
@book{oudot_persistence_2015,
	location = {Providence, Rhode Island},
	title = {Persistence theory: from quiver representations to data analysis},
	isbn = {978-1-4704-2545-6},
	series = {Mathematical surveys and monographs},
	shorttitle = {Persistence theory},
	pagetotal = {218},
	number = {volume 209},
	publisher = {American Mathematical Society},
	author = {Oudot, Steve Y.},
	date = {2015},
	keywords = {Algebraic topology, Algebraic topology -- Applied homological algebra and category theory -- Simplicial sets and complexes, Associative rings and algebras -- Representation theory of rings and algebras -- Representations of quivers and partially ordered sets, Computer science -- Computing methodologies and applications -- Computer graphics; computational geometry, Homology theory, Statistics -- Data analysis},
	file = {Steve_Oudot_Persistence_Theory.pdf:/home/dimitri/Zotero/storage/ALZW577G/Steve_Oudot_Persistence_Theory.pdf:application/pdf}
}

@article{carlsson_topology_2009,
	title = {Topology and data},
	volume = {46},
	issn = {0273-0979},
	url = {http://www.ams.org/journal-getitem?pii=S0273-0979-09-01249-X},
	doi = {10.1090/S0273-0979-09-01249-X},
	pages = {255--308},
	number = {2},
	journaltitle = {Bulletin of the American Mathematical Society},
	author = {Carlsson, Gunnar},
	urldate = {2017-11-03},
	date = {2009-01-29},
	langid = {english},
	file = {carlsson2009.pdf:/home/dimitri/Zotero/storage/WYT52FA5/carlsson2009.pdf:application/pdf}
}

@article{chazal_introduction_2017,
	title = {An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists},
	shorttitle = {An introduction to Topological Data Analysis},
	journaltitle = {{arXiv} preprint {arXiv}:1710.04019},
	author = {Chazal, Frédéric and Michel, Bertrand},
	date = {2017},
	file = {chazal2017.pdf:/home/dimitri/Zotero/storage/CH8YWVM3/chazal2017.pdf:application/pdf}
}

@article{xu_hierarchical_2017,
	title = {Hierarchical Segmentation Using Tree-Based Shape Spaces},
	volume = {39},
	issn = {0162-8828, 2160-9292},
	url = {http://ieeexplore.ieee.org/document/7452658/},
	doi = {10.1109/TPAMI.2016.2554550},
	pages = {457--469},
	number = {3},
	journaltitle = {{IEEE} Transactions on Pattern Analysis and Machine Intelligence},
	author = {Xu, Yongchao and Carlinet, Edwin and Geraud, Thierry and Najman, Laurent},
	urldate = {2017-11-03},
	date = {2017-03-01},
	file = {xu2016.pdf:/home/dimitri/Zotero/storage/X49E35AC/xu2016.pdf:application/pdf}
}

@article{tierny_generalized_2012,
	title = {Generalized topological simplification of scalar fields on surfaces},
	volume = {18},
	pages = {2005--2013},
	number = {12},
	journaltitle = {{IEEE} transactions on visualization and computer graphics},
	author = {Tierny, Julien and Pascucci, Valerio},
	date = {2012},
	file = {tierny2012.pdf:/home/dimitri/Zotero/storage/ID96MTE2/tierny2012.pdf:application/pdf}
}

@article{tierny_loop_2009,
	title = {Loop surgery for volumetric meshes: Reeb graphs reduced to contour trees},
	volume = {15},
	shorttitle = {Loop surgery for volumetric meshes},
	number = {6},
	journaltitle = {{IEEE} Transactions on Visualization and Computer Graphics},
	author = {Tierny, Julien and Gyulassy, Attila and Simon, Eddie and Pascucci, Valerio},
	date = {2009},
	file = {tierny2009.pdf:/home/dimitri/Zotero/storage/9VGB22UH/tierny2009.pdf:application/pdf}
}

@inproceedings{monasse_scale-space_1999,
	title = {Scale-space from a level lines tree},
	volume = {99},
	pages = {175--186},
	booktitle = {Scale-Space},
	publisher = {Springer},
	author = {Monasse, Pascal and Guichard, Frédéric},
	date = {1999},
	file = {Scale-Space_from_a_Level_Lines_Tree.pdf:/home/dimitri/Zotero/storage/GXHUMD2G/Scale-Space_from_a_Level_Lines_Tree.pdf:application/pdf}
}

@article{robins_theory_2011,
	title = {Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images},
	volume = {33},
	issn = {0162-8828},
	url = {http://ieeexplore.ieee.org/document/5766002/},
	doi = {10.1109/TPAMI.2011.95},
	pages = {1646--1658},
	number = {8},
	journaltitle = {{IEEE} Transactions on Pattern Analysis and Machine Intelligence},
	author = {Robins, V and Wood, P J and Sheppard, A P},
	urldate = {2017-11-03},
	date = {2011-08},
	file = {robins2011.pdf:/home/dimitri/Zotero/storage/D4PMIRGY/robins2011.pdf:application/pdf}
}

@article{rieck_persistent_2015,
	title = {Persistent Homology for the Evaluation of Dimensionality Reduction Schemes},
	volume = {34},
	issn = {01677055},
	url = {http://doi.wiley.com/10.1111/cgf.12655},
	doi = {10.1111/cgf.12655},
	pages = {431--440},
	number = {3},
	journaltitle = {Computer Graphics Forum},
	author = {Rieck, B. and Leitte, H.},
	urldate = {2017-11-03},
	date = {2015-06},
	langid = {english},
	file = {rieck2015.pdf:/home/dimitri/Zotero/storage/4VEXZ4DG/rieck2015.pdf:application/pdf}
}

@inproceedings{reininghaus_stable_2015,
	title = {A stable multi-scale kernel for topological machine learning},
	pages = {4741--4748},
	booktitle = {Proceedings of the {IEEE} conference on computer vision and pattern recognition},
	author = {Reininghaus, Jan and Huber, Stefan and Bauer, Ulrich and Kwitt, Roland},
	date = {2015},
	file = {reininghaus2015.pdf:/home/dimitri/Zotero/storage/H6VIHWWS/reininghaus2015.pdf:application/pdf}
}

@book{harvey_understanding_2012,
	title = {Understanding high-dimensional data using Reeb graphs},
	publisher = {The Ohio State University},
	author = {Harvey, William John},
	date = {2012},
	file = {osu1342614959.pdf:/home/dimitri/Zotero/storage/M4SPU65W/osu1342614959.pdf:application/pdf}
}

@inproceedings{li_persistence-based_2014,
	title = {Persistence-Based Structural Recognition},
	isbn = {978-1-4799-5118-5},
	url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6909654},
	doi = {10.1109/CVPR.2014.257},
	pages = {2003--2010},
	publisher = {{IEEE}},
	author = {Li, Chunyuan and Ovsjanikov, Maks and Chazal, Frederic},
	urldate = {2017-11-03},
	date = {2014-06},
	file = {li2014.pdf:/home/dimitri/Zotero/storage/9JSHF2C4/li2014.pdf:application/pdf}
}

@article{krim_discovering_2016,
	title = {Discovering the Whole by the Coarse: A topological paradigm for data analysis},
	volume = {33},
	issn = {1053-5888},
	url = {http://ieeexplore.ieee.org/document/7426571/},
	doi = {10.1109/MSP.2015.2510703},
	shorttitle = {Discovering the Whole by the Coarse},
	pages = {95--104},
	number = {2},
	journaltitle = {{IEEE} Signal Processing Magazine},
	author = {Krim, Hamid and Gentimis, Thanos and Chintakunta, Harish},
	urldate = {2017-11-03},
	date = {2016-03},
	file = {krim2016.pdf:/home/dimitri/Zotero/storage/379FA7KH/krim2016.pdf:application/pdf}
}

@collection{pascucci_topological_2011,
	location = {Berlin, Heidelberg},
	title = {Topological Methods in Data Analysis and Visualization},
	isbn = {978-3-642-15013-5 978-3-642-15014-2},
	url = {http://link.springer.com/10.1007/978-3-642-15014-2},
	series = {Mathematics and Visualization},
	publisher = {Springer Berlin Heidelberg},
	editor = {Pascucci, Valerio and Tricoche, Xavier and Hagen, Hans and Tierny, Julien},
	urldate = {2017-11-03},
	date = {2011},
	doi = {10.1007/978-3-642-15014-2},
	file = {(Mathematics and Visualization) Kirk E. Jordan, Lance E. Miller (auth.), Valerio Pascucci, Xavier Tricoche, Hans Hagen, Julien Tierny (eds.)-Topological Methods in Data Analysis and Visualization_ The.pdf:/home/dimitri/Zotero/storage/IMYEDN4S/(Mathematics and Visualization) Kirk E. Jordan, Lance E. Miller (auth.), Valerio Pascucci, Xavier Tricoche, Hans Hagen, Julien Tierny (eds.)-Topological Methods in Data Analysis and Visualization_ The.pdf:application/pdf}
}

