This document summarizes Dror Atariah's PhD defense presentation on parameterizations in configuration spaces and approximations of related surfaces for motion planning. It covered parameterizing contact between robots and obstacles using vertex-edge and edge-vertex models. It also discussed approximating saddle surfaces encountered in configuration spaces to obtain locally optimal approximations with minimal vertical distance. The approximations were interpolating or non-interpolating and the document provided analysis of the vertical distance for both cases on simple saddles.
Presentation used by Pedro Prieto-Martín, the founding president of the association, for the defense of his Doctoral Thesis ("Creating the 'symbiotic city': A proposal for the interdisciplinary co-design and co-creation of Civic Software Systems"), 29th of October 2012 in the University of Alcalá.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
Sensors and Samples: A Homological ApproachDon Sheehy
In their seminal work on homological sensor networks, de Silva and Ghrist showed the surprising fact that its possible to certify the coverage of a coordinate free sensor network even with very minimal knowledge of the space to be covered. We give a new, simpler proof of the de Silva-Ghrist Topological Coverage Criterion that eliminates any assumptions about the smoothness of the boundary of the underlying space, allowing the results to be applied to much more general problems. The new proof factors the geometric, topological, and combinatorial aspects of this approach. This factoring reveals an interesting new connection between the topological coverage condition and the notion of weak feature size in geometric sampling theory. We then apply this connection to the problem of showing that for a given scale, if one knows the number of connected components and the distance to the boundary, one can also infer the higher betti numbers or provide strong evidence that more samples are needed. This is in contrast to previous work which merely assumed a good sample and gives no guarantees if the sampling condition is not met.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Presentation used by Pedro Prieto-Martín, the founding president of the association, for the defense of his Doctoral Thesis ("Creating the 'symbiotic city': A proposal for the interdisciplinary co-design and co-creation of Civic Software Systems"), 29th of October 2012 in the University of Alcalá.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
Sensors and Samples: A Homological ApproachDon Sheehy
In their seminal work on homological sensor networks, de Silva and Ghrist showed the surprising fact that its possible to certify the coverage of a coordinate free sensor network even with very minimal knowledge of the space to be covered. We give a new, simpler proof of the de Silva-Ghrist Topological Coverage Criterion that eliminates any assumptions about the smoothness of the boundary of the underlying space, allowing the results to be applied to much more general problems. The new proof factors the geometric, topological, and combinatorial aspects of this approach. This factoring reveals an interesting new connection between the topological coverage condition and the notion of weak feature size in geometric sampling theory. We then apply this connection to the problem of showing that for a given scale, if one knows the number of connected components and the distance to the boundary, one can also infer the higher betti numbers or provide strong evidence that more samples are needed. This is in contrast to previous work which merely assumed a good sample and gives no guarantees if the sampling condition is not met.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
In general dimension, there is no known total polynomial algorithm for either convex hull or vertex enumeration, i.e. an algorithm whose complexity depends polynomially on the input and output sizes. It is thus important to identify problems (and polytope representations) for which total polynomial-time algorithms can be obtained. We offer the first total polynomial-time algorithm for computing the edge-skeleton (including vertex enumeration) of a polytope given by an optimization or separation oracle, where we are also given a superset of its edge directions. We also offer a space-efficient variant of our algorithm by employing reverse search. All complexity bounds refer to the (oracle) Turing machine model. There is a number of polytope classes naturally defined by oracles; for some of them neither vertex nor facet representation is obvious. We consider two main applications, where we obtain (weakly) total polynomial-time algorithms: Signed Minkowski sums of convex polytopes, where polytopes can be subtracted provided the signed sum is a convex polytope, and computation of secondary, resultant, and discriminant polytopes. Further applications include convex combinatorial optimization and convex integer programming, where we offer a new approach, thus removing the complexity's exponential dependence in the dimension.
BEADS : filtrage asymétrique de ligne de base (tendance) et débruitage pour d...Laurent Duval
This paper jointly addresses the problems of chromatogram baseline correction and noise reduction. The proposed approach is based on modeling the series of chromatogram peaks as sparse with sparse derivatives, and on modeling the baseline as a low-pass signal. A convex optimization problem is formulated so as to encapsulate these non-parametric models. To account for the positivity of chromatogram peaks, an asymmetric penalty functions is utilized. A robust, computationally efficient, iterative algorithm is developed that is guaranteed to converge to the unique optimal solution. The approach, termed Baseline Estimation And Denoising with Sparsity (BEADS), is evaluated and compared with two state-of-the-art methods using both simulated and real chromatogram data.
Sampling Spectrahedra: Volume Approximation and OptimizationApostolos Chalkis
My talk to SIAM Conference on Applied Algebraic Geometry (AG21) on volume approximation of spectrahedra and convex optimization with randomized methods based on MCMC sampling with geometric random walks
2017-03, ICASSP, Projection-based Dual Averaging for Stochastic Sparse Optimi...asahiushio1
We present a variant of the regularized dual averaging (RDA) algorithm for stochastic sparse optimization. Our approach differs from the previous studies of RDA in two respects. First, a sparsity-promoting metric is employed, originated from the proportionate-type adaptive filtering algorithms. Second, the squared-distance function to a closed convex set is employed as a part of the objective functions. In the particular application of online regression, the squared-distance function is reduced to a normalized version of the typical squared-error (least square) function. The two differences yield a better sparsity-seeking capability, leading to improved convergence properties. Numerical examples show the advantages of the proposed algorithm over the existing methods including ADAGRAD and adaptive proximal forward-backward splitting (APFBS).
