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  • 1. Adams, Henry
    et al.
    Tausz, Andrew
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP. Institut Jozef Stefan, Slovenia .
    javaPlex: A Research Software Package for Persistent (Co) Homology2014Conference paper (Refereed)
    Abstract [en]

    The computation of persistent homology has proven a fundamental component of the nascent field of topological data analysis and computational topology. We describe a new software package for topological computation, with design focus on needs of the research community. This tool, replacing previous jPlex and Plex, enables researchers to access state of the art algorithms for persistent homology, cohomology, hom complexes, filtered simplicial complexes, filtered cell complexes, witness complex constructions, and many more essential components of computational topology. We describe, herewithin, the design goals we have chosen, as well as the resulting software package, and some of its more novel capabilities.

  • 2.
    Carvalho, Joao Frederico
    et al.
    KTH, School of Electrical Engineering and Computer Science (EECS), Robotics, perception and learning, RPL. KTH, School of Electrical Engineering and Computer Science (EECS), Centres, Centre for Autonomous Systems, CAS.
    Vejdemo-Johansson, Mikael
    CUNY College of Staten Island, Mathematics Department, New York, USA.
    Kragic, Danica
    KTH, School of Electrical Engineering and Computer Science (EECS), Robotics, perception and learning, RPL. KTH, School of Electrical Engineering and Computer Science (EECS), Centres, Centre for Autonomous Systems, CAS.
    Pokorny, Florian T.
    KTH, School of Electrical Engineering and Computer Science (EECS), Robotics, perception and learning, RPL. KTH, School of Electrical Engineering and Computer Science (EECS), Centres, Centre for Autonomous Systems, CAS.
    Path Clustering with Homology Area2018In: 2018 IEEE International Conference on Robotics and Automation (ICRA), IEEE conference proceedings, 2018, p. 7346-7353Conference paper (Refereed)
    Abstract [en]

    Path clustering has found many applications in recent years. Common approaches to this problem use aggregates of the distances between points to provide a measure of dissimilarity between paths which do not satisfy the triangle inequality. Furthermore, they do not take into account the topology of the space where the paths are embedded. To tackle this, we extend previous work in path clustering with relative homology, by employing minimum homology area as a measure of distance between homologous paths in a triangulated mesh. Further, we show that the resulting distance satisfies the triangle inequality, and how we can exploit the properties of homology to reduce the amount of pairwise distance calculations necessary to cluster a set of paths. We further compare the output of our algorithm with that of DTW on a toy dataset of paths, as well as on a dataset of real-world paths.

  • 3. Chambers, E. W.
    et al.
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP. University of Minnesota, United States; Jozef Stefan Institute, Slovenia.
    Computing Minimum Area Homologies2015In: Computer graphics forum (Print), ISSN 0167-7055, E-ISSN 1467-8659, Vol. 34, no 6, p. 13-21Article in journal (Refereed)
    Abstract [en]

    Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure.

  • 4. de Silva, Vin
    et al.
    Morozov, Dmitriy
    Vejdemo-Johansson, Mikael
    St Andrews University, UK.
    Dualities in persistent (co) homology2011In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 27, no 12Article in journal (Refereed)
    Abstract [en]

    We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent information. We explain how one can use the existing algorithm for persistent homology to process any of the four modules, and relate it to a recently introduced persistent cohomology algorithm. We present experimental evidence for the practical efficiency of the latter algorithm.

  • 5. de Silva, Vin
    et al.
    Vejdemo-Johansson, Mikael
    Department of Mathematics, Stanford University, California.
    Persistent cohomology and circular coordinates2009In: Proceedings of the 25th annual symposium on Computational geometry, Association for Computing Machinery (ACM), 2009, p. 227-236Conference paper (Refereed)
    Abstract [en]

    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.

  • 6. De Silva, Vin
    et al.
    Vejdemo-Johansson, Mikael
    Stanford University, USA.
    Morozov, Dmitriy
    Persistent cohomology and circular coordinates2011In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 45, p. 737-759Article in journal (Refereed)
    Abstract [en]

    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.

  • 7. Dotsenko, Vladimir
    et al.
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    Implementing Gröbner bases for operads2009In: Séminaires et Congrès, Vol. 26, p. 77-98Article in journal (Refereed)
    Abstract [en]

    We present an implementation of the algorithm for computing Grobner bases for operads due to the rst author and A. Khoroshkin. We discuss the actual algorithms, the choices made for the implementation platform and the data representation, and strengths and weaknesses of our approach.