@book{edelsbrunner_computational_2010,
	location = {Providence, R.I},
	title = {Computational topology: an introduction},
	isbn = {978-0-8218-4925-5},
	shorttitle = {Computational topology},
	pagetotal = {241},
	publisher = {American Mathematical Society},
	author = {Edelsbrunner, Herbert and Harer, J.},
	date = {2010},
	note = {{OCLC}: ocn427757156},
	keywords = {Algorithms, Computational complexity, Data processing, Geometry, Topology},
	file = {Herbert Edelsbrunner, John L. Harer-Computational Topology_ An Introduction-American Mathematical Society (2009).pdf:/home/dimitri/Zotero/storage/FWGR5NJ3/Herbert Edelsbrunner, John L. Harer-Computational Topology_ An Introduction-American Mathematical Society (2009).pdf:application/pdf}
}

@article{monasse_fast_2000,
	title = {Fast computation of a contrast-invariant image representation},
	volume = {9},
	issn = {10577149},
	url = {http://ieeexplore.ieee.org/document/841532/},
	doi = {10.1109/83.841532},
	pages = {860--872},
	number = {5},
	journaltitle = {{IEEE} Transactions on Image Processing},
	author = {Monasse, P. and Guichard, F.},
	urldate = {2017-11-03},
	date = {2000-05},
	file = {monasse2000.pdf:/home/dimitri/Zotero/storage/3UDY8L47/monasse2000.pdf:application/pdf}
}

@article{stolz_persistent_2017,
	title = {Persistent homology of time-dependent functional networks constructed from coupled time series},
	volume = {27},
	issn = {1054-1500},
	url = {http://aip.scitation.org/doi/full/10.1063/1.4978997},
	doi = {10.1063/1.4978997},
	abstract = {We use topological data analysis to study “functional networks” that we           construct from time-series data from both experimental and synthetic sources. We use           persistent homology with a weight rank clique filtration to gain insights into these           functional networks, and we use persistence landscapes to interpret our results.           Our first example uses time-series output from networks of coupled Kuramoto oscillators. Our second example           consists of biological data in the form of functional magnetic resonance imaging data that           were acquired from human subjects during a simple motor-learning task in which subjects           were monitored for three days during a five-day period. With these examples, we           demonstrate that (1) using persistent homology to study functional networks provides           fascinating insights into their properties and (2) the position of the features in a           filtration can sometimes play a more vital role than persistence in the interpretation of           topological features, even though conventionally the latter is used to distinguish between           signal and noise. We find that persistent homology can detect differences in           synchronization patterns in our data sets over time, giving insight both on changes in           community structure in the networks and on increased synchronization between brain regions that form loops           in a functional network during motor learning. For the motor-learning data, persistence           landscapes also reveal that on average the majority of changes in the network loops take place           on the second of the three days of the learning process.},
	pages = {047410},
	number = {4},
	journaltitle = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
	shortjournal = {Chaos},
	author = {Stolz, Bernadette J. and Harrington, Heather A. and Porter, Mason A.},
	urldate = {2018-01-18},
	date = {2017-04-01},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/A2BD6EHP/Stolz et al. - 2017 - Persistent homology of time-dependent functional n.pdf:application/pdf;sichaostimeseries-april2017-corrected-v4-4.pdf:/home/dimitri/Zotero/storage/2W4IQ5TQ/sichaostimeseries-april2017-corrected-v4-4.pdf:application/pdf}
}

@article{taylor_topological_2015,
	title = {Topological data analysis of contagion maps for examining spreading processes on networks},
	volume = {6},
	issn = {2041-1723},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4566922/},
	doi = {10.1038/ncomms8723},
	abstract = {Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth’s surface; however, in modern contagions long-range edges—for example, due to airline transportation or communication media—allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct “contagion maps” that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modeling, forecast, and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring low-dimensional structure in networks.},
	pages = {7723},
	journaltitle = {Nature communications},
	shortjournal = {Nat Commun},
	author = {Taylor, Dane and Klimm, Florian and Harrington, Heather A. and Kramár, Miroslav and Mischaikow, Konstantin and Porter, Mason A. and Mucha, Peter J.},
	urldate = {2018-01-18},
	date = {2015-07-21},
	pmid = {26194875},
	pmcid = {PMC4566922},
	file = {PubMed Central Full Text PDF:/home/dimitri/Zotero/storage/BRA55ZPK/Taylor et al. - 2015 - Topological data analysis of contagion maps for ex.pdf:application/pdf}
}

@article{stolz_topological_2016,
	title = {The Topological "Shape" of Brexit},
	url = {http://arxiv.org/abs/1610.00752},
	abstract = {Persistent homology is a method from computational algebraic topology that can be used to study the "shape" of data. We illustrate two filtrations --- the weight rank clique filtration and the Vietoris--Rips ({VR}) filtration --- that are commonly used in persistent homology, and we apply these filtrations to a pair of data sets that are both related to the 2016 European Union "Brexit" referendum in the United Kingdom. These examples consider a topical situation and give useful illustrations of the strengths and weaknesses of these methods.},
	journaltitle = {{arXiv}:1610.00752 [physics]},
	author = {Stolz, Bernadette J. and Harrington, Heather A. and Porter, Mason A.},
	urldate = {2018-01-18},
	date = {2016-09-15},
	eprinttype = {arxiv},
	eprint = {1610.00752},
	keywords = {Computer Science - Computational Geometry, Mathematics - Algebraic Topology, Physics - Physics and Society},
	file = {arXiv\:1610.00752 PDF:/home/dimitri/Zotero/storage/9MIPK9ZY/Stolz et al. - 2016 - The Topological Shape of Brexit.pdf:application/pdf;arXiv.org Snapshot:/home/dimitri/Zotero/storage/EGR5HLE4/1610.html:text/html}
}

@article{schaub_graph_2016,
	title = {Graph partitions and cluster synchronization in networks of oscillators},
	volume = {26},
	issn = {1054-1500},
	url = {http://aip.scitation.org/doi/full/10.1063/1.4961065},
	doi = {10.1063/1.4961065},
	abstract = {Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators.},
	pages = {094821},
	number = {9},
	journaltitle = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
	shortjournal = {Chaos},
	author = {Schaub, Michael T. and O'Clery, Neave and Billeh, Yazan N. and Delvenne, Jean-Charles and Lambiotte, Renaud and Barahona, Mauricio},
	urldate = {2018-02-13},
	date = {2016-08-19},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/QDQY8L8M/Schaub et al. - 2016 - Graph partitions and cluster synchronization in ne.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/JP2SXD5G/1.html:text/html}
}

@article{noulas_mining_2015,
	title = {Mining open datasets for transparency in taxi transport in metropolitan environments},
	volume = {4},
	rights = {2015 Noulas et al.},
	issn = {2193-1127},
	url = {https://epjdatascience.springeropen.com/articles/10.1140/epjds/s13688-015-0060-2},
	doi = {10.1140/epjds/s13688-015-0060-2},
	abstract = {Uber has recently been introducing novel practices in urban taxi transport. Journey prices can change dynamically in almost real time and also vary geographically from one area to another in a city, a strategy known as surge pricing. In this paper, we explore the power of the new generation of open datasets towards understanding the impact of the new disruption technologies that emerge in the area of public transport. With our primary goal being a more transparent economic landscape for urban commuters, we provide a direct price comparison between Uber and the Yellow Cab company in New York. We discover that Uber, despite its lower standard pricing rates, effectively charges higher fares on average, especially during short in length, but frequent in occurrence, taxi journeys. Building on this insight, we develop a smartphone application, {OpenStreetCab}, that offers a personalized consultation to mobile users on which taxi provider is cheaper for their journey. Almost five months after its launch, the app has attracted more than three thousand users in a single city. Their journey queries have provided additional insights on the potential savings similar technologies can have for urban commuters, with a highlight being that on average, a user in New York saves 6 U.S. Dollars per taxi journey if they pick the cheapest taxi provider. We run extensive experiments to show how Uber’s surge pricing is the driving factor of higher journey prices and therefore higher potential savings for our application’s users. Finally, motivated by the observation that Uber’s surge pricing is occurring more frequently that intuitively expected, we formulate a prediction task where the aim becomes to predict a geographic area’s tendency to surge. Using exogenous to Uber data, in particular Yellow Cab and Foursquare data, we show how it is possible to estimate customer demand within an area, and by extension surge pricing, with high accuracy.},
	pages = {23},
	number = {1},
	journaltitle = {{EPJ} Data Science},
	author = {Noulas, Anastasios and Salnikov, Vsevolod and Lambiotte, Renaud and Mascolo, Cecilia},
	urldate = {2018-02-13},
	date = {2015-12},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/N6P7THVK/Noulas et al. - 2015 - Mining open datasets for transparency in taxi tran.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/H3R7HWMH/s13688-015-0060-2.html:text/html}
}

