Prime Rigid Graphs and Multidimensional Scaling with Missing Data

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In this paper we investigate the problem of embedding a number of points given certain (but typically not all) inter-pair distance measurements. This problem is relevant for multi-dimensional scaling problems with missing data, and is applicable within anchor-free sensor network node calibration and anchor-free node localization using radio or sound TOA measurements. There are also applications within chemistry for deducing molecular 3D structure given inter-atom distance measurements and within machine learning and visualization of data, where only similarity measures between sample points are provided. The problem has been studied previously within the field of rigid graph theory. Our aim is here to construct numerically stable and efficient solvers for finding all embeddings of such minimal rigid graphs. The method is based on the observation that all graphs are either irreducibly rigid, here called prime rigid graphs, or contain smaller rigid graphs. By solving the embedding problem for the prime rigid graphs and for ways of assembling such graphs to other minimal rigid graphs, we show how to (i) calculate the number of embeddings and (ii) construct numerically stable and efficient algorithms for obtaining all embeddings given inter-node measurements. The solvers are verified with experiments on simulated data.


Enheter & grupper

Ämnesklassifikation (UKÄ) – OBLIGATORISK

  • Matematik
  • Datorseende och robotik (autonoma system)
Titel på värdpublikationPattern Recognition (ICPR), 2014 22nd International Conference on
FörlagIEEE - Institute of Electrical and Electronics Engineers Inc.
Antal sidor6
StatusPublished - 2014
Peer review utfördJa
Evenemang22nd International Conference on Pattern Recognition (ICPR 2014) - Stockholm, Sverige
Varaktighet: 2014 aug 242014 aug 28
Konferensnummer: 22


ISSN (tryckt)1051-4651


Konferens22nd International Conference on Pattern Recognition (ICPR 2014)