  • 8. Dotsenko, Vladimir
    et al.
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    Operadic Gröbner bases: an implementation2010In: Mathematical Software–ICMS 2010, Springer Berlin/Heidelberg, 2010, p. 249-252Chapter in book (Refereed)
  • 9. Hirsch, D.
    et al.
    Markström, Ingemar
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Patterson, M. L.
    Sandberg, A.
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP. Jožef Štefan Institute, Slovenia.
    More ties than we thought2015In: PeerJ, ISSN 2167-8359, E-ISSN 2167-8359, Vol. 2015, no 1, article id e2Article in journal (Refereed)
    Abstract [en]

    We extend the existing enumeration of neck tie-knots to include tie-knots with a textured front, tied with the narrow end of a tie. These tie-knots have gained popularity in recent years, based on reconstructions of a costume detail from The Matrix Reloaded, and are explicitly ruled out in the enumeration by Fink & Mao (2000). We show that the relaxed tie-knot description language that comprehensively describes these extended tie-knot classes is context free. It has a regular sub-language that covers all the knots that originally inspired the work. From the full language, we enumerate 266,682 distinct tie-knots that seem tie-able with a normal neck-tie. Out of these 266,682, we also enumerate 24,882 tie-knots that belong to the regular sub-language.

  • 10.
    Johansson, Mikael
    Friedrich-Schiller-Univesität, JGermany.
    Computation of Poincare-Betti series for monomial rings2005In: Rendiconti dell'Istituto di Matematica dell'Università di Trieste, ISSN 0049-4704, E-ISSN 2464-8728, Vol. 37, no 1-2, p. 85-94Article in journal (Refereed)
  • 11. Lum, P Y
    et al.
    Singh, G
    Lehman, A
    Ishkanov, T
    Vejdemo-Johansson, Mikael
    School of Computer Science@University of St Andrews, Scotland.
    Alagappan, M
    Carlsson, J
    Carlsson, G
    Extracting insights from the shape of complex data using topology2013In: Scientific Reports, ISSN 2045-2322, E-ISSN 2045-2322, Vol. 3Article in journal (Refereed)
    Abstract [en]

    This paper applies topological methods to study complex high dimensional data sets by extracting shapes (patterns) and obtaining insights about them. Our method combines the best features of existing standard methodologies such as principal component and cluster analyses to provide a geometric representation of complex data sets. Through this hybrid method, we often find subgroups in data sets that traditional methodologies fail to find. Our method also permits the analysis of individual data sets as well as the analysis of relationships between related data sets. We illustrate the use of our method by applying it to three very different kinds of data, namely gene expression from breast tumors, voting data from the United States House of Representatives and player performance data from the NBA, in each case finding stratifications of the data which are more refined than those produced by standard methods.

  • 12. Sexton, H.
    et al.
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    JPlex simplicial complex library2011Other (Other academic)
  • 13.
    Skraba, Primoz
    et al.
    Jozef Stefan Institute.
    de Silva, Vin
    Pomona College.
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    Topological Analysis of Recurrent Systems2012Conference paper (Refereed)
    Abstract [en]

    We propose a new framework for the experimental study of periodic, quasi- periodic and recurrent dynamical systems. These behaviors express themselves as topological features which we detect using persistent cohomology. The result- ing 1-cocycles yield circle-valued coordinates associated to the recurrent behavior. We demonstrate how to use these coordinates to perform fundamental tasks like period recovery and parameter choice for delay embeddings. 

  • 14. Tausz, Andrew
    et al.
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    Adams, Henry
    javaPlex: a research platform for persistent homology2012In: Book of Abstracts: Minisymposium on Publicly Available Geometric/Topological Software, 2012, p. 7-12Conference paper (Refereed)
  • 15.
    Vejdemo Johansson, Mikael
    et al.
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP. AI Laboratory, Jožef Stefan Institute, Ljubljana, Slovenia .
    Pokorny, Florian T.
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    Skraba, Primoz
    Kragic, Danica
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    Cohomological learning of periodic motion2015In: Applicable Algebra in Engineering, Communication and Computing, ISSN 0938-1279, E-ISSN 1432-0622, Vol. 26, no 1-2, p. 5-26Article in journal (Refereed)
    Abstract [en]

    This work develops a novel framework which can automatically detect, parameterize and interpolate periodic motion patterns obtained from a motion capture sequence. Using our framework, periodic motions such as walking and running gaits or any motion sequence with periodic structure such as cleaning, dancing etc. can be detected automatically and without manual marking of the period start and end points. Our approach constructs an intrinsic parameterization of the motion and is computationally fast. Using this parameterization, we are able generate prototypical periodic motions. Additionally, we are able to interpolate between various motions, yielding a rich class of 'mixed' periodic actions. Our approach is based on ideas from applied algebraic topology. In particular, we apply a novel persistent cohomology based method for the first time in a graphics application which enables us to recover circular coordinates of motions. We also develop a suitable notion of homotopy which can be used to interpolate between periodic motion patterns. Our framework is directly applicable to the construction of walk cycles for animating character motions with motion graphs or state machine driven animation engines and processed our examples at an average speed of 11.78 frames per second.