@article{tierny_topology_2017,
	title = {The Topology {ToolKit}},
	url = {https://hal.archives-ouvertes.fr/hal-01499905/document},
	abstract = {This system paper presents the Topology {ToolKit} ({TTK}), a software platform designed for topological data analysis in scientific visualization. While topological data analysis has gained in popularity over the last two decades, it has not yet been widely adopted as a standard data analysis tool for end users or developers. {TTK} aims at addressing this problem by providing a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. {TTK} is easily accessible to end users due to a tight integration with {ParaView}. It is also easily accessible to developers through a variety of bindings (Python, {VTK}/C++) for fast prototyping or through direct, dependence-free, C++, to ease integration into pre-existing complex systems. While developing {TTK}, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by {TTK}, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to {TTK} features, while still allowing for researchers powerful and easy bindings and extensions. {TTK} is open source ({BSD} license) and its code, online documentation and video tutorials are available on {TTK}'s website (https://topology-tool-kit.github.io/).},
	journaltitle = {{IEEE} Transactions on Visualization and Computer Graphics},
	author = {Tierny, Julien and Favelier, Guillaume and Levine, Joshua and Gueunet, Charles and Michaux, Michael},
	urldate = {2018-02-15},
	date = {2017-10-01},
	langid = {english},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/TGURBQBF/Tierny et al. - 2017 - The Topology ToolKit.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/JAIQUA5K/hal-01499905v2.html:text/html}
}

@inproceedings{maria_gudhi_2014,
	title = {The Gudhi Library: Simplicial Complexes and Persistent Homology},
	isbn = {978-3-662-44198-5},
	url = {https://link.springer.com/chapter/10.1007/978-3-662-44199-2_28},
	doi = {10.1007/978-3-662-44199-2_28},
	series = {Lecture Notes in Computer Science},
	shorttitle = {The Gudhi Library},
	abstract = {We present the main algorithmic and design choices that have been made to represent complexes and compute persistent homology in the Gudhi library. The Gudhi library (Geometric Understanding in Higher Dimensions) is a generic C++ library for computational topology. Its goal is to provide robust, efficient, flexible and easy to use implementations of state-of-the-art algorithms and data structures for computational topology. We present the different components of the software, their interaction and the user interface. We justify the algorithmic and design decisions made in Gudhi and provide benchmarks for the code. The software, which has been developped by the first author, will be available soon at project.inria.fr/gudhi/software/ .},
	eventtitle = {International Congress on Mathematical Software},
	pages = {167--174},
	booktitle = {Mathematical Software – {ICMS} 2014},
	publisher = {Springer, Berlin, Heidelberg},
	author = {Maria, Clément and Boissonnat, Jean-Daniel and Glisse, Marc and Yvinec, Mariette},
	urldate = {2018-02-15},
	date = {2014-08-05},
	langid = {english},
	file = {Snapshot:/home/dimitri/Zotero/storage/3YRXLXZL/978-3-662-44199-2_28.html:text/html}
}

@online{oudot_inf556_2017,
	title = {{INF}556 -- Topological Data Analysis},
	url = {http://www.enseignement.polytechnique.fr/informatique/INF556/},
	author = {Oudot, Steve Y.},
	urldate = {2018-02-16},
	date = {2017},
	file = {INF556 -- Topological Data Analysis:/home/dimitri/Zotero/storage/TNRU945Q/INF556.html:text/html}
}

@article{salnikov_co-occurrence_2018,
	title = {Co-occurrence simplicial complexes in mathematics: identifying the holes of knowledge},
	url = {http://arxiv.org/abs/1803.04410},
	shorttitle = {Co-occurrence simplicial complexes in mathematics},
	abstract = {In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. Holes die when a subset of their concepts appear in the same article, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the dimension of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We also show that authors' conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.},
	journaltitle = {{arXiv}:1803.04410 [physics]},
	author = {Salnikov, Vsevolod and Cassese, Daniele and Lambiotte, Renaud and Jones, Nick S.},
	urldate = {2018-04-12},
	date = {2018-03-11},
	eprinttype = {arxiv},
	eprint = {1803.04410},
	keywords = {Physics - Physics and Society, Computer Science - Digital Libraries, Mathematics - History and Overview},
	file = {arXiv\:1803.04410 PDF:/home/dimitri/Zotero/storage/HVHFGEJV/Salnikov et al. - 2018 - Co-occurrence simplicial complexes in mathematics.pdf:application/pdf;arXiv.org Snapshot:/home/dimitri/Zotero/storage/NZ7QXYNU/1803.html:text/html}
}

@article{otter_roadmap_2017,
	title = {A roadmap for the computation of persistent homology},
	volume = {6},
	issn = {2193-1127},
	url = {https://link.springer.com/article/10.1140/epjds/s13688-017-0109-5},
	doi = {10.1140/epjds/s13688-017-0109-5},
	abstract = {Persistent homology ({PH}) is a method used in topological data analysis ({TDA}) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of {PH} is an open area with numerous important and fascinating challenges. The field of {PH} computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for {PH} to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of {PH}. We give a friendly introduction to {PH}, navigate the pipeline for the computation of {PH} with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of {PH}. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of {PH}. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.},
	pages = {17},
	number = {1},
	journaltitle = {{EPJ} Data Science},
	shortjournal = {{EPJ} Data Sci.},
	author = {Otter, Nina and Porter, Mason A. and Tillmann, Ulrike and Grindrod, Peter and Harrington, Heather A.},
	urldate = {2018-04-12},
	date = {2017-12-01},
	langid = {english},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/UJRUWEUA/Otter et al. - 2017 - A roadmap for the computation of persistent homolo.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/6XMV77X9/s13688-017-0109-5.html:text/html}
}

@article{zomorodian_computing_2005,
	title = {Computing Persistent Homology},
	volume = {33},
	issn = {0179-5376, 1432-0444},
	url = {https://link.springer.com/article/10.1007/s00454-004-1146-y},
	doi = {10.1007/s00454-004-1146-y},
	abstract = {We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This result generalizes and extends the previously known algorithm that was restricted to subcomplexes of S3 and Z2 coefficients. Finally, our study implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domain in any dimension.},
	pages = {249--274},
	number = {2},
	journaltitle = {Discrete \& Computational Geometry},
	shortjournal = {Discrete Comput Geom},
	author = {Zomorodian, Afra and Carlsson, Gunnar},
	urldate = {2018-04-16},
	date = {2005-02-01},
	langid = {english},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/TB6ZGPWL/Zomorodian and Carlsson - 2005 - Computing Persistent Homology.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/HQ6KTAAF/s00454-004-1146-y.html:text/html}
}

@software{reininghaus_dipha_2018,
	title = {{DIPHA} (A Distributed Persistent Homology Algorithm)},
	rights = {{LGPL}-3.0},
	url = {https://github.com/DIPHA/dipha},
	publisher = {{DIPHA}},
	author = {Reininghaus, Jan},
	urldate = {2018-04-16},
	date = {2018-04-03},
	note = {original-date: 2015-12-25T17:23:32Z},
	file = {Snapshot:/home/dimitri/Zotero/storage/VSIEADNZ/dipha.html:text/html}
}

@software{bauer_ripser:_2018,
	title = {ripser: Ripser: a lean C++ code for the computation of Vietoris–Rips persistence barcodes},
	rights = {{GPL}-3.0},
	url = {https://github.com/Ripser/ripser},
	shorttitle = {ripser},
	publisher = {Ripser},
	author = {Bauer, Ulrich},
	urldate = {2018-04-16},
	date = {2018-04-03},
	note = {original-date: 2015-10-27T21:43:59Z},
	file = {Snapshot:/home/dimitri/Zotero/storage/HZRP5QNK/ripser.html:text/html}
}