In general dimension, there is no known total polynomial algorithm for either convex hull or vertex enumeration, i.e. an algorithm whose complexity depends polynomially on the input and output sizes. It is thus important to identify problems (and polytope representations) for which total polynomial-time algorithms can be obtained. We offer the first total polynomial-time algorithm for computing the edge-skeleton (including vertex enumeration) of a polytope given by an optimization or separation oracle, where we are also given a superset of its edge directions. We also offer a space-efficient variant of our algorithm by employing reverse search. All complexity bounds refer to the (oracle) Turing machine model. There is a number of polytope classes naturally defined by oracles; for some of them neither vertex nor facet representation is obvious. We consider two main applications, where we obtain (weakly) total polynomial-time algorithms: Signed Minkowski sums of convex polytopes, where polytopes can be subtracted provided the signed sum is a convex polytope, and computation of secondary, resultant, and discriminant polytopes. Further applications include convex combinatorial optimization and convex integer programming, where we offer a new approach, thus removing the complexity's exponential dependence in the dimension.
BEADS : filtrage asymétrique de ligne de base (tendance) et débruitage pour d...Laurent Duval
This paper jointly addresses the problems of chromatogram baseline correction and noise reduction. The proposed approach is based on modeling the series of chromatogram peaks as sparse with sparse derivatives, and on modeling the baseline as a low-pass signal. A convex optimization problem is formulated so as to encapsulate these non-parametric models. To account for the positivity of chromatogram peaks, an asymmetric penalty functions is utilized. A robust, computationally efficient, iterative algorithm is developed that is guaranteed to converge to the unique optimal solution. The approach, termed Baseline Estimation And Denoising with Sparsity (BEADS), is evaluated and compared with two state-of-the-art methods using both simulated and real chromatogram data.
Sampling Spectrahedra: Volume Approximation and OptimizationApostolos Chalkis
My talk to SIAM Conference on Applied Algebraic Geometry (AG21) on volume approximation of spectrahedra and convex optimization with randomized methods based on MCMC sampling with geometric random walks
2017-03, ICASSP, Projection-based Dual Averaging for Stochastic Sparse Optimi...asahiushio1
We present a variant of the regularized dual averaging (RDA) algorithm for stochastic sparse optimization. Our approach differs from the previous studies of RDA in two respects. First, a sparsity-promoting metric is employed, originated from the proportionate-type adaptive filtering algorithms. Second, the squared-distance function to a closed convex set is employed as a part of the objective functions. In the particular application of online regression, the squared-distance function is reduced to a normalized version of the typical squared-error (least square) function. The two differences yield a better sparsity-seeking capability, leading to improved convergence properties. Numerical examples show the advantages of the proposed algorithm over the existing methods including ADAGRAD and adaptive proximal forward-backward splitting (APFBS).
15. Approximation
Results
Theorem (Non-Interpolating)
Let > 0 be fixed and set T ⊂ R2 to be the triangle with the following
vertices:
p0 = (0, 0)
p1 = 2
√
1 +
1
√
3
, 1 −
1
√
3
, p2 = 2
√
1 −
1
√
3
, 1 +
1
√
3
If ˆT is the lifting of the triangle T to the offset saddle S3
, then the
following hold
• Vertical distance: distV (S, ˆT ) =
• Area of projection to the plane:
area(T ) =
8
√
3
≈ 4.6188 > 4.4721 ≈ 2
√
5 = area(TI
)
PhD. Defense (D. Atariah) 12
17. CGAL
Arrangements of Polylines
Motivation
• Arrangements of rational curves
• Extend the support of 2D Arrangements to unbounded polylines
Contribution
• Improvement of the implementation
• 5% speedup of the runtime
• Code is more generic
PhD. Defense (D. Atariah) 14
23. Parameterization
Differential Geometry of E-V Case
• Surface Normal: Ns(t, φ) =
RφnA
i
ai,t, EA
i
• First Fundamental Forms: E = ai − ai+1
2, F = det(ai , ai+1) and
G = 1 + ai,t
2
• Gaussian Curvature: K(t) = −E2
ν4 , where ν =
√
EG − F2.
• Mean Curvature: H(t) = EF
2ν3
• supt∈R |K(t)| = |K(t )| = 1 where t = ai ,ai −ai+1
E and ai,t is the
closest point on the support line of EA
i to the origin
• Normal Curvature, Principal Curvatures and Principal Curvature
Directions
PhD. Defense (D. Atariah) 20
26. Approximation
The Vertical Distance
• For p, q ∈ S = { (x, y, z) : z = F(x, y) } we have
distV ( pq, S) =
1
4
|a11∆2
x + 2a12∆x ∆y + a22∆2
y |,
where ∆x = px − qx , ∆y = py − qy and F(x, y) is a quadratic form.
• distH(A, B) ≤ distV (A, B)
h
vx
y
PhD. Defense (D. Atariah) 23