  • 16.
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    A partial A-infinityˆž-structure on the cohomology of C n x C m2008In: Journal of Homotopy & Related Structures (JHRS), ISSN 1512-2891, Vol. 3, no 1, p. 1-11Article in journal (Refereed)
    Abstract [en]

    Suppose k is a finite field, and n, m >= 4 multiples of the field characteristic. Then the A(infinity)-structure of the group cohomology algebras H* (C(n), k) and H* (C(m), k) are well known. We give results characterizing an A(infinity)-structure on H* (C(n) x C(m), k) including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.

  • 17.
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    Blackbox computation of A-infinityˆž-algebras2010In: Georgian Mathematical Journal, ISSN 1072-947X, E-ISSN 1572-9176, Vol. 17, no 2, p. 391-404Article in journal (Refereed)
    Abstract [en]

    Kadeishvili's proof of the minimality theorem [T. Kadeishvili, On the homology theory of fiber spaces, Russ. Math. Surv. 35:3 (1980), 231-238] induces an algorithm for the inductive computation of an A(infinity)-algebra structure on the homology of a dg-algebra.In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete A(infinity)-algebra structure after a finite amount of computational work.

  • 18.
    Vejdemo-Johansson, Mikael
    Friedrich-Schiller-Universität Jena, Germany.
    Computation of A-infinity algebras in group cohomology2008Doctoral thesis, monograph (Other academic)
  • 19.
    Vejdemo-Johansson, Mikael
    Friedrich Schiller University Jena .
    Enumerating the Saneblidze-Umble diagonal in Haskell2008In: ACM Communications in Computer Algebra, ISSN 1932-2240, Vol. 42, no 1-2, p. 20-20Article in journal (Refereed)
    Abstract [en]

    In 2004, Saneblidze and Umble gave a construction of a diagonal on the permutahedron, thus providing important tools to several disciplines using the permutahedra and associated combinatorial polyhedra. There has since been a few, mostly unpublished, attempts at writing computer programs to perform the highly combinatorial construction. The author presents an implementation of the construction in the programming language Haskell: a language which allows one to write computer code that stays very close to the mathematical expression of the construction itself. The author wishes to emphasize the perceived naturality of translating mathematics into Haskell code by giving examples from this program.

  • 20.
    Vejdemo-Johansson, Mikael
    School of Computer Science, University of St Andrews.
    GAP persistence–a computational topology package for GAP2012Conference paper (Refereed)
    Abstract [en]

    We introduce a recently built package for computing persistent homology within the computer algebra system gap. A native gap implementation of the persistent homology algorithm allows for easier access to the group-theoretic primitives in gap while simultaneously working on computational topology questions – enabling for instance the use of symmetry groups to study symmetric point clouds. The strongest innovation here is in the combination of persistent homology and point cloud algorithms with the group theoretic tools in gap. This package is different from the several existing homological algebra and simplicial topology packages already in gap: simpcomp, homalg, Hap, et c., in that the focus here is on point cloud topology, and not combinatorial complexes or homological algebra.

  • 21.
    Vejdemo-Johansson, Mikael
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP. Jozef Stefan Inst, Slovenia.
    Sketches of a platypus: persistent homology and its algebraic foundations2014In: ALGEBRAIC TOPOLOGY: APPLICATIONS AND NEW DIRECTIONS, American Mathematical Society (AMS), 2014, p. 295-320Conference paper (Refereed)
    Abstract [en]

    The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are rooted in choosing a particular algebraic foundational theory to describe persistent homology, and applying results from that theory to prove useful and important results.

    In this survey paper, we shall examine the various choices in use, and what they allow us to prove. We shall also discuss the inherent differences between the choices people use, and speculate on potential directions of research to resolve these differences.

  • 22.
    Vejdemo-Johansson, Mikael
    et al.
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    Carlsson, Gunnar
    Stanford University.
    Lum, Pek Yee
    Ayasdi Inc..
    Lehman, Alan
    Ayasdi Inc..
    Singh, Gurjeet
    Ayasdi Inc..
    Ishkhanov, Tigran
    Ayasdi Inc..
    The topology of politics: voting connectivity in the US House of Representatives2012Conference paper (Refereed)
    Abstract [en]

    Time-varying topological simplifications of the space of votes in the US House of Representatives (US HoR) display several interesting features unavailable with classical methods of machine learning. In this paper we demonstrate how a re- cently developed topological simplification method, MAPPER, can detect changes in collaboration structures within the US HoR over time.

