Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
On Wiedemann's Method of Solving Sparse Linear Systems
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Discrete & Computational Geometry
Vines and vineyards by updating persistence in linear time
Proceedings of the twenty-second annual symposium on Computational geometry
Retrieval of trademark images by means of size functions
Graphical Models - Special issue on the vision, video and graphics conference 2005
Stability of Persistence Diagrams
Discrete & Computational Geometry
Faster inversion and other black box matrix computations using efficient block projections
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
On the Local Behavior of Spaces of Natural Images
International Journal of Computer Vision
Extending Persistence Using Poincaré and Lefschetz Duality
Foundations of Computational Mathematics
Persistent cohomology and circular coordinates
Proceedings of the twenty-fifth annual symposium on Computational geometry
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Zigzag persistent homology and real-valued functions
Proceedings of the twenty-fifth annual symposium on Computational geometry
The Theory of Multidimensional Persistence
Discrete & Computational Geometry - 23rd Annual Symposium on Computational Geometry
Measuring and computing natural generators for homology groups
Computational Geometry: Theory and Applications
Persistence Diagrams of Cortical Surface Data
IPMI '09 Proceedings of the 21st International Conference on Information Processing in Medical Imaging
The tidy set: a minimal simplicial set for computing homology of clique complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
A randomized O(m log m) time algorithm for computing Reeb graphs of arbitrary simplicial complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
Topological inference via meshing
Proceedings of the twenty-sixth annual symposium on Computational geometry
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Computing Robustness and Persistence for Images
IEEE Transactions on Visualization and Computer Graphics
Zigzag persistent homology in matrix multiplication time
Proceedings of the twenty-seventh annual symposium on Computational geometry
Diffusion runs low on persistence fast
ICCV '11 Proceedings of the 2011 International Conference on Computer Vision
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In this paper, we present the first output-sensitive algorithm to compute the persistence diagram of a filtered simplicial complex. For any @C0, it returns only those homology classes with persistence at least @C. Instead of the classical reduction via column operations, our algorithm performs rank computations on submatrices of the boundary matrix. For an arbitrary constant @d@?(0,1), the running time is O(C"("1"-"@d")"@CR"d(n)logn), where C"("1"-"@d")"@C is the number of homology classes with persistence at least (1-@d)@C, n is the total number of simplices in the complex, d its dimension, and R"d(n) is the complexity of computing the rank of an nxn matrix with O(dn) nonzero entries. Depending on the choice of the rank algorithm, this yields a deterministic O(C"("1"-"@d")"@Cn^2^.^3^7^6) algorithm, an O(C"("1"-"@d")"@Cn^2^.^2^8) Las-Vegas algorithm, or an O(C"("1"-"@d")"@Cn^2^+^@e) Monte-Carlo algorithm for an arbitrary @e0. The space complexity of the Monte-Carlo version is bounded by O(dn)=O(nlogn).