Convex Hulls of Orbits and Orientations of a Moving Protein Domain

Longinetti, Marco; Sgheri, Luca; Sottile, Frank

doi:10.1007/s00454-008-9076-8

Cited by 8 publications

(16 citation statements)

References 15 publications

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“…Sanyal et al raise the general question of whether conv SO(n) is a spectrahedron for all n (which we answer in the affirmative), and more broadly ask for a classification of the SO(n)-orbitopes that are spectrahedra. Earlier work on orbitopes in the context of convex geometry includes the work of Barvinok and Vershik [3] who consider orbitopes of finite groups in the context of combinatorial optimization, Barvinok and Blekherman [2], who used asymptotic volume computations to show that there are many more non-negative polynomials than sums of squares (among other things), and Longinetti et al [21] who studied SO(3)-orbitopes with a view to applications in protein structure determination. More recently Sinn [32] has studied in detail the algebraic boundary of four-dimensional SO(2)orbitopes as well as the Barvinok-Novik orbitopes.…”

Section: Related Workmentioning

confidence: 99%

“…In some situations, especially those arising from physical problems, we require the additional constraint that the decision variables be in the set of rotation matrices SO(n) := {X ∈ R n×n : X T X = I, det(X) = 1} (2) representing Euclidean isometries that also preserve orientation. For example, these additional constraints arise in problems involving attitude estimation for spacecraft [25] or pose estimation in computer vision applications [17], or in understanding protein folding [21]. The unit determinant constraint is important in these situations because we typically cannot reflect physical objects such as spacecraft or molecules.…”

Section: Introductionmentioning

confidence: 99%

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Semidefinite Descriptions of the Convex Hull of Rotation Matrices

Saunderson¹,

Parrilo²,

Willsky³

2015

SIAM J. Optim.

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We study the convex hull of SO(n), thought of as the set of n × n orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of SO(n) is doubly spectrahedral, i.e. both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit, and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Semidefinite Descriptions of the Convex Hull of Rotation Matrices

Saunderson¹,

Parrilo²,

Willsky³

2015

SIAM J. Optim.

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“…For the proof we refer to [23]. Proposition 2.1 holds also for PCS, thus the maximum number of independent metal ions is again 5.…”

Section: Theorymentioning

confidence: 89%

Looking for central tendencies in the conformational freedom of proteins using NMR measurements

Clarelli

Sgheri

2017

Inverse Problems

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We study the conformational freedom of a protein made by two rigid domains connected by a flexible linker. The conformational freedom is represented as an unknown probability distribution on the space of allowed states. A new algorithm for the calculation of the Maximum Allowable Probability is proposed, which can be extended to any type of measurements. In this paper we use Pseudo Contact Shifts and Residual Dipolar Coupling. We reconstruct a single central tendency in the distribution and discuss in depth the results. arXiv:1602.05953v1 [q-bio.BM]

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“…Optimization, coordination, and consensus over the group of rotation matrices (i.e. over elements of SO(n)) is a problem of fundamental importance in a wide range of applications, from satellite attitude and spin estimation [1], [2], vehicle coordination [3], frequency synchronization [4], (distributed) visual pose estimation [5], [6], [7] and protein folding [8].…”

Section: Introductionmentioning

confidence: 99%

A convex approach to consensus on SO(n)

Matni

Horowitz

2014

2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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This paper introduces several new algorithms for consensus over the special orthogonal group. By relying on a convex relaxation of the space of rotation matrices, consensus over rotation elements is reduced to solving a convex problem with a unique global solution. The consensus protocol is then implemented as a distributed optimization using (i) dual decomposition, and (ii) both semi and fully distributed variants of the alternating direction method of multipliers technique -all with strong convergence guarantees. The convex relaxation is shown to be exact at all iterations of the dual decomposition based method, and exact once consensus is reached in the case of the alternating direction method of multipliers. Further, analytic and/or efficient solutions are provided for each iteration of these distributed computation schemes, allowing consensus to be reached without any online optimization. Examples in satellite attitude alignment with up to 100 agents, an estimation problem from computer vision, and a rotation averaging problem on SO(6) validate the approach. I. INTRODUCTIONOptimization, coordination, and consensus over the group of rotation matrices (i.e. over elements of SO(n)) is a problem of fundamental importance in a wide range of applications, from satellite attitude and spin estimation [1], [2], vehicle coordination [3], frequency synchronization [4], (distributed) visual pose estimation [5], [6], [7] and protein folding [8].Traditional consensus in Euclidean space has a rich history [9], with results offering strong convergence guarantees under mild and realistic assumptions using purely local protocols. Further, with the resurfacing of distributed optimization methods such as dual decomposition [10] and the alternating direction method of multipliers [11], these methods have been successfully applied to achieve so-called "fast consensus" in a distributed [12] and semi-distributed (i.e. sensor fusion) [11] setting. Unfortunately, generalizing these approaches to manifolds, even those with as much structure as the group of rotation matrices, has proven nontrivial.Fortunately, although the space of rotation matrices is highly non-linear, it is a Lie Group. This structure has allowed for sophisticated consensus methods to be created with so-called "almost-global" convergence -that is to say the only stable stationary points of the protocol are global minimizers of the underlying optimization. These consensus schemes fall under two broad categories, namely intrinsic and extrinsic approaches.

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Convex Hulls of Orbits and Orientations of a Moving Protein Domain

Cited by 8 publications

References 15 publications

Semidefinite Descriptions of the Convex Hull of Rotation Matrices

Semidefinite Descriptions of the Convex Hull of Rotation Matrices

Looking for central tendencies in the conformational freedom of proteins using NMR measurements

A convex approach to consensus on SO(n)

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