@book{zomorodian_topology_2009,
	location = {New York, {NY}, {USA}},
	title = {Topology for Computing},
	isbn = {978-0-521-13609-9},
	abstract = {Written by a computer scientist for computer scientists, this book teaches topology from a computational point of view, and shows how to solve real problems that have topological aspects involving computers. Such problems arise in many areas, such as computer graphics, robotics, structural biology, and chemistry. The author starts from the basics of topology, assuming no prior exposure to the subject, and moves rapidly up to recent advances in the area, including topological persistence and hierarchical Morse complexes. Algorithms and data structures are presented when appropriate.},
	publisher = {Cambridge University Press},
	author = {Zomorodian, Afra J.},
	date = {2009},
	file = {Zomorodian - 2009 - Topology for Computing.pdf:/home/dimitri/Zotero/storage/4JNUZVQS/Zomorodian - 2009 - Topology for Computing.pdf:application/pdf}
}

@book{jonsson_simplicial_2008,
	location = {Berlin},
	title = {Simplicial complexes of graphs},
	isbn = {978-3-540-75859-4},
	url = {https://cds.cern.ch/record/1691716},
	series = {Lecture Notes in Mathematics},
	abstract = {A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics, including commutative algebra, geometry, and knot theory. Identifying each graph with its edge set, one may view a graph complex as a simplicial complex and hence interpret it as a geometric object. This volume examines topological properties of graph complexes, focusing on homotopy type and homology. Many of the proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this volume also provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees; it is successfully applied to a large number of graph and digraph complexes.},
	publisher = {Springer},
	author = {Jonsson, Jakob},
	urldate = {2018-04-16},
	date = {2008},
	langid = {english},
	doi = {10.1007/978-3-540-75859-4, 10.1007/978-3-540-75859-4},
	file = {Jonsson - 2008 - Simplicial complexes of graphs.pdf:/home/dimitri/Zotero/storage/689R2YHC/Jonsson - 2008 - Simplicial complexes of graphs.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/7CIVG53B/1691716.html:text/html}
}

@article{horak_persistent_2009,
	title = {Persistent homology of complex networks},
	volume = {2009},
	issn = {1742-5468},
	url = {http://stacks.iop.org/1742-5468/2009/i=03/a=P03034},
	doi = {10.1088/1742-5468/2009/03/P03034},
	abstract = {Long-lived topological features are distinguished from short-lived ones (considered as topological noise) in simplicial complexes constructed from complex networks. A new topological invariant, persistent homology, is determined and presented as a parameterized version of a Betti number. Complex networks with distinct degree distributions exhibit distinct persistent topological features. Persistent topological attributes, shown to be related to the robust quality of networks, also reflect the deficiency in certain connectivity properties of networks. Random networks, networks with exponential connectivity distribution and scale-free networks were considered for homological persistency analysis.},
	pages = {P03034},
	number = {3},
	journaltitle = {Journal of Statistical Mechanics: Theory and Experiment},
	shortjournal = {J. Stat. Mech.},
	author = {Horak, Danijela and Maletić, Slobodan and Rajković, Milan},
	urldate = {2018-04-16},
	date = {2009},
	langid = {english},
	file = {IOP Full Text PDF:/home/dimitri/Zotero/storage/IT5PKTTS/Horak et al. - 2009 - Persistent homology of complex networks.pdf:application/pdf}
}

@software{morozov_dionysus:_2018,
	title = {dionysus: Library for computing persistent homology},
	url = {https://github.com/mrzv/dionysus},
	shorttitle = {dionysus},
	author = {Morozov, Dimitriy},
	urldate = {2018-04-16},
	date = {2018-04-11},
	note = {original-date: 2017-07-14T19:02:35Z},
	file = {Snapshot:/home/dimitri/Zotero/storage/BBVYF9D2/dionysus.html:text/html}
}

@article{turner_frechet_2014,
	title = {Fréchet Means for Distributions of Persistence Diagrams},
	volume = {52},
	issn = {0179-5376, 1432-0444},
	url = {https://link.springer.com/article/10.1007/s00454-014-9604-7},
	doi = {10.1007/s00454-014-9604-7},
	abstract = {Given a distribution ρ{\textbackslash}rho on persistence diagrams and observations X1,…,Xn∼{iidρX}\_\{1\},{\textbackslash}ldots ,X\_\{n\} {\textbackslash}mathop \{{\textbackslash}sim \}{\textbackslash}limits {\textasciicircum}\{iid\} {\textbackslash}rho we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1,…,{XnX}\_\{1\},{\textbackslash}ldots ,X\_\{n\}. If the underlying measure ρ{\textbackslash}rho is a combination of Dirac masses ρ=1m∑mi=1{δZi}{\textbackslash}rho = {\textbackslash}frac\{1\}\{m\} {\textbackslash}sum \_\{i=1\}{\textasciicircum}\{m\} {\textbackslash}delta \_\{Z\_\{i\}\} then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from ρ{\textbackslash}rho . We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.},
	pages = {44--70},
	number = {1},
	journaltitle = {Discrete \& Computational Geometry},
	shortjournal = {Discrete Comput Geom},
	author = {Turner, Katharine and Mileyko, Yuriy and Mukherjee, Sayan and Harer, John},
	urldate = {2018-04-20},
	date = {2014-07-01},
	langid = {english},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/WFNRGRL6/Turner et al. - 2014 - Fréchet Means for Distributions of Persistence Dia.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/TIA4XC3D/s00454-014-9604-7.html:text/html}
}

@article{munch_probabilistic_2015,
	title = {Probabilistic Fréchet means for time varying persistence diagrams},
	volume = {9},
	issn = {1935-7524},
	url = {https://projecteuclid.org/euclid.ejs/1433195858},
	doi = {10.1214/15-EJS1030},
	abstract = {Project Euclid - mathematics and statistics online},
	pages = {1173--1204},
	number = {1},
	journaltitle = {Electronic Journal of Statistics},
	author = {Munch, Elizabeth and Turner, Katharine and Bendich, Paul and Mukherjee, Sayan and Mattingly, Jonathan and Harer, John},
	urldate = {2018-04-20},
	date = {2015},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/HRY5Z3E2/Munch et al. - 2015 - Probabilistic Fréchet means for time varying persi.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/DEP25GGC/1433195858.html:text/html}
}

@article{bubenik_statistical_2015,
	title = {Statistical Topological Data Analysis using Persistence Landscapes},
	volume = {16},
	url = {http://www.jmlr.org/papers/v16/bubenik15a.html},
	pages = {77--102},
	journaltitle = {Journal of Machine Learning Research},
	author = {Bubenik, Peter},
	urldate = {2018-04-20},
	date = {2015},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/IQ3T72BZ/Bubenik - 2015 - Statistical Topological Data Analysis using Persis.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/94GGQHGV/bubenik15a.html:text/html}
}

@incollection{kwitt_statistical_2015,
	title = {Statistical Topological Data Analysis - A Kernel Perspective},
	url = {http://papers.nips.cc/paper/5887-statistical-topological-data-analysis-a-kernel-perspective.pdf},
	pages = {3070--3078},
	booktitle = {Advances in Neural Information Processing Systems 28},
	publisher = {Curran Associates, Inc.},
	author = {Kwitt, Roland and Huber, Stefan and Niethammer, Marc and Lin, Weili and Bauer, Ulrich},
	editor = {Cortes, C. and Lawrence, N. D. and Lee, D. D. and Sugiyama, M. and Garnett, R.},
	urldate = {2018-04-20},
	date = {2015},
	file = {NIPS Full Text PDF:/home/dimitri/Zotero/storage/9NRWV859/Kwitt et al. - 2015 - Statistical Topological Data Analysis - A Kernel P.pdf:application/pdf;NIPS Snapshort:/home/dimitri/Zotero/storage/G7LF48UM/5887-statistical-topological-data-analysis-a-kernel-perspective.html:text/html}
}

