| 1. |
- Pinto Basto de Carvalho, Joao Frederico, 1988-, et al.
(författare)
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Long-term Prediction of Motion Trajectories Using Path Homology Clusters
- 2019
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Ingår i: 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). - : Institute of Electrical and Electronics Engineers (IEEE).
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Konferensbidrag (refereegranskat)abstract
- In order for robots to share their workspace with people, they need to reason about human motion efficiently. In this work we leverage large datasets of paths in order to infer local models that are able to perform long-term predictions of human motion. Further, since our method is based on simple dynamics, it is conceptually simple to understand and allows one to interpret the predictions produced, as well as to extract a cost function that can be used for planning. The main difference between our method and similar systems, is that we employ a map of the space and translate the motion of groups of paths into vector fields on that map. We test our method on synthetic data and show its performance on the Edinburgh forum pedestrian long-term tracking dataset [1] where we were able to outperform a Gaussian Mixture Model tasked with extracting dynamics from the paths.
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| 2. |
- Vejdemo-Johansson, Mikael, 1980-, et al.
(författare)
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Comparing Distributions of Color Words : Pitfalls and Metric Choices
- 2014
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Ingår i: PLOS ONE. - : Public Library of Science. - 1932-6203. ; 9:2
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Adams, Henry, et al.
(författare)
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javaPlex : A Research Software Package for Persistent (Co) Homology
- 2014
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 4. |
- Berwald, J. J., et al.
(författare)
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Automatic recognition and tagging of topologically different regimes in dynamical systems
- 2014
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Ingår i: Discontinuity, Nonlinearity, and Complexity. - : L and H Scientific Publishing, LLC. - 2164-6376. ; 3:4, s. 413-426
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Tidskriftsartikel (refereegranskat)abstract
- Complex systems are commonly modeled using nonlinear dynamical systems. These models are often high-dimensional and chaotic. An important goal in studying physical systems through the lens of mathematical models is to determine when the system undergoes changes in qualitative behavior. A detailed description of the dynamics can be difficult or impossible to obtain for high-dimensional and chaotic systems. Therefore, a more sensible goal is to recognize and mark transitions of a system between qualitatively different regimes of behavior. In practice, one is interested in developing techniques for detection of such transitions from sparse observations, possibly contaminated by noise. In this paper we develop a framework to accurately tag different regimes of complex systems based on topological features. In particular, our framework works with a high degree of success in picking out a cyclically orbiting regime from a stationary equilibrium regime in high-dimensional stochastic dynamical systems.
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| 5. |
- Carvalho, Joao Frederico, 1988-, et al.
(författare)
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An algorithm for calculating top-dimensional bounding chains
- 2018
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Ingår i: PEERJ COMPUTER SCIENCE. - : PEERJ INC. - 2376-5992.
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Tidskriftsartikel (refereegranskat)abstract
- We describe the Coefficient-Flow algorithm for calculating the bounding chain of an (n-1)-boundary on an n-manifold-like simplicial complex S. We prove its correctness and show that it has a computational time complexity of O(vertical bar S(n-1)vertical bar) (where S(n-1) is the set of (n-1)-faces of S). We estimate the big-O coefficient which depends on the dimension of S and the implementation. We present an implementation, experimentally evaluate the complexity of our algorithm, and compare its performance with that of solving the underlying linear system.
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| 6. |
- Carvalho, Joao Frederico, et al.
(författare)
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Path Clustering with Homology Area
- 2018
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Ingår i: 2018 IEEE INTERNATIONAL CONFERENCE ON ROBOTICS AND AUTOMATION (ICRA). - : IEEE Computer Society. - 9781538630815 ; , s. 7346-7353
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 7. |
- de Silva, Vin, et al.
(författare)
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Dualities in persistent (co) homology
- 2011
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Ingår i: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; 27:12
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 8. |
- de Silva, Vin, et al.
(författare)
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Persistent cohomology and circular coordinates
- 2009
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Ingår i: Proceedings of the 25th annual symposium on Computational geometry. - New York, NY, USA : Association for Computing Machinery (ACM). ; , s. 227-236
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Konferensbidrag (refereegranskat)abstract
- Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle, for example, is one-dimensional but its faithful representation requires two real coordinates. In this work, we present a strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise NLDR analysis of a broader range of realistic data sets.
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| 9. |
- De Silva, Vin, et al.
(författare)
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Persistent cohomology and circular coordinates
- 2011
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Ingår i: Discrete & Computational Geometry. - : Springer Publishing Company. - 0179-5376 .- 1432-0444. ; 45, s. 737-759
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Tidskriftsartikel (refereegranskat)abstract
- Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE, and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle, for example, is one-dimensional, but its faithful representation requires two real coordinates. In this work, we present a strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise NLDR analysis of a broader range of realistic data sets.
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| 10. |
- Dotsenko, Vladimir, et al.
(författare)
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Implementing Gröbner bases for operads
- 2009
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Ingår i: Séminaires et Congrès. ; 26, s. 77-98
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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