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- The rigidity transition in random graphs

only on the combinatorics of the graph formed by the bars We show that if this graph is an Erdős Rényi random graph then there exists a sharp threshold for a giant rigid component to emerge For w h p all rigid components span one two or three vertices and when w h p there is a giant rigid component The constant is the threshold for 2 orientability discovered independently

Original URL path: http://theran.lt/papers/c006-rigiditytransition.html (2016-04-26)

Open archived version from archive - Rigid components of random graphs

2009 Full text arXiv URL We study the emergence of rigid components in an Erdős Rényi random graph using the parameterization for a fixed constant We show that for all almost surely all rigid components have size 2 3 or

Original URL path: http://theran.lt/papers/c005-randomcomps.html (2016-04-26)

Open archived version from archive - Combinatorial genericitity and minimal rigidity

related problem which sets of distances are minimal with the property that they allow for the reconstruction of up to a finite set of possibilities In the planar case the answer is known generically via the landmark Maxwell Laman Theorem from Rigidity Theory and it leads to a combinatorial answer the underlying structure of such a generic minimal collection of distances is a minimally rigid or Laman graph for which very efficient combinatorial decision algorithms exist For non generic cases the situation appears to be dramatically different with the best known algorithms relying on exponential time Gröbner base methods and some specific instances known to be NP hard Understanding what makes a point set generic emerges as an intriguing geometric question with practical algorithmic consequences Several definitions some but not all equivalent of genericity appear in the rigidity literature and they have either a measure theoretic topologic or algebraic geometric flavor Some generic point sets appear to be highly degenerate and still turn out to be generic All existing proofs of Laman s Theorem make use at some point of one or another of these geometric genericity assumptions The main result of this paper is the first purely combinatorial proof

Original URL path: http://theran.lt/papers/c004-sliders.html (2016-04-26)

Open archived version from archive - Analyzing rigidity with pebble games

many pair wise distances must be prescribed between an unknown set of points and how should they be distributed to determine only a discrete set of possible solutions These questions and related generalizations are central in a variety of applications Combinatorial rigidity shows that in two dimensions one can get the answer generically via an efficiently testable sparse graph property We present a video and a web site illustrating algorithmic

Original URL path: http://theran.lt/papers/c003-pgvideo.html (2016-04-26)

Open archived version from archive - The slider-pinning problem

07 2007 Full text URL A Laman mechanism is a flexible planar bar and joint framework with edges and exactly degrees of freedom The slider pinning problem is to eliminate all the degrees of freedom of a Laman mechanism in

Original URL path: http://theran.lt/papers/c002-slideralgos.html (2016-04-26)

Open archived version from archive - Finding and maintaining rigid components

2005 Full text URL We give the first complete analysis that the complexity of finding and maintaining rigid components of planar bar and joint frameworks and arbitrary dimensional body and bar frameworks using a family of algorithms called pebble games is To this end we introduce a new data structure problem called union pair find which maintains disjoint edge sets and supports pair find queries of whether two vertices are

Original URL path: http://theran.lt/papers/c001-components.html (2016-04-26)

Open archived version from archive - Algebraic-combinatorial methods for low-rank matrix completion with application to athletic performance prediction

prediction Authors Duncan Blythe Louis Theran and Franz J Király Preprint 1406 2864 2014 Full text arXiv This paper presents novel algorithms which exploit the intrinsic algebraic and combinatorial structure of the matrix completion task for estimating missing en tries in the general low rank setting For positive data we achieve results out performing the state of the art nuclear norm both in accuracy and computational efficiency in simulations and

Original URL path: http://theran.lt/papers/p011-running.html (2016-04-26)

Open archived version from archive - Learning with Cross-Kernels and Ideal PCA

main potential of cross kernels lies in the fact that a only one side of the matrix scales with the number of data points and b cross kernels as opposed to the usual kernel matrices can be used to certify for the data manifold Our theoretical framework which is based on a duality involving the feature space and vanishing ideals indicates that cross kernels have the potential to be used

Original URL path: http://theran.lt/papers/p010-ipca.html (2016-04-26)

Open archived version from archive