@article{petri_topological_2013,
	title = {Topological Strata of Weighted Complex Networks},
	volume = {8},
	url = {http://adsabs.harvard.edu/abs/2013PLoSO...866506P},
	doi = {10.1371/journal.pone.0066506},
	abstract = {The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and --more recently-- correlations between neighboring nodes. However, statistical methods quickly become
cumbersome when dealing with many-body properties and do not capture the precise mesoscopic structure of complex networks. Here we introduce a novel method, based on persistent homology, to detect particular non-local structures, akin to weighted holes within the link-weight network fabric, which are invisible to existing methods. Their
properties divide weighted networks in two broad classes: one is characterized by small hierarchically nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasilocal network properties, because of the intrinsic non-locality of homological properties, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. Moreover, this new method creates the first bridge between network theory and algebraic topology, which will allow to import the toolset of algebraic methods to complex systems.},
	pages = {e66506},
	journaltitle = {{PLoS} {ONE}},
	shortjournal = {{PLoS} {ONE}},
	author = {Petri, Giovanni and Scolamiero, Martina and Donato, Irene and Vaccarino, Francesco},
	urldate = {2018-04-20},
	date = {2013-06-01},
	file = {Topological Strata of Weighted Complex Networks.PDF:/home/dimitri/Zotero/storage/X43JU3GL/Topological Strata of Weighted Complex Networks.PDF:application/pdf}
}

@inproceedings{carlsson_zigzag_2009,
	location = {New York, {NY}, {USA}},
	title = {Zigzag Persistent Homology and Real-valued Functions},
	isbn = {978-1-60558-501-7},
	url = {http://doi.acm.org/10.1145/1542362.1542408},
	doi = {10.1145/1542362.1542408},
	series = {{SCG} '09},
	abstract = {We study the problem of computing zigzag persistence of a sequence of homology groups and study a particular sequence derived from the levelsets of a real-valued function on a topological space. The result is a local, symmetric interval descriptor of the function. Our structural results establish a connection between the zigzag pairs in this sequence and extended persistence, and in the process resolve an open question associated with the latter. Our algorithmic results not only provide a way to compute zigzag persistence for any sequence of homology groups, but combined with our structural results give a novel algorithm for computing extended persistence. This algorithm is easily parallelizable and uses (asymptotically) less memory.},
	pages = {247--256},
	booktitle = {Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry},
	publisher = {{ACM}},
	author = {Carlsson, Gunnar and de Silva, Vin and Morozov, Dmitriy},
	urldate = {2018-04-20},
	date = {2009},
	keywords = {algorithms, extended persistence, levelset zigzag, Mayer-Vietoris pyramid, zigzag persistent homology},
	file = {Carlsson et al. - 2009 - Zigzag Persistent Homology and Real-valued Functio.pdf:/home/dimitri/Zotero/storage/WNIUXA7Y/Carlsson et al. - 2009 - Zigzag Persistent Homology and Real-valued Functio.pdf:application/pdf}
}

@inproceedings{dey_computing_2014,
	location = {New York, {NY}, {USA}},
	title = {Computing Topological Persistence for Simplicial Maps},
	isbn = {978-1-4503-2594-3},
	url = {http://doi.acm.org/10.1145/2582112.2582165},
	doi = {10.1145/2582112.2582165},
	series = {{SOCG}'14},
	abstract = {Algorithms for persistent homology are well-studied where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under Z2 coefficients for a (monotone) sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis. A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses--two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally. Finally, we exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.},
	pages = {345:345--345:354},
	booktitle = {Proceedings of the Thirtieth Annual Symposium on Computational Geometry},
	publisher = {{ACM}},
	author = {Dey, Tamal K. and Fan, Fengtao and Wang, Yusu},
	urldate = {2018-04-20},
	date = {2014},
	keywords = {cohomology, homology, simplicial maps, topological data analysis, Topological persistence},
	file = {Dey et al. - 2014 - Computing Topological Persistence for Simplicial M.pdf:/home/dimitri/Zotero/storage/X6R4GKRU/Dey et al. - 2014 - Computing Topological Persistence for Simplicial M.pdf:application/pdf}
}

@article{carlsson_theory_2009,
	title = {The Theory of Multidimensional Persistence},
	volume = {42},
	issn = {0179-5376, 1432-0444},
	url = {https://link.springer.com/article/10.1007/s00454-009-9176-0},
	doi = {10.1007/s00454-009-9176-0},
	abstract = {Persistent homology captures the topology of a filtration—a one-parameter family of increasing spaces—in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence. Instead, we propose the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension.},
	pages = {71--93},
	number = {1},
	journaltitle = {Discrete \& Computational Geometry},
	shortjournal = {Discrete Comput Geom},
	author = {Carlsson, Gunnar and Zomorodian, Afra},
	urldate = {2018-04-30},
	date = {2009-07-01},
	langid = {english},
	file = {10.1.1.86.1620.pdf:/home/dimitri/Zotero/storage/4EEYB2MK/10.1.1.86.1620.pdf:application/pdf;Full Text PDF:/home/dimitri/Zotero/storage/RBN5LKT6/Carlsson and Zomorodian - 2009 - The Theory of Multidimensional Persistence.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/PT7Q4KVU/10.html:text/html}
}

@article{perea_sliding_2017,
	title = {Sliding windows and persistence},
	volume = {141},
	issn = {0001-4966},
	url = {https://asa.scitation.org/doi/abs/10.1121/1.4987655},
	doi = {10.1121/1.4987655},
	pages = {3585--3585},
	number = {5},
	journaltitle = {The Journal of the Acoustical Society of America},
	shortjournal = {The Journal of the Acoustical Society of America},
	author = {Perea, Jose and Traile, Chris},
	urldate = {2018-05-02},
	date = {2017-05-01},
	file = {Snapshot:/home/dimitri/Zotero/storage/NAPFXSEL/1.html:text/html}
}

@article{perea_sliding_2015,
	title = {Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis},
	volume = {15},
	issn = {1615-3375, 1615-3383},
	url = {https://link.springer.com/article/10.1007/s10208-014-9206-z},
	doi = {10.1007/s10208-014-9206-z},
	shorttitle = {Sliding Windows and Persistence},
	abstract = {We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method {SW}1PerS, which stands for Sliding Windows and 1-Dimensional Persistence Scoring.},
	pages = {799--838},
	number = {3},
	journaltitle = {Foundations of Computational Mathematics},
	shortjournal = {Found Comput Math},
	author = {Perea, Jose A. and Harer, John},
	urldate = {2018-05-02},
	date = {2015-06-01},
	langid = {english},
	file = {Perea and Harer - 2015 - Sliding Windows and Persistence An Application of.pdf:/home/dimitri/Zotero/storage/2PSQQ8F4/Perea and Harer - 2015 - Sliding Windows and Persistence An Application of.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/UDRR3GRW/s10208-014-9206-z.html:text/html}
}

@article{perea_sw1pers:_2015,
	title = {{SW}1PerS: Sliding windows and 1-persistence scoring; discovering periodicity in gene expression time series data},
	volume = {16},
	issn = {1471-2105},
	url = {https://doi.org/10.1186/s12859-015-0645-6},
	doi = {10.1186/s12859-015-0645-6},
	shorttitle = {{SW}1PerS},
	abstract = {Identifying periodically expressed genes across different processes (e.g. the cell and metabolic cycles, circadian rhythms, etc) is a central problem in computational biology. Biological time series may contain (multiple) unknown signal shapes of systemic relevance, imperfections like noise, damping, and trending, or limited sampling density. While there exist methods for detecting periodicity, their design biases (e.g. toward a specific signal shape) can limit their applicability in one or more of these situations.},
	pages = {257},
	journaltitle = {{BMC} Bioinformatics},
	shortjournal = {{BMC} Bioinformatics},
	author = {Perea, Jose A. and Deckard, Anastasia and Haase, Steve B. and Harer, John},
	urldate = {2018-05-02},
	date = {2015-08-16},
	keywords = {Gene expression, Periodicity, Persistent homology, Sliding windows, Time series},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/YH7DK289/Perea et al. - 2015 - SW1PerS Sliding windows and 1-persistence scoring.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/2N8XI4AJ/s12859-015-0645-6.html:text/html}
}

