Dissertation: final update
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Dimitri Lozeve
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@ -738,18 +738,6 @@ novel application of the discriminatory power of {PIs}.},
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file = {Fulltext PDF:/home/dimitri/Zotero/storage/EUWNMLQF/Adams et al. - 2017 - Persistence Images A Stable Vector Representation.pdf:application/pdf}
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}
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@article{bubenik_statistical_2015-1,
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title = {Statistical Topological Data Analysis using Persistence Landscapes},
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volume = {16},
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url = {http://www.jmlr.org/papers/v16/bubenik15a.html},
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pages = {77--102},
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journaltitle = {Journal of Machine Learning Research},
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author = {Bubenik, Peter},
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urldate = {2018-06-12},
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date = {2015},
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file = {Full Text PDF:/home/dimitri/Zotero/storage/CJW9F5XG/Bubenik - 2015 - Statistical Topological Data Analysis using Persis.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/E2DN26NP/bubenik15a.html:text/html}
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}
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@article{kalisnik_tropical_2018,
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title = {Tropical Coordinates on the Space of Persistence Barcodes},
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issn = {1615-3375, 1615-3383},
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@ -873,4 +861,187 @@ novel application of the discriminatory power of {PIs}.},
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urldate = {2018-07-31},
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date = {2010},
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keywords = {computational topology, simplicial set, vietoris-rips complex, witness complex}
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}
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@inproceedings{zeppelzauer_topological_2016,
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title = {Topological Descriptors for 3D Surface Analysis},
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isbn = {978-3-319-39440-4 978-3-319-39441-1},
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url = {https://link.springer.com/chapter/10.1007/978-3-319-39441-1_8},
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doi = {10.1007/978-3-319-39441-1_8},
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series = {Lecture Notes in Computer Science},
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abstract = {We investigate topological descriptors for 3D surface analysis, i.e. the classification of surfaces according to their geometric fine structure. On a dataset of high-resolution 3D surface reconstructions we compute persistence diagrams for a 2D cubical filtration. In the next step we investigate different topological descriptors and measure their ability to discriminate structurally different 3D surface patches. We evaluate their sensitivity to different parameters and compare the performance of the resulting topological descriptors to alternative (non-topological) descriptors. We present a comprehensive evaluation that shows that topological descriptors are (i) robust, (ii) yield state-of-the-art performance for the task of 3D surface analysis and (iii) improve classification performance when combined with non-topological descriptors.},
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eventtitle = {International Workshop on Computational Topology in Image Context},
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pages = {77--87},
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booktitle = {Computational Topology in Image Context},
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publisher = {Springer, Cham},
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author = {Zeppelzauer, Matthias and Zieliński, Bartosz and Juda, Mateusz and Seidl, Markus},
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urldate = {2018-08-16},
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date = {2016-06-15},
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langid = {english},
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file = {Snapshot:/home/dimitri/Zotero/storage/JH8QTE5R/978-3-319-39441-1_8.html:text/html}
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}
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@article{muandet_kernel_2017,
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title = {Kernel Mean Embedding of Distributions: A Review and Beyond},
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volume = {10},
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issn = {1935-8237, 1935-8245},
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url = {https://www.nowpublishers.com/article/Details/MAL-060},
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doi = {10.1561/2200000060},
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shorttitle = {Kernel Mean Embedding of Distributions},
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abstract = {Kernel Mean Embedding of Distributions: A Review and Beyond},
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pages = {1--141},
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number = {1},
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journaltitle = {Foundations and Trends® in Machine Learning},
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shortjournal = {{MAL}},
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author = {Muandet, Krikamol and Fukumizu, Kenji and Sriperumbudur, Bharath and Schölkopf, Bernhard},
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urldate = {2018-08-30},
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date = {2017-06-28},
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file = {Full Text PDF:/home/dimitri/Zotero/storage/87JN65NM/Muandet et al. - 2017 - Kernel Mean Embedding of Distributions A Review a.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/7DS27M8D/MAL-060.html:text/html}
