Filters








4,784 Hits in 6.4 sec

Manifold reconstruction in arbitrary dimensions using witness complexes

Jean-Daniel Boissonnat, Leonidas J. Guibas, Steve Y. Oudot
2007 Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07  
We also use our structural results to devise an algorithm that reconstructs manifolds of any arbitrary dimension or co-dimension at different scales.  ...  It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions.  ...  The first author was supported in part by the Program of the EU as Shared-cost RTD (FET Open) Project under Contract No IST-006413 (ACS Algorithms for Complex Shapes).  ... 
doi:10.1145/1247069.1247106 dblp:conf/compgeom/BoissonnatGO07 fatcat:cdawlpqrwfddfl46p2yd65iszi

Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes

Jean-Daniel Boissonnat, Leonidas J. Guibas, Steve Y. Oudot
2009 Discrete & Computational Geometry  
We also use our structural results to devise an algorithm that reconstructs manifolds of any arbitrary dimension or co-dimension at different scales.  ...  It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions.  ...  The first author was supported in part by the Program of the EU as Shared-cost RTD (FET Open) Project under Contract No IST-006413 (ACS Algorithms for Complex Shapes).  ... 
doi:10.1007/s00454-009-9175-1 fatcat:4crybknpqzh6xo3wkndc2b62xy

Reconstruction Using Witness Complexes

Leonidas J. Guibas, Steve Y. Oudot
2008 Discrete & Computational Geometry  
To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d.  ...  At a high level, our method is very similar in spirit to Chew's surface meshing algorithm, with one notable difference: the restricted Delaunay triangulation is replaced by the witness complex, which makes  ...  Application to reconstruction 4.1 Algorithm. The algorithm works in any arbitrary metric space.  ... 
doi:10.1007/s00454-008-9094-6 pmid:21643440 pmcid:PMC3105630 fatcat:4ubzmcuo6ve4hhh6zyr3buafua

Towards Persistence-Based Reconstruction in Euclidean Spaces [article]

Frédéric Chazal, Steve Oudot (INRIA Sophia Antipolis)
2007 arXiv   pre-print
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions.  ...  In this paper, we introduce a novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data.  ...  It is therefore appealing to consider other filtrations that are easy to compute in arbitrary dimensions, such as the Rips and witness complex filtrations.  ... 
arXiv:0712.2638v2 fatcat:m5pblzah4vhz7ngi4tm6dzunuy

Only distances are required to reconstruct submanifolds [article]

Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, Steve Y. Oudot
2017 arXiv   pre-print
Our algorithm uses the witness complex and relies on the stability of power protection, a notion introduced in this paper.  ...  The complexity of the algorithm depends exponentially on the intrinsic dimension of the manifold, rather than the dimension of ambient space, and linearly on the dimension of the ambient space.  ...  Partial support has also been provided by the Advanced Grant of the European Research Council GUDHI (Geometric Understanding in Higher Dimensions).  ... 
arXiv:1410.7012v2 fatcat:6xtvydtimvep3nlj7o7jypvwdu

Exploring the topology of dynamical reconstructions

Joshua Garland, Elizabeth Bradley, James D. Meiss
2016 Physica D : Non-linear phenomena  
Using trajectories from the classic Lorenz system, we reconstruct the dynamics using the method of delays, then build a parsimonious simplicial complex---the "witness complex"---to compute its homology  ...  Surprisingly, we find that the witness complex correctly resolves the homology of the underlying invariant set from noisy samples of that set even if the reconstruction dimension is well below the thresholds  ...  The authors thank the IMA for hosting the Algebraic Topology and Dynamics conference in February 2014, and K. Mischaikow for numerous useful discussions.  ... 
doi:10.1016/j.physd.2016.03.006 fatcat:xpydtqkpg5c3ppdicsdyy4m3bi

A weak characterisation of the Delaunay triangulation

Vin de Silva
2008 Geometriae Dedicata  
We give a short empirical demonstration that weak Delaunay complexes can lead to dramatically clean results in the problem of estimating the homology groups of a manifold represented by a finite point  ...  There are weighted and approximate versions of the weak and strong complexes in all these geometries, and we prove equality theorems in those cases also.  ...  in Section 3; and Gunnar Carlsson, for his unfailing support during much of the time this work was being carried out.  ... 
doi:10.1007/s10711-008-9261-1 fatcat:5xyik2euyja5jew7a3qgiikp3q

Towards persistence-based reconstruction in euclidean spaces

Frédéric Chazal, Steve Yann Oudot
2008 Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08  
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions.  ...  In this paper, we introduce a novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data.  ...  data in arbitrary dimensions.  ... 
doi:10.1145/1377676.1377719 dblp:conf/compgeom/ChazalO08 fatcat:vge7xs7oprdythqpvvwad3blla