@inproceedings{seversky_time-series_2016,
	title = {On Time-Series Topological Data Analysis: New Data and Opportunities},
	doi = {10.1109/CVPRW.2016.131},
	shorttitle = {On Time-Series Topological Data Analysis},
	abstract = {This work introduces a new dataset and framework for the exploration of topological data analysis ({TDA}) techniques applied to time-series data. We examine the end-toend {TDA} processing pipeline for persistent homology applied to time-delay embeddings of time series - embeddings that capture the underlying system dynamics from which time series data is acquired. In particular, we consider stability with respect to time series length, the approximation accuracy of sparse filtration methods, and the discriminating ability of persistence diagrams as a feature for learning. We explore these properties across a wide range of time-series datasets spanning multiple domains for single source multi-segment signals as well as multi-source single segment signals. Our analysis and dataset captures the entire {TDA} processing pipeline and includes time-delay embeddings, persistence diagrams, topological distance measures, as well as kernels for similarity learning and classification tasks for a broad set of time-series data sources. We outline the {TDA} framework and rationale behind the dataset and provide insights into the role of {TDA} for time-series analysis as well as opportunities for new work.},
	eventtitle = {2016 {IEEE} Conference on Computer Vision and Pattern Recognition Workshops ({CVPRW})},
	pages = {1014--1022},
	booktitle = {2016 {IEEE} Conference on Computer Vision and Pattern Recognition Workshops ({CVPRW})},
	author = {Seversky, L. M. and Davis, S. and Berger, M.},
	date = {2016-06},
	keywords = {Topology, topological data analysis, approximation accuracy, approximation theory, Context, data analysis, embedded systems, Kernel, learning feature, Pipelines, signal processing, single source multisegment signals, sparse filtration methods, Support vector machines, system dynamics, {TDA}, Three-dimensional displays, time series, Time series analysis, time-delay embeddings, time-series data},
	file = {IEEE Xplore Abstract Record:/home/dimitri/Zotero/storage/BINURQ6P/7789621.html:text/html;Seversky et al. - 2016 - On Time-Series Topological Data Analysis New Data.pdf:/home/dimitri/Zotero/storage/7BBZHTTF/Seversky et al. - 2016 - On Time-Series Topological Data Analysis New Data.pdf:application/pdf}
}

@article{cang_evolutionary_2018,
	title = {Evolutionary homology on coupled dynamical systems},
	url = {http://arxiv.org/abs/1802.04677},
	abstract = {Time dependence is a universal phenomenon in nature, and a variety of mathematical models in terms of dynamical systems have been developed to understand the time-dependent behavior of real-world problems. Originally constructed to analyze the topological persistence over spatial scales, persistent homology has rarely been devised for time evolution. We propose the use of a new filtration function for persistent homology which takes as input the adjacent oscillator trajectories of a dynamical system. We also regulate the dynamical system by a weighted graph Laplacian matrix derived from the network of interest, which embeds the topological connectivity of the network into the dynamical system. The resulting topological signatures, which we call evolutionary homology ({EH}) barcodes, reveal the topology-function relationship of the network and thus give rise to the quantitative analysis of nodal properties. The proposed {EH} is applied to protein residue networks for protein thermal fluctuation analysis, rendering the most accurate B-factor prediction of a set of 364 proteins. This work extends the utility of dynamical systems to the quantitative modeling and analysis of realistic physical systems.},
	journaltitle = {{arXiv}:1802.04677 [math, q-bio]},
	author = {Cang, Zixuan and Munch, Elizabeth and Wei, Guo-Wei},
	urldate = {2018-05-02},
	date = {2018-02-13},
	eprinttype = {arxiv},
	eprint = {1802.04677},
	keywords = {Mathematics - Algebraic Topology, Mathematics - Dynamical Systems, Quantitative Biology - Quantitative Methods},
	file = {arXiv\:1802.04677 PDF:/home/dimitri/Zotero/storage/6RNFZZ93/Cang et al. - 2018 - Evolutionary homology on coupled dynamical systems.pdf:application/pdf;arXiv.org Snapshot:/home/dimitri/Zotero/storage/984CQS7D/1802.html:text/html}
}

@article{umeda_time_2017,
	title = {Time Series Classification via Topological Data Analysis},
	volume = {32},
	issn = {1346-0714},
	url = {http://adsabs.harvard.edu/abs/2017TJSAI..32G..72U},
	doi = {10.1527/tjsai.D-G72},
	abstract = {Not Available},
	journaltitle = {Transactions of the Japanese Society for Artificial Intelligence},
	shortjournal = {Transactions of the Japanese Society for Artificial Intelligence},
	author = {Umeda, Yuhei},
	urldate = {2018-05-02},
	date = {2017},
	file = {Umeda - 2017 - Time Series Classification via Topological Data An.pdf:/home/dimitri/Zotero/storage/YK5UQZ4D/Umeda - 2017 - Time Series Classification via Topological Data An.pdf:application/pdf}
}

@article{kusano_kernel_2017,
	title = {Kernel method for persistence diagrams via kernel embedding and weight factor},
	url = {http://arxiv.org/abs/1706.03472},
	abstract = {Topological data analysis is an emerging mathematical concept for characterizing shapes in multi-scale data. In this field, persistence diagrams are widely used as a descriptor of the input data, and can distinguish robust and noisy topological properties. Nowadays, it is highly desired to develop a statistical framework on persistence diagrams to deal with practical data. This paper proposes a kernel method on persistence diagrams. A theoretical contribution of our method is that the proposed kernel allows one to control the effect of persistence, and, if necessary, noisy topological properties can be discounted in data analysis. Furthermore, the method provides a fast approximation technique. The method is applied into several problems including practical data in physics, and the results show the advantage compared to the existing kernel method on persistence diagrams.},
	journaltitle = {{arXiv}:1706.03472 [physics, stat]},
	author = {Kusano, Genki and Fukumizu, Kenji and Hiraoka, Yasuaki},
	urldate = {2018-06-12},
	date = {2017-06-12},
	eprinttype = {arxiv},
	eprint = {1706.03472},
	keywords = {Statistics - Machine Learning, Mathematics - Algebraic Topology, Physics - Data Analysis, Statistics and Probability},
	file = {arXiv\:1706.03472 PDF:/home/dimitri/Zotero/storage/ISLEIEE9/Kusano et al. - 2017 - Kernel method for persistence diagrams via kernel .pdf:application/pdf;arXiv.org Snapshot:/home/dimitri/Zotero/storage/6FIXWZVF/1706.html:text/html}
}

@article{adams_persistence_2017,
	title = {Persistence Images: A Stable Vector Representation of Persistent Homology},
	volume = {18},
	url = {http://jmlr.org/papers/v18/16-337.html},
	shorttitle = {Persistence Images},
	abstract = {Many data sets can be viewed as a noisy sampling of an
underlying space, and tools from topological data analysis can
characterize this structure for the purpose of knowledge
discovery. One such tool is persistent homology, which provides
a multiscale description of the homological features within a
data set. A useful representation of this homological
information is a persistence diagram ({PD}). Efforts have
been made to map {PDs} into spaces with additional structure
valuable to machine learning tasks. We convert a {PD} to a finite-
dimensional vector representation which we call a
persistence image ({PI}), and prove the stability of this
transformation with respect to small perturbations in the
inputs. The discriminatory power of {PIs} is compared against
existing methods, showing significant performance gains. We
explore the use of {PIs} with vector-based machine learning tools,
such as linear sparse support vector machines, which identify
features containing discriminating topological information.
Finally, high accuracy inference of parameter values from the
dynamic output of a discrete dynamical system (the linked
twist map) and a partial differential equation (the
anisotropic Kuramoto-Sivashinsky equation) provide a
novel application of the discriminatory power of {PIs}.},
	pages = {1--35},
	number = {8},
	journaltitle = {Journal of Machine Learning Research},
	author = {Adams, Henry and Emerson, Tegan and Kirby, Michael and Neville, Rachel and Peterson, Chris and Shipman, Patrick and Chepushtanova, Sofya and Hanson, Eric and Motta, Francis and Ziegelmeier, Lori},
	urldate = {2018-06-12},
	date = {2017},
	file = {Fulltext PDF:/home/dimitri/Zotero/storage/EUWNMLQF/Adams et al. - 2017 - Persistence Images A Stable Vector Representation.pdf:application/pdf}
}

@article{kalisnik_tropical_2018,
	title = {Tropical Coordinates on the Space of Persistence Barcodes},
	issn = {1615-3375, 1615-3383},
	url = {https://link.springer.com/article/10.1007/s10208-018-9379-y},
	doi = {10.1007/s10208-018-9379-y},
	abstract = {The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e., a collection of intervals, to a finite metric space. Because of the nature of the invariant, barcodes are not well adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes, and these outputs can then be used as input to standard algorithms. It is the purpose of this paper to identify tropical coordinates on the space of barcodes and prove that they are stable with respect to the bottleneck distance and Wasserstein distances.},
	pages = {1--29},
	journaltitle = {Foundations of Computational Mathematics},
	shortjournal = {Found Comput Math},
	author = {Kališnik, Sara},
	urldate = {2018-06-13},
	date = {2018-01-30},
	langid = {english},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/VIP5PCKK/Kališnik - 2018 - Tropical Coordinates on the Space of Persistence B.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/U2VKTXMW/10.html:text/html}
}

