An application-based characterization of dynamical distance geometry problems

Mucherino, Antonio; Omer, Jérémy; Hoyet, Ludovic; Giordano, Paolo Robuffo; Multon, Franck

doi:10.1007/s11590-018-1302-6

Cited by 10 publications

(12 citation statements)

References 31 publications

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“…The DGP is NP-hard [26], and has several different applications, including: (i) protein structure determination [7] (this is the application we will consider in our experiments in Section IV); (ii) sensor network localization [27]; (iii) multidimensional scaling [13]; (iv) clock synchronization [8]; (v) motion adaptation [25]; and others.…”

Section: Definition 1 Given a Simple Weighted Undirected Graph G = (Vmentioning

confidence: 99%

“…SPG can terminate because of different criteria: either when the objective function value becomes smaller than 10 −6 , or when the norm of the search direction becomes smaller than 10 −6 , or when it reaches the maximum number of allowed interactions, which is set to 10000 in our experiments. Notice that the objective function optimized by SPG is not the MDE function above (MDE is in fact not differentiable in its entire definition domain): more information about SPG and its implementation can be found in [22], [25].…”

Section: Some Initial Experimentsmentioning

confidence: 99%

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An Efficient Exhaustive Search for the Discretizable Distance Geometry Problem with Interval Data

Mucherino

Lin

2019

Proceedings of the 2019 Federated Conference on Computer Science and Information Systems

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The Distance Geometry Problem (DGP) asks whether a simple weighted undirected graph can be realized in a given space (generally Euclidean) so that a given set of distance constraints (associated to the edges of the graph) is satisfied. The Discretizable DGP (DDGP) represents a subclass of instances where the search space can be reduced to a discrete domain having the structure of a tree. In the ideal case where all distances are precise, the tree is binary and one singleton, representing one possible position for a vertex of the graph, is associated to every tree node. When the distance information is however not precise, the uncertainty on the distance values implies that a three-dimensional region of the search space needs to be assigned to some nodes of the tree. By using a recently proposed coarse-grained representation for DDGP solutions, we extend in this work the branch-and-prune (BP) algorithm so that it can efficiently perform an exhaustive search of the search domain, even when the uncertainty on the distances is important. Instead of associating singletons to nodes, we consider a pair consisting of a box and of a most-likely position for the vertex in this box. Initial estimations of the vertex positions in every box can be subsequently refined by using local optimization. The aim of this paper is twofold: (i) we propose a new simple method for the computation of the three-dimensional boxes to be associated to the nodes of the search tree; (ii) we introduce the resolution parameter ρ, with the aim of controling the similarity between pairs of solutions in the solution set. Some initial computational experiments show that our algorithm extension, differently from previously proposed variants of the BP algorithm, is actually able to terminate the enumeration of the solution set by providing solutions that differ from one another accordingly to the given resolution parameter.

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Section: Definition 1 Given a Simple Weighted Undirected Graph G = (Vmentioning

confidence: 99%

Section: Some Initial Experimentsmentioning

confidence: 99%

An Efficient Exhaustive Search for the Discretizable Distance Geometry Problem with Interval Data

Mucherino

Lin

2019

Proceedings of the 2019 Federated Conference on Computer Science and Information Systems

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“…Examples are action recognition from dynamic interjoint distance skeleton data [22] and more generally data structures for describing kinetic point sets [14]. Applications in multi-robot coordination, crowd simulations, and motion retargeting are explored in [23], [24], where the authors introduce the dynamical distance geometry problem (dynDGP). 1 Even in applications to proteins and molecules, the atoms move (for example, proteins fold) in specific ways [25].…”

Section: Introductionmentioning

confidence: 99%

Kinetic Euclidean Distance Matrices

Tabaghi

Vetterli

2020

IEEE Trans. Signal Process.

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Euclidean distance matrices (EDMs) are a major tool for localization from distances, with applications ranging from protein structure determination to global positioning and manifold learning. They are, however, static objects which serve to localize points from a snapshot of distances. If the objects move, one expects to do better by modeling the motion. In this paper, we introduce Kinetic Euclidean Distance Matrices (KEDMs)-a new kind of time-dependent distance matrices that incorporate motion. The entries of KEDMs become functions of time, the squared time-varying distances. We study two smooth trajectory models-polynomial and bandlimited trajectoriesand show that these trajectories can be reconstructed from incomplete, noisy distance observations, scattered over multiple time instants. Our main contribution is a semidefinite relaxation (SDR), inspired by SDRs for static EDMs. Similarly to the static case, the SDR is followed by a spectral factorization step; however, because spectral factorization of polynomial matrices is more challenging than for constant matrices, we propose a new factorization method that uses anchor measurements.Extensive numerical experiments show that KEDMs and the new semidefinite relaxation accurately reconstruct trajectories from noisy, incomplete distance data and that, in fact, motion improves rather than degrades localization if properly modeled. This makes KEDMs a promising tool for problems in geometry of dynamic points sets.

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“…Given a weighted undirected graph = ( × , , { , }), the dynamical Distance Geometry Problem (dynDGP) [16,20] in dimension 3 consists in finding a realization : ( , ) ∈ × −→ ( , , ) ∈ R 3 (1) of in the Euclidean space R 3 such that the following objective is minimized:…”

Section: Introductionmentioning

confidence: 99%

“…In other words, a dynDGP with = {1} is a "standard" distance geometry problem [12]. This article focuses on a particular application of the dynDGP, where the main interest is in analyzing human motions through the study of animated skeletal structures which resemble the human body [20]. Some initial works have shown that tools for distance geometry can be employed in this context [2,17], but they consider, as in the majority of the applications in distance geometry, the Euclidean distance as a metric.…”

Section: Introductionmentioning

confidence: 99%