models.”
Topological Consistency via Kernel Estimation (2014, methodology): “Asymptotic and non-asymptotic analysis of consistency and convergence of topological summaries as the number of observations increase has been examined for a variety of geometric objects using a variety of statistical and probabilistic tools [1, 2, 3, 4, 5, 8, 9, 12, 16, 30, 31, 36, 37].”