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}
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@book{berlinet_reproducing_2011,
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title = {Reproducing Kernel Hilbert Spaces in Probability and Statistics},
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isbn = {978-1-4419-9096-9},
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abstract = {The reproducing kernel Hilbert space construction is a bijection or transform theory which associates a positive definite kernel (gaussian processes) with a Hilbert space offunctions. Like all transform theories (think Fourier), problems in one space may become transparent in the other, and optimal solutions in one space are often usefully optimal in the other. The theory was born in complex function theory, abstracted and then accidently injected into Statistics; Manny Parzen as a graduate student at Berkeley was given a strip of paper containing his qualifying exam problem- It read "reproducing kernel Hilbert space"- In the 1950's this was a truly obscure topic. Parzen tracked it down and internalized the subject. Soon after, he applied it to problems with the following fla vor: consider estimating the mean functions of a gaussian process. The mean functions which cannot be distinguished with probability one are precisely the functions in the Hilbert space associated to the covariance kernel of the processes. Parzen's own lively account of his work on re producing kernels is charmingly told in his interview with H. Joseph Newton in Statistical Science, 17, 2002, p. 364-366. Parzen moved to Stanford and his infectious enthusiasm caught Jerry Sacks, Don Ylvisaker and Grace Wahba among others. Sacks and Ylvis aker applied the ideas to design problems such as the following. Sup pose ({XdO}},
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pagetotal = {369},
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publisher = {Springer Science \& Business Media},
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author = {Berlinet, Alain and Thomas-Agnan, Christine},
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date = {2011-06-28},
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langid = {english},
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note = {Google-Books-{ID}: {bX}3TBwAAQBAJ},
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keywords = {Business \& Economics / Economics / General, Business \& Economics / Economics / Theory, Business \& Economics / General, Business \& Economics / Statistics, Mathematics / Probability \& Statistics / General}
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}
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@thesis{price-wright_topological_2015,
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title = {A Topological Approach to Temporal Networks},
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abstract = {“Temporal networks” are a mathematical tool to represent systems that change
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over time. Research on temporal networks is very active, and limited theoreti-
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cal work has been done to study them. One approach to is to construct a series
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of static subgraphs called snapshots. Existing techniques attempt to find the
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temporal structure of a network to inform its partitioning into snapshots. An
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important goal of such methods is to uncover meaningful temporal structure
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that corresponds to actual features of the underlying system.
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We investigate existing methods used to partition temporal networks based
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on di↵erent ways of identifying temporal structure. Such methods have never
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previously been compared directly to each other, so we examine and evaluate
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their performance side-by-side on a suite of random-graph ensembles. We
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show that without prior knowledge about a network’s temporal structure,
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these existing methods have limitations producing meaningful partitions.
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To tackle the problem of finding temporal structure in a network, we ap-
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ply methods from computational topology. Such methods have begun to be
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employed in the study of static networks and provide a summary of global
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features in data sets. We use them here to track the topology of a network
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over time and distinguish important temporal features from trivial ones. We
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define two types of topological spaces derived from temporal networks and use
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persistent homology to generate a temporal profile for a network. We then
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present di↵erent ways to use this to understand a network’s temporal struc-
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ture with limited prior knowledge. We show that the methods we apply from
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computational topology can distinguish temporal distributions and provide a
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high-level summary of temporal structure. These combined can be used to
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inform a meaningful network partitioning and a deeper understanding of a
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temporal network itself.},
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institution = {University of Oxford},
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type = {{MSc} dissertation in Mathematics and Foundations of Computer Science},
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author = {Price-Wright, Erin},
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date = {2015},
