skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Discovering the Whole by the Coarse: A topological paradigm for data analysis

Journal Article · · IEEE Signal Processing Magazine
 [1];  [2];  [3]
  1. North Carolina State Univ., Raleigh, NC (United States). Dept. of Electrical and Computer Engineering
  2. National and Kapodistrian Univ. of Athens (Greece)
  3. Pomona College, Claremont, CA (United States). Mathematics Dept.

The increasing interest in big data applications is ushering in a large effort in seeking new, efficient, and adapted data models to reduce complexity, while preserving maximal intrinsic information. Graph-based models have recently been getting a lot of attention on account of their intuitive and direct connection to the data. The cost of these models, however, is to some extent giving up geometric insight as well as algebraic flexibility. Topology, as an intermediate analysis medium, focuses on coarse structures of an object/signal in general. It affords a formalism of transitioning from a local to a global description of an object, while providing significant information, which respects the local structure of measurements. It may also support a global visualization (e.g., data variation trends) and enhances the understanding of the underlying phenomenon. Throughout this article, measurements may be considered proximal in various ways, depending on the specific application, and the selected proximity metrics are subsequently used in the construction of a graph structure, thus highlighting various groups of neighboring data. As described later, these groups, often based on proximity criteria, are represented by n-simplices, where n + 1 corresponds to the number of points in an associated group. Collectively, they form what is (and will be throughout) referred to as a simplicial complex, a special representation of a topological space. We find the key advantage of computational topology is its inherently algebraic structure on the basic elements of the resulting graph-like structure.

Research Organization:
North Carolina State University, Raleigh, NC (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA), Office of Nonproliferation and Verification Research and Development (NA-22); Defense Threat Reduction Agency (DTRA)
Grant/Contract Number:
NA0002576
OSTI ID:
1438398
Journal Information:
IEEE Signal Processing Magazine, Vol. 33, Issue 2; ISSN 1053-5888
Publisher:
IEEECopyright Statement
Country of Publication:
United States
Language:
English

References (31)

Statistical ranking and combinatorial Hodge theory journal November 2010
A Distributed Topological Camera Network Representation for Tracking Applications journal October 2010
Dynamic Coverage Verification in Mobile Sensor Networks Via Switched Higher Order Laplacians conference June 2007
Human Gait Identification Using Persistent Homology book January 2012
Computational Topology in Text Mining book January 2012
Digital Imaging: A Unified Topological Framework journal July 2011
Persistent Homology of Delay Embeddings and its Application to Wheeze Detection journal April 2014
Target Enumeration via Euler Characteristic Integrals journal January 2009
Cohomological learning of periodic motion journal March 2015
Persistence-based segmentation of deformable shapes conference June 2010
Local persistent homology based distance between maps
  • Ahmed, Mahmuda; Fasy, Brittany Terese; Wenk, Carola
  • Proceedings of the 22nd ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems https://doi.org/10.1145/2666310.2666390
conference November 2014
Coordinate-free Coverage in Sensor Networks with Controlled Boundaries via Homology journal December 2006
Distributed merge trees journal August 2013
Topological mapping of unknown environments using an unlocalized robotic swarm conference November 2013
Simplifying the homology of networks via strong collapses conference May 2013
Real time detection of harmonic structure: A case for topological signal analysis conference May 2014
Distributed computation of coverage in sensor networks by homological methods journal April 2012
A graph, non-tree representation of the topology of a gray scale image conference February 2011
Topological Localization Via Signals of Opportunity journal May 2012
Distributed Coverage Verification in Sensor Networks Without Location Information journal August 2010
A Robust Topology-Based Algorithm for Gene Expression Profiling journal November 2012
Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival journal April 2011
Discrete texture traces: Topological representation of geometric context conference June 2012
Coverage and hole-detection in sensor networks via homology conference June 2005
Topology and data journal January 2009
Stability of Persistence Diagrams journal December 2006
Distributed Localization of Coverage Holes Using Topological Persistence journal May 2014
Finding the Homology of Submanifolds with High Confidence from Random Samples journal March 2008
Randomized incremental construction of Delaunay and Voronoi diagrams journal June 1992
Gromov-Hausdorff Stable Signatures for Shapes using Persistence journal July 2009
Topological Signal Processing book January 2014