Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm

Ferrez, Jean-Albert; Fukuda, Komei; Liebling, Th. M.

doi:10.1016/j.ejor.2003.04.011

Cited by 55 publications

(49 citation statements)

References 18 publications

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“…We note that the required optimal binary vector in (22) can, alternatively, be computed through the algorithm in [38], [39] with time complexity and space complexity at least based on the reverse search for cell enumeration in arrangements [40] or with time complexity but space complexity proportional to based on the incremental algorithm for cell enumeration in arrangements [41], [42]. Another algorithm that can solve (22) with polynomial complexity is in [43].…”

Section: ) Casementioning

confidence: 99%

Optimal Algorithms for <formula formulatype="inline"> <tex Notation="TeX">$L_{1}$</tex></formula>-subspace Signal Processing

Markopoulos

Karystinos

Pados

2014

IEEE Trans. Signal Process.

141

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We describe ways to define and calculate -norm signal subspaces that are less sensitive to outlying data than -calculated subspaces. We start with the computation of the maximum-projection principal component of a data matrix containing signal samples of dimension . We show that while the general problem is formally NP-hard in asymptotically large , , the case of engineering interest of fixed dimension and asymptotically large sample size is not. In particular, for the case where the sample size is less than the fixed dimension , we present in explicit form an optimal algorithm of computational cost . For the case , we present an optimal algorithm of complexity . We generalize to multiple -max-projection components and present an explicit optimal subspace calculation algorithm of complexity where is the desired number of principal components (subspace rank). We conclude with illustrations of -subspace signal processing in the fields of data dimensionality reduction, direction-of-arrival estimation, and image conditioning/restoration.

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Section: ) Casementioning

confidence: 99%

Optimal Algorithms for <formula formulatype="inline"> <tex Notation="TeX">$L_{1}$</tex></formula>-subspace Signal Processing

Markopoulos

Karystinos

Pados

2014

IEEE Trans. Signal Process.

141

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“…A binary quadratic programming problem is NP-hard in general. The structure of our problem does not fall into one of the solvable cases; our quadratic matrix has positive offdiagonal elements [30], is a non-singular matrix [3,14], and cannot be represented by a tri-/five-diagonal matrix [16]. Also, the underlying graph structure is not series parallel [6].…”

Section: Initialization Of Equation (5)mentioning

confidence: 99%

3D reconstruction of a smooth articulated trajectory from a monocular image sequence

Park

Sheikh

2011

2011 International Conference on Computer Vision

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An articulated trajectory is defined as a trajectory that remains at a fixed distance with respect to a parent trajectory. In this paper, we present a method to reconstruct an articulated trajectory in three dimensions given the two dimensional projection of the articulated trajectory, the 3D parent trajectory, and the camera pose at each time instant. This is a core challenge in reconstructing the 3D motion of articulated structures such as the human body because endpoints of each limb form articulated trajectories. We simultaneously apply activity-independent spatial and temporal constraints, in the form of fixed 3D distance to the parent trajectory and smooth 3D motion. There exist two solutions that satisfy each instantaneous 2D projection and articulation constraint (a ray intersects a sphere at up to two locations) and we show that resolving this ambiguity by enforcing smoothness is equivalent to solving a binary quadratic programming problem. A geometric analysis of the reconstruction of articulated trajectories is also presented and a measure of the reconstructibility of an articulated trajectory is proposed.

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“…In the literature on computational geometry, several algorithms for enumeration of vertices and facets of a zonotope given by the set of generators are known. Moreover, there are methods with computation time which is bounded by a polynomial in the size of input and the size of output; see [30] and [31]. In Corollary 1 we have shown that if p is fixed then the size of the output is polynomially bounded in the size of the input.…”

Section: Proof We Havementioning

confidence: 99%

On the Possibilistic Approach to Linear Regression with Rounded or Interval-Censored Data

Černý¹,

Rada²

2011

Measurement Science Review

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Consider a linear regression model where some or all of the observations of the dependent variable have been either rounded or interval-censored and only the resulting interval is available. Given a linear estimator β of the vector of regression parameters, we consider its possibilistic generalization for the model with rounded/censored data, which is called the OLS-set in the special case β = Ordinary Least Squares. We derive a geometric characterization of the set: we show that it is a zonotope in the parameter space. We show that even for models with a small number of regression parameters and a small number of observations, the combinatorial complexity of the polyhedron can be high. We therefore derive simple bounds on the OLS-set. These bounds allow to quantify the worst-case impact of rounding/censoring on the estimator β . This approach is illustrated by an example. We also observe that the method can be used for quantification of the rounding/censoring effect in advance, before the experiment is made, and hence can provide information on the choice of measurement precision when the experiment is being planned.

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Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm

Cited by 55 publications

References 18 publications

Optimal Algorithms for <formula formulatype="inline"> <tex Notation="TeX">$L_{1}$</tex></formula>-subspace Signal Processing

Optimal Algorithms for <formula formulatype="inline"> <tex Notation="TeX">$L_{1}$</tex></formula>-subspace Signal Processing

3D reconstruction of a smooth articulated trajectory from a monocular image sequence

On the Possibilistic Approach to Linear Regression with Rounded or Interval-Censored Data

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