  • 23.
    Vejdemo-Johansson, Mikael
    et al.
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    Skraba, Primoz
    Jozef Stefan Institute.
    Parallel & scalable zig-zag persistent homology2012Conference paper (Refereed)
    Abstract [en]

    By computing repeated pullbacks, we are able to compute zig-zag persistent homology in a way that easily parallelizes. In this paper, we demonstrate this algorithm together with its underlying mathematical foundation. We can parallelizeand scale the computation scheme.

  • 24.
    Vejdemo-Johansson, Mikael
    et al.
    St. Andrews University, UK.
    Vejdemo, Susanne
    Stockholm University.
    Ek, Carl-Henrik
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP.
    Comparing Distributions of Color Words: Pitfalls and Metric Choices2014In: PLoS ONE, ISSN 1932-6203, E-ISSN 1932-6203, Vol. 9, no 2Article in journal (Refereed)
    Abstract [en]

    Computational methods have started playing a significant role in semantic analysis. One particularly accessible area for developing good computational methods for linguistic semantics is in color naming, where perceptual dissimilarity measures provide a geometric setting for the analyses. This setting has been studied first by Berlin & Kay in 1969, and then later on by a large data collection effort: the World Color Survey (WCS). From the WCS, a dataset on color naming by 2 616 speakers of 110 different languages is made available for further research. In the analysis of color naming from WCS, however, the choice of analysis method is an important factor of the analysis. We demonstrate concrete problems with the choice of metrics made in recent analyses of WCS data, and offer approaches for dealing with the problems we can identify. Picking a metric for the space of color naming distributions that ignores perceptual distances between colors assumes a decorrelated system, where strong spatial correlations in fact exist. We can demonstrate that the corresponding issues are significantly improved when using Earth Mover's Distance, or Quadratic Χ-square Distance, and we can approximate these solutions with a kernel-based analysis method.

  • 25. Vejdemo-Johansson, Mikael
    et al.
    Vejdemo, Susanne
    Stockholm University, Faculty of Humanities, Department of Linguistics, General Linguistics.
    Ek, Carl-Henrik
    Comparing Distributions of Color Words: Pitfalls and Metric Choices2014In: PLoS ONE, ISSN 1932-6203, E-ISSN 1932-6203, Vol. 9, no 2, p. e89184-Article in journal (Refereed)
    Abstract [en]

    Computational methods have started playing a significant role in semantic analysis. One particularly accessible area for developing good computational methods for linguistic semantics is in color naming, where perceptual dissimilarity measures provide a geometric setting for the analyses. This setting has been studied first by Berlin & Kay in 1969, and then later on by a large data collection effort: the World Color Survey (WCS). From the WCS, a dataset on color naming by 2 616 speakers of 110 different languages is made available for further research. In the analysis of color naming from WCS, however, the choice of analysis method is an important factor of the analysis. We demonstrate concrete problems with the choice of metrics made in recent analyses of WCS data, and offer approaches for dealing with the problems we can identify. Picking a metric for the space of color naming distributions that ignores perceptual distances between colors assumes a decorrelated system, where strong spatial correlations in fact exist. We can demonstrate that the corresponding issues are significantly improved when using Earth Mover's Distance, or Quadratic x-square Distance, and we can approximate these solutions with a kernel-based analysis method.

  • 26. Wang, Bei
    et al.
    Summa, Brian
    Pascucci, Valerio
    Vejdemo-Johansson, Mikael
    Stanford University, USA.
    Branching and Circular Features in High Dimensional Data2011In: IEEE Transactions on Visualization and Computer Graphics, ISSN 1077-2626, E-ISSN 1941-0506, Vol. 17, no 12, p. 1902-1911Article in journal (Refereed)
    Abstract [en]

    Large observations and simulations in scientific research give rise to high-dimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in application domains such as engineering, computational biology, climate study, imaging and motion capture are faced with the problem of how to discover compact representations of highdimensional data while preserving their intrinsic structure. In many applications, the original data is projected onto low-dimensional space via dimensionality reduction techniques prior to modeling. One problem with this approach is that the projection step in the process can fail to preserve structure in the data that is only apparent in high dimensions. Conversely, such techniques may create structural illusions in the projection, implying structure not present in the original high-dimensional data. Our solution is to utilize topological techniques to recover important structures in high-dimensional data that contains non-trivial topology. Specifically, we are interested in high-dimensional branching structures. We construct local circle-valued coordinate functions to represent such features. Subsequently, we perform dimensionality reduction on the data while ensuring such structures are visually preserved. Additionally, we study the effects of global circular structures on visualizations. Our results reveal never-before-seen structures on real-world data sets from a variety of applications.

1 - 26 of 26
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