@article{le_persistence_2018,
	title = {Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams},
	url = {http://arxiv.org/abs/1802.03569},
	shorttitle = {Persistence Fisher Kernel},
	abstract = {Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data such as shapes, linked twist maps, and material data. Among them, {\textbackslash}textit\{persistent homology\} is a well-known tool to extract robust topological features, and outputs as {\textbackslash}textit\{persistence diagrams\} ({PDs}). However, {PDs} are point multi-sets which can not be used in machine learning algorithms for vector data. To deal with it, an emerged approach is to use kernel methods, and an appropriate geometry for {PDs} is an important factor to measure the similarity of {PDs}. A popular geometry for {PDs} is the {\textbackslash}textit\{Wasserstein metric\}. However, Wasserstein distance is not {\textbackslash}textit\{negative definite\}. Thus, it is limited to build positive definite kernels upon the Wasserstein distance {\textbackslash}textit\{without approximation\}. In this work, we rely upon the alternative {\textbackslash}textit\{Fisher information geometry\} to propose a positive definite kernel for {PDs} {\textbackslash}textit\{without approximation\}, namely the Persistence Fisher ({PF}) kernel. Then, we analyze eigensystem of the integral operator induced by the proposed kernel for kernel machines. Based on that, we derive generalization error bounds via covering numbers and Rademacher averages for kernel machines with the {PF} kernel. Additionally, we show some nice properties such as stability and infinite divisibility for the proposed kernel. Furthermore, we also propose a linear time complexity over the number of points in {PDs} for an approximation of our proposed kernel with a bounded error. Throughout experiments with many different tasks on various benchmark datasets, we illustrate that the {PF} kernel compares favorably with other baseline kernels for {PDs}.},
	journaltitle = {{arXiv}:1802.03569 [cs, math, stat]},
	author = {Le, Tam and Yamada, Makoto},
	urldate = {2018-06-18},
	date = {2018-02-10},
	eprinttype = {arxiv},
	eprint = {1802.03569},
	keywords = {Computer Science - Learning, Statistics - Machine Learning, Mathematics - Algebraic Topology},
	file = {arXiv\:1802.03569 PDF:/home/dimitri/Zotero/storage/RYBJTQ9F/Le and Yamada - 2018 - Persistence Fisher Kernel A Riemannian Manifold K.pdf:application/pdf;arXiv.org Snapshot:/home/dimitri/Zotero/storage/4P6MP6RX/1802.html:text/html}
}

@inproceedings{reininghaus_stable_2015-1,
	title = {A stable multi-scale kernel for topological machine learning},
	doi = {10.1109/CVPR.2015.7299106},
	abstract = {Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel {SVMs} or kernel {PCA}. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.},
	eventtitle = {2015 {IEEE} Conference on Computer Vision and Pattern Recognition ({CVPR})},
	pages = {4741--4748},
	booktitle = {2015 {IEEE} Conference on Computer Vision and Pattern Recognition ({CVPR})},
	author = {Reininghaus, J. and Huber, S. and Bauer, U. and Kwitt, R.},
	date = {2015-06},
	keywords = {1-Wasserstein distance, 3D shape classification, 3D shape retrieval, image classification, image retrieval, image texture, learning (artificial intelligence), multiscale kernel, persistence diagrams, shape recognition, texture recognition, topological machine learning, Yttrium},
	file = {IEEE Xplore Abstract Record:/home/dimitri/Zotero/storage/R29X3VFL/7299106.html:text/html;Reininghaus et al. - 2015 - A stable multi-scale kernel for topological machin.pdf:/home/dimitri/Zotero/storage/K3FZI79D/Reininghaus et al. - 2015 - A stable multi-scale kernel for topological machin.pdf:application/pdf}
}

@inproceedings{carriere_sliced_2017,
	title = {Sliced Wasserstein Kernel for Persistence Diagrams},
	url = {http://proceedings.mlr.press/v70/carriere17a.html},
	abstract = {Persistence diagrams ({PDs}) play a key role in topological data analysis ({TDA}), in which they are routinely used to describe succinctly complex topological properties of complicated shapes. {PDs} enjo...},
	eventtitle = {International Conference on Machine Learning},
	pages = {664--673},
	booktitle = {International Conference on Machine Learning},
	author = {Carrière, Mathieu and Cuturi, Marco and Oudot, Steve},
	urldate = {2018-06-20},
	date = {2017-07-17},
	langid = {english},
	file = {arXiv\:1706.03358 PDF:/home/dimitri/Zotero/storage/NWMEA95P/Carrière et al. - 2017 - Sliced Wasserstein Kernel for Persistence Diagrams.pdf:application/pdf;Full Text PDF:/home/dimitri/Zotero/storage/7FZZJDKP/Carrière et al. - 2017 - Sliced Wasserstein Kernel for Persistence Diagrams.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/VDXI2J8D/carriere17a.html:text/html}
}

@book{adler_persistent_2010,
	title = {Persistent homology for random fields and complexes},
	isbn = {978-0-940600-79-9},
	url = {https://projecteuclid.org/euclid.imsc/1288099016},
	abstract = {Project Euclid - mathematics and statistics online},
	publisher = {Institute of Mathematical Statistics},
	author = {Adler, Robert J. and Bobrowski, Omer and Borman, Matthew S. and Subag, Eliran and Weinberger, Shmuel},
	urldate = {2018-07-30},
	date = {2010},
	doi = {10.1214/10-IMSCOLL609},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/N8UHEK9G/Adler et al. - 2010 - Persistent homology for random fields and complexe.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/29PXN97Q/1288099016.html:text/html}
}

@article{cohen-steiner_stability_2007,
	title = {Stability of Persistence Diagrams},
	volume = {37},
	issn = {0179-5376, 1432-0444},
	url = {https://link.springer.com/article/10.1007/s00454-006-1276-5},
	doi = {10.1007/s00454-006-1276-5},
	abstract = {The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes.},
	pages = {103--120},
	number = {1},
	journaltitle = {Discrete \& Computational Geometry},
	shortjournal = {Discrete Comput Geom},
	author = {Cohen-Steiner, David and Edelsbrunner, Herbert and Harer, John},
	urldate = {2018-07-31},
	date = {2007-01-01},
	langid = {english},
	file = {Full Text PDF:/home/dimitri/Zotero/storage/4WEUZ4B5/Cohen-Steiner et al. - 2007 - Stability of Persistence Diagrams.pdf:application/pdf;Snapshot:/home/dimitri/Zotero/storage/5P323WWZ/s00454-006-1276-5.html:text/html}
}

@article{chazal_persistence_2014,
	title = {Persistence stability for geometric complexes},
	volume = {173},
	issn = {0046-5755, 1572-9168},
	url = {https://link.springer.com/article/10.1007/s10711-013-9937-z},
	doi = {10.1007/s10711-013-9937-z},
	abstract = {In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Čech complexes built on top of compact spaces.},
	pages = {193--214},
	number = {1},
	journaltitle = {Geometriae Dedicata},
	shortjournal = {Geom Dedicata},
	author = {Chazal, Frédéric and Silva, Vin de and Oudot, Steve},
	urldate = {2018-07-31},
	date = {2014-12-01},
	langid = {english},
	file = {Snapshot:/home/dimitri/Zotero/storage/7EESRFL3/s10711-013-9937-z.html:text/html}
}

@inproceedings{zomorodian_tidy_2010,
	location = {New York, {NY}, {USA}},
	title = {The Tidy Set: A Minimal Simplicial Set for Computing Homology of Clique Complexes},
	isbn = {978-1-4503-0016-2},
	url = {http://doi.acm.org/10.1145/1810959.1811004},
	doi = {10.1145/1810959.1811004},
	series = {{SoCG} '10},
	shorttitle = {The Tidy Set},
	abstract = {We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the Vietoris-Rips complex and the weak witness complex, methods that are popular in topological data analysis. The key feature of our approach is that it skips constructing the clique complex. We give algorithms for constructing tidy sets, implement them, and present experiments. Our preliminary results show that tidy sets are orders of magnitude smaller than clique complexes, giving us a homology engine with small memory requirements.},
	pages = {257--266},
	booktitle = {Proceedings of the Twenty-sixth Annual Symposium on Computational Geometry},
	publisher = {{ACM}},
	author = {Zomorodian, Afra},
	urldate = {2018-07-31},
	date = {2010},
	keywords = {computational topology, simplicial set, vietoris-rips complex, witness complex}
}