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file = {Price-Wrigt - 2015 - A Topological Approach to Temporal Networks.pdf:/home/dimitri/Zotero/storage/6YI5RC6K/Price-Wrigt - 2015 - A Topological Approach to Temporal Networks.pdf:application/pdf}
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}
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@inproceedings{edelsbrunner_topological_2000,
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title = {Topological persistence and simplification},
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doi = {10.1109/SFCS.2000.892133},
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abstract = {We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise, depending on its life-time or persistence within the filtration. We give fast algorithms for completing persistence and experimental evidence for their speed and utility.},
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eventtitle = {Proceedings 41st Annual Symposium on Foundations of Computer Science},
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pages = {454--463},
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booktitle = {Proceedings 41st Annual Symposium on Foundations of Computer Science},
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author = {Edelsbrunner, H. and Letscher, D. and Zomorodian, A.},
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date = {2000-11},
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keywords = {Topology, History, computational topology, algorithm theory, alpha shapes, computational geometry, Computational geometry, Computer graphics, Computer science, Density functional theory, fast algorithms, filtration, Filtration, growing complex, homology groups, Mathematics, Noise shaping, Shape, topological change, topological persistence, topological simplification, topology},
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file = {IEEE Xplore Abstract Record:/home/dimitri/Zotero/storage/5LPIWG5Z/892133.html:text/html}
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}
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@article{morozov_persistence_2005,
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title = {Persistence algorithm takes cubic time in worst case},
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abstract = {Given a sequence of N simplices, we consider the sequence of sets Ki consisting of the first i simplices, for 1 ≤ i ≤ N. We call the sequence of Ki a filtration if all the Ki are simplicial complexes. In this note, we describe a filtration of a simplicial complex of N simplices on which the algorithm Pair-Simplices of Edelsbrunner, Letscher and Zomorodian [1] performs Ω(N 3) operations. The existence of this filtration should be contrasted to the experimentally observed only slightly super-linear running time for filtrations that arise from applications. We describe the space as well as the ordering on the simplices. Let n = ⌊(N + 29)/7⌋, v = ⌊(n − 1)/2⌋, and note that both n and v are in Ω(N). In our filtration, all vertices appear before all edges in the filtration, and all edges appear before all triangles. The indices that we assign to the simplices will be within their respective classes (e.g., edge labeled n will appear before the triangle labeled 1). Some edges will receive a negative index, which is done for simplicity to indicate that they appear before the edges with positive labels (see Figure 2). Figure 1 illustrates the construction of our space as well as the assignment of indices to the simplices. Starting with triangle {ABC}, we add v vertices inside the triangle in the following manner: we place the first vertex V1 near the middle of edge {AB}, the second vertex V2 near the middle of},
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journaltitle = {{BioGeometry} News, Dept. Comput. Sci., Duke Univ},
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author = {Morozov, Dmitriy},
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date = {2005},
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file = {Citeseer - Full Text PDF:/home/dimitri/Zotero/storage/I6HKHZQ5/Morozov - 2005 - Persistence algorithm takes cubic time in worst ca.pdf:application/pdf;Citeseer - Snapshot:/home/dimitri/Zotero/storage/WN8ZVZH9/summary.html:text/html}
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}
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@article{de_silva_persistent_2011,
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title = {Persistent Cohomology and Circular Coordinates},
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volume = {45},
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issn = {1432-0444},
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url = {https://doi.org/10.1007/s00454-011-9344-x},
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doi = {10.1007/s00454-011-9344-x},
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abstract = {Nonlinear dimensionality reduction ({NLDR}) algorithms such as Isomap, {LLE}, and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle, for example, is one-dimensional, but its faithful representation requires two real coordinates. In this work, we present a strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise {NLDR} analysis of a broader range of realistic data sets.},
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pages = {737--759},
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number = {4},
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journaltitle = {Discrete \& Computational Geometry},
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shortjournal = {Discrete Comput Geom},
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author = {de Silva, Vin and Morozov, Dmitriy and Vejdemo-Johansson, Mikael},
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urldate = {2018-09-05},
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date = {2011-06-01},
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langid = {english},
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keywords = {Persistent homology, Computational topology, Dimensionality reduction, Persistent cohomology},
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file = {Springer Full Text PDF:/home/dimitri/Zotero/storage/EX9L3F7F/de Silva et al. - 2011 - Persistent Cohomology and Circular Coordinates.pdf:application/pdf}
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}
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@article{de_silva_dualities_2011,