Graph Induced Complex on Point Data [article]

Tamal K. Dey, Fengtao Fan, Yusu Wang
2013 arXiv   pre-print
We show that, using the graph induced complex one can (i) infer the one dimensional homology of a manifold from a very lean subsample, (ii) reconstruct a surface in three dimension from a sparse subsample  ...  The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structures.  ...  Reconstruction of M using G α (P, Q, d E ) In this subsection, we observe that the graph induced complexes can also be used for surface reconstruction.  ... 
arXiv:1304.0662v1 fatcat:6sjfinlcffhhtprhahnt4i7wzi

Geometry Score: A Method For Comparing Generative Adversarial Networks [article]

Valentin Khrulkov, Ivan Oseledets
2018 arXiv   pre-print
In this work, we construct a novel measure of performance of a GAN by comparing geometrical properties of the underlying data manifold and the generated one, which provides both qualitative and quantitative  ...  Our algorithm can be applied to datasets of an arbitrary nature and is not limited to visual data.  ...  In order to reconstruct the underlying manifold using a simplicial complex several methods exist.  ... 
arXiv:1802.02664v3 fatcat:6jdrxgl56rgrbm65sjl7la77bu

Graph induced complex on point data

Tamal Krishna Dey, Fengtao Fan, Yusu Wang
2013 Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13  
We show that, using the graph induced complex one can (i) infer the one dimensional homology of a manifold from a lean subsample, (ii) reconstruct a surface in three dimension from a sparse subsample without  ...  The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structure.  ...  Reconstruction of M using G α (P, Q, d E ) In this subsection, we observe that the graph induced complexes can also be used for surface reconstruction.  ... 
doi:10.1145/2493132.2462387 fatcat:mv54hjqxnveabbdsugrpxde6iy

Graph induced complex on point data

Tamal Krishna Dey, Fengtao Fan, Yusu Wang
2013 Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13  
We show that, using the graph induced complex one can (i) infer the one dimensional homology of a manifold from a lean subsample, (ii) reconstruct a surface in three dimension from a sparse subsample without  ...  The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structure.  ...  Reconstruction of M using G α (P, Q, d E ) In this subsection, we observe that the graph induced complexes can also be used for surface reconstruction.  ... 
doi:10.1145/2462356.2462387 dblp:conf/compgeom/DeyFW13 fatcat:6fo5ffsnqnhdzcdezamzlagxaa

Graph induced complex on point data

Tamal K. Dey, Fengtao Fan, Yusu Wang
2015 Computational geometry  
We show that, using the graph induced complex one can (i) infer the one dimensional homology of a manifold from a lean subsample, (ii) reconstruct a surface in three dimension from a sparse subsample without  ...  The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structure.  ...  Reconstruction of M using G α (P, Q, d E ) In this subsection, we observe that the graph induced complexes can also be used for surface reconstruction.  ... 
doi:10.1016/j.comgeo.2015.04.003 fatcat:brrjrpdz2fbxzh2k3d4rfstmnq

Computational Topology [chapter]

Afra Zomorodian
2009 Algorithms and Theory of Computation Handbook, Second Edition, Volume 2  
Recently, Boissonnat et al. [2007] use witness complexes to reconstruct manifolds in arbitrary dimensions. There is also a new focus on finding homotopy equivalent spaces.  ...  de Silva and Carlsson [2004] define the witness complex using a relaxation of the Delaunay test. The complex is built on landmarks, a subset of the point sets.  ... 
doi:10.1201/9781584888215-c3 fatcat:viuyqsyftfbhfgvlg7vkwbguce

DeepVisualInsight: Time-Travelling Visualization for Spatio-Temporal Causality of Deep Classification Training

Xianglin Yang, Yun Lin, Ruofan Liu, Zhenfeng He, Chao Wang, Jin Song Dong, Hong Mei
2022 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
Such causality allows us to observe and analyze the whole learning process in the visible low dimensional space.  ...  In this work, we propose a time-travelling visual solution DeepVisualInsight (DVI), aiming to manifest the spatio-temporal causality while training a deep learning image classifier.  ...  T1-251RES1901), the National Research Foundation Singapore via its National Satellite of Excellence in Trustworthy Software Systems office (NSOE-TSS2019-05).  ... 
doi:10.1609/aaai.v36i5.20473 fatcat:7bhaegzyhrh5fdtvk2kfd3wqvi
« Previous Showing results 1 — 15 out of 4,784 results