@inproceedings{zeppelzauer_topological_2016,
	title = {Topological Descriptors for 3D Surface Analysis},
	isbn = {978-3-319-39440-4 978-3-319-39441-1},
	url = {https://link.springer.com/chapter/10.1007/978-3-319-39441-1_8},
	doi = {10.1007/978-3-319-39441-1_8},
	series = {Lecture Notes in Computer Science},
	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.},
	eventtitle = {International Workshop on Computational Topology in Image Context},
	pages = {77--87},
	booktitle = {Computational Topology in Image Context},
	publisher = {Springer, Cham},
	author = {Zeppelzauer, Matthias and Zieliński, Bartosz and Juda, Mateusz and Seidl, Markus},
	urldate = {2018-08-16},
	date = {2016-06-15},
	langid = {english},
	file = {Snapshot:/home/dimitri/Zotero/storage/JH8QTE5R/978-3-319-39441-1_8.html:text/html}
}

@article{muandet_kernel_2017,
	title = {Kernel Mean Embedding of Distributions: A Review and Beyond},
	volume = {10},
	issn = {1935-8237, 1935-8245},
	url = {https://www.nowpublishers.com/article/Details/MAL-060},
	doi = {10.1561/2200000060},
	shorttitle = {Kernel Mean Embedding of Distributions},
	abstract = {Kernel Mean Embedding of Distributions: A Review and Beyond},
	pages = {1--141},
	number = {1},
	journaltitle = {Foundations and Trends® in Machine Learning},
	shortjournal = {{MAL}},
	author = {Muandet, Krikamol and Fukumizu, Kenji and Sriperumbudur, Bharath and Schölkopf, Bernhard},
	urldate = {2018-08-30},
	date = {2017-06-28},
	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}
}

@book{berlinet_reproducing_2011,
	title = {Reproducing Kernel Hilbert Spaces in Probability and Statistics},
	isbn = {978-1-4419-9096-9},
	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}},
	pagetotal = {369},
	publisher = {Springer Science \& Business Media},
	author = {Berlinet, Alain and Thomas-Agnan, Christine},
	date = {2011-06-28},
	langid = {english},
	note = {Google-Books-{ID}: {bX}3TBwAAQBAJ},
	keywords = {Business \& Economics / Economics / General, Business \& Economics / Economics / Theory, Business \& Economics / General, Business \& Economics / Statistics, Mathematics / Probability \& Statistics / General}
}

@thesis{price-wright_topological_2015,
	title = {A Topological Approach to Temporal Networks},
	abstract = {“Temporal networks” are a mathematical tool to represent systems that change
over time. Research on temporal networks is very active, and limited theoreti-
cal work has been done to study them. One approach to is to construct a series
of static subgraphs called snapshots. Existing techniques attempt to find the
temporal structure of a network to inform its partitioning into snapshots. An
important goal of such methods is to uncover meaningful temporal structure
that corresponds to actual features of the underlying system.
We investigate existing methods used to partition temporal networks based
on di↵erent ways of identifying temporal structure. Such methods have never
previously been compared directly to each other, so we examine and evaluate
their performance side-by-side on a suite of random-graph ensembles. We
show that without prior knowledge about a network’s temporal structure,
these existing methods have limitations producing meaningful partitions.
To tackle the problem of finding temporal structure in a network, we ap-
ply methods from computational topology. Such methods have begun to be
employed in the study of static networks and provide a summary of global
features in data sets. We use them here to track the topology of a network
over time and distinguish important temporal features from trivial ones. We
define two types of topological spaces derived from temporal networks and use
persistent homology to generate a temporal profile for a network. We then
present di↵erent ways to use this to understand a network’s temporal struc-
ture with limited prior knowledge. We show that the methods we apply from
computational topology can distinguish temporal distributions and provide a
high-level summary of temporal structure. These combined can be used to
inform a meaningful network partitioning and a deeper understanding of a
temporal network itself.},
	institution = {University of Oxford},
	type = {{MSc} dissertation in Mathematics and Foundations of Computer Science},
	author = {Price-Wright, Erin},
	date = {2015},
	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}
}

@inproceedings{edelsbrunner_topological_2000,
	title = {Topological persistence and simplification},
	doi = {10.1109/SFCS.2000.892133},
	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.},
	eventtitle = {Proceedings 41st Annual Symposium on Foundations of Computer Science},
	pages = {454--463},
	booktitle = {Proceedings 41st Annual Symposium on Foundations of Computer Science},
	author = {Edelsbrunner, H. and Letscher, D. and Zomorodian, A.},
	date = {2000-11},
	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},
	file = {IEEE Xplore Abstract Record:/home/dimitri/Zotero/storage/5LPIWG5Z/892133.html:text/html}
}

@article{morozov_persistence_2005,
	title = {Persistence algorithm takes cubic time in worst case},
	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},
	journaltitle = {{BioGeometry} News, Dept. Comput. Sci., Duke Univ},
	author = {Morozov, Dmitriy},
	date = {2005},
	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}
}

@article{de_silva_persistent_2011,
	title = {Persistent Cohomology and Circular Coordinates},
	volume = {45},
	issn = {1432-0444},
	url = {https://doi.org/10.1007/s00454-011-9344-x},
	doi = {10.1007/s00454-011-9344-x},
	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.},
	pages = {737--759},
	number = {4},
	journaltitle = {Discrete \& Computational Geometry},
	shortjournal = {Discrete Comput Geom},
	author = {de Silva, Vin and Morozov, Dmitriy and Vejdemo-Johansson, Mikael},
	urldate = {2018-09-05},
	date = {2011-06-01},
	langid = {english},
	keywords = {Persistent homology, Computational topology, Dimensionality reduction, Persistent cohomology},
	file = {Springer Full Text PDF:/home/dimitri/Zotero/storage/EX9L3F7F/de Silva et al. - 2011 - Persistent Cohomology and Circular Coordinates.pdf:application/pdf}
}

@article{de_silva_dualities_2011,
	title = {Dualities in persistent (co)homology},
	volume = {27},
	issn = {0266-5611, 1361-6420},
	url = {http://stacks.iop.org/0266-5611/27/i=12/a=124003?key=crossref.1f4b24ef80c9b1fc789ecdc6221097de},
	doi = {10.1088/0266-5611/27/12/124003},
	pages = {124003},
	number = {12},
	journaltitle = {Inverse Problems},
	author = {de Silva, Vin and Morozov, Dmitriy and Vejdemo-Johansson, Mikael},
	urldate = {2018-09-05},
	date = {2011-12-01}
}

@incollection{mcgeoch_distributed_2014,
	location = {Philadelphia, {PA}},
	title = {Distributed Computation of Persistent Homology},
	isbn = {978-1-61197-319-8},
	url = {http://epubs.siam.org/doi/abs/10.1137/1.9781611973198.4},
	pages = {31--38},
	booktitle = {2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments ({ALENEX})},
	publisher = {Society for Industrial and Applied Mathematics},
	author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan},
	editor = {{McGeoch}, Catherine C. and Meyer, Ulrich},
	urldate = {2018-09-05},
	date = {2014-05},
	langid = {english},
	doi = {10.1137/1.9781611973198.4}
}

@article{carlsson_zigzag_2008,
	title = {Zigzag Persistence},
	url = {http://arxiv.org/abs/0812.0197},
	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.},
	journaltitle = {{arXiv}:0812.0197 [cs]},
	author = {Carlsson, Gunnar and de Silva, Vin},
	urldate = {2018-09-08},
	date = {2008-11-30},
	eprinttype = {arxiv},
	eprint = {0812.0197},
	keywords = {Computer Science - Computational Geometry, I.3.5},
	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}
}

@article{maria_computing_2016,
	title = {Computing Zigzag Persistent Cohomology},
	url = {http://arxiv.org/abs/1608.06039},
	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.},
	journaltitle = {{arXiv}:1608.06039 [cs]},
	author = {Maria, Clément and Oudot, Steve},
	urldate = {2018-09-08},
	date = {2016-08-21},
	eprinttype = {arxiv},
	eprint = {1608.06039},
	keywords = {Computer Science - Computational Geometry},
	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}
}