Open AccessBook Chapter10.1007/978-3-319-78199-0_5

Bragg Diffraction Patterns as Graph Characteristics

- 30 Oct 2017

- pp 62-75

TL;DR: This paper compute the spatio-temporal Fourier transform of the Schrodinger operator and then pack its spherical representation in a point of a Stiefel manifold and shows that when the temporal interval of analysis is set according to quantum efficiency principles the proposed approach outperforms the alternatives in graph discrimination.

Abstract: In this paper we establish a link between diffraction theory and graph characterization through the Schrodinger operator. This provides a natural way of characterizing wave propagation on a graph. In order to do so, we compute the spatio-temporal Fourier transform of the operator and then pack its spherical representation in a point of a Stiefel manifold. We show that when the temporal interval of analysis is set according to quantum efficiency principles the proposed approach outperforms the alternatives in graph discrimination.

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1. What have the authors contributed in "Bragg diffraction patterns as graph characteristics" ?

In this paper the authors establish a link between diffraction theory and graph characterization through the Schrödinger operator.. This provides a natural way of characterizing wave propagation on a graph.. In order to do so, the authors compute the spatio-temporal Fourier transform of the operator and then pack its spherical representation in a point of a Stiefel manifold.. The authors show that when the temporal interval of analysis is set according to quantum efficiency principles the proposed approach outperforms the alternatives in graph discrimination.

2. What future works have the authors mentioned in the paper "Bragg diffraction patterns as graph characteristics" ?

Future works include the formulation of finding the optimal T that maximizes quantum transport efficiency.

Fig. 1. Evolution with time (t = 1, 25 and 100). From top to bottom: heat kernel for a 100 node line graph, Schrödinger operator for a 100 node line graph, Schrödinger operator for a 100 node circle graph, Schrödinger operator for a 10×10 grid graph with 4 neighbour connectivity and Schrödinger operator for a 10×10 grid graph with 8 neighbour connectivity. (Courtesy of Pablo Suau)

Fig. 2. Power spectra of the Schrödinger operator for different graphs of 400 nodes at t = 25: circle (loop) graph (top-left), line graph (top-middle), 20×20 grid graph with 4 neighbor connectivity (bottom-left) and 20×20 grid graph with 8 neighbor connectivity. For the latter graphs we also show their spectra (top-right) and the multiplicity of each value of ∆kl (bottom-right).

Fig. 4. Experimental Results. Top: Delaunay graphs of Gatorbait - Gator#1 (left) vsGator#20 (right). Middle: Quantum Efficiency for Gator#1 (left) vs QE for Gator#20 (right); The intercept for the first one is smaller almost one order of magnitude that that of the second. Bottom: Average Recall vs Retrieval curves. Best performance is achieved with T = 64 on average, even when larger values of T are assumed (left). The result outperforms state-of-the-art graph matching algorithms (right). In all cases classes with only one element are not considered for measuring the performance.

Fig. 3. Spatio-Temporal Power spectra of the Schrödinger operator for a 20×20 grid graph with 4 neighbor connectivity. Top-left: planes ωu = 0, ωv = 0, ωt′ = 0. Topcenter/right: detail of ωt′ = 0 and ωt′ = 3 showing parallel high pooled lines. Bottomleft: spherical coordinates of log-amplitudes. Bottom-right: 10 principal eigenvectors of the θ − φ space.

References

Journal Article•10.1038/171737A0

Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid

James D. Watson, +1 more

- 25 Apr 1953

- Nature

TL;DR: The determination in 1953 of the structure of deoxyribonucleic acid (DNA), with its two entwined helices and paired organic bases, was a tour de force in X-ray crystallography and opened the way for a deeper understanding of perhaps the most important biological process.

...read moreread less

10.3K

Journal Article•10.1023/A:1011139631724

Modeling the Shape of the Scene: A Holistic Representation of the Spatial Envelope

Aude Oliva, +1 more

- 01 May 2001

- International Journal of Computer Vision

TL;DR: The performance of the spatial envelope model shows that specific information about object shape or identity is not a requirement for scene categorization and that modeling a holistic representation of the scene informs about its probable semantic category.

...read moreread less

7.5K

•Journal Article•10.1137/S0895479895290954

The Geometry of Algorithms with Orthogonality Constraints

Alan Edelman, +2 more

- 01 Apr 1999

- SIAM Journal on Matrix Analysis and Appl...

TL;DR: The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms and developers of new algorithms and perturbation theories will benefit from the theory.

...read moreread less

2.9K

•Journal Article•10.1111/J.1467-8659.2009.01515.X

A concise and provably informative multi-scale signature based on heat diffusion

Jian Sun, +2 more

- 15 Jul 2009

TL;DR: The Heat Kernel Signature, called the HKS, is obtained by restricting the well‐known heat kernel to the temporal domain and shows that under certain mild assumptions, HKS captures all of the information contained in the heat kernel, and characterizes the shape up to isometry.

...read moreread less

1.7K

•Journal Article•10.1103/PHYSREVA.58.915

Quantum computation and decision trees

Edward Farhi, +1 more

- 01 Aug 1998

- Physical Review A

TL;DR: This work devise a quantum-mechanical algorithm that evolves a state, initially localized at the root, through the tree, and proves that if the classical strategy succeeds in reaching level $n$ in time polynomial in $n,$ then so does the quantum algorithm.

...read moreread less

1.4K

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Bragg Diffraction Patterns as Graph Characteristics

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Most frequently asked questions

1. What have the authors contributed in "Bragg diffraction patterns as graph characteristics" ?

2. What future works have the authors mentioned in the paper "Bragg diffraction patterns as graph characteristics" ?

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References

Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid

Modeling the Shape of the Scene: A Holistic Representation of the Spatial Envelope

The Geometry of Algorithms with Orthogonality Constraints

A concise and provably informative multi-scale signature based on heat diffusion

Quantum computation and decision trees

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