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title = {Dualities in persistent (co)homology},
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volume = {27},
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issn = {0266-5611, 1361-6420},
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url = {http://stacks.iop.org/0266-5611/27/i=12/a=124003?key=crossref.1f4b24ef80c9b1fc789ecdc6221097de},
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doi = {10.1088/0266-5611/27/12/124003},
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pages = {124003},
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number = {12},
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journaltitle = {Inverse Problems},
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author = {de Silva, Vin and Morozov, Dmitriy and Vejdemo-Johansson, Mikael},
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urldate = {2018-09-05},
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date = {2011-12-01}
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}
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@incollection{mcgeoch_distributed_2014,
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location = {Philadelphia, {PA}},
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title = {Distributed Computation of Persistent Homology},
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isbn = {978-1-61197-319-8},
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url = {http://epubs.siam.org/doi/abs/10.1137/1.9781611973198.4},
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pages = {31--38},
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booktitle = {2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments ({ALENEX})},
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publisher = {Society for Industrial and Applied Mathematics},
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author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan},
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editor = {{McGeoch}, Catherine C. and Meyer, Ulrich},
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urldate = {2018-09-05},
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date = {2014-05},
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langid = {english},
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doi = {10.1137/1.9781611973198.4}
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}
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@article{carlsson_zigzag_2008,
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title = {Zigzag Persistence},
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url = {http://arxiv.org/abs/0812.0197},
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abstract = {We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by that theory. In this paper we develop theoretical and algorithmic foundations with a view towards applications in topological statistics.},
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journaltitle = {{arXiv}:0812.0197 [cs]},
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author = {Carlsson, Gunnar and de Silva, Vin},
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urldate = {2018-09-08},
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date = {2008-11-30},
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eprinttype = {arxiv},
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eprint = {0812.0197},
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keywords = {Computer Science - Computational Geometry, I.3.5},
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file = {arXiv\:0812.0197 PDF:/home/dimitri/Zotero/storage/PKSM89FF/Carlsson and de Silva - 2008 - Zigzag Persistence.pdf:application/pdf;arXiv.org Snapshot:/home/dimitri/Zotero/storage/QF37EI5F/0812.html:text/html}
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}
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@article{maria_computing_2016,
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title = {Computing Zigzag Persistent Cohomology},
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url = {http://arxiv.org/abs/1608.06039},
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abstract = {Zigzag persistent homology is a powerful generalisation of persistent homology that allows one not only to compute persistence diagrams with less noise and using less memory, but also to use persistence in new fields of application. However, due to the increase in complexity of the algebraic treatment of the theory, most algorithmic results in the field have remained of theoretical nature. This article describes an efficient algorithm to compute zigzag persistence, emphasising on its practical interest. The algorithm is a zigzag persistent cohomology algorithm, based on the dualisation of reflections and transpositions transformations within the zigzag sequence. We provide an extensive experimental study of the algorithm. We study the algorithm along two directions. First, we compare its performance with zigzag persistent homology algorithm and show the interest of cohomology in zigzag persistence. Second, we illustrate the interest of zigzag persistence in topological data analysis by comparing it to state of the art methods in the field, specifically optimised algorithm for standard persistent homology and sparse filtrations. We compare the memory and time complexities of the different algorithms, as well as the quality of the output persistence diagrams.},
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journaltitle = {{arXiv}:1608.06039 [cs]},
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author = {Maria, Clément and Oudot, Steve},
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urldate = {2018-09-08},
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date = {2016-08-21},
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eprinttype = {arxiv},
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eprint = {1608.06039},
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keywords = {Computer Science - Computational Geometry},
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file = {arXiv\:1608.06039 PDF:/home/dimitri/Zotero/storage/LJBHWTMY/Maria and Oudot - 2016 - Computing Zigzag Persistent Cohomology.pdf:application/pdf;arXiv.org Snapshot:/home/dimitri/Zotero/storage/TCKJGZET/1608.html:text/html}
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}
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