The Experts below are selected from a list of 34653 Experts worldwide ranked by ideXlab platform
Gerhard Wunder - One of the best experts on this subject based on the ideXlab platform.
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Hierarchical restricted isometry property for Kronecker product measurements
arXiv: Information Theory, 2018Co-Authors: Ingo Roth, Axel Flinth, Richard Kueng, Jens Eisert, Gerhard WunderAbstract:Hierarchically sparse signals and Kronecker product structured measurements arise naturally in a variety of applications. The simplest example of a hierarchical sparsity structure is two-level $(s,\sigma)$-hierarchical sparsity which features $s$-block-sparse signals with $\sigma$-sparse blocks. For a large class of algorithms recovery guarantees can be derived based on the restricted isometry property (RIP) of the measurement matrix and model-based variants thereof. We show that given two matrices $\mathbf{A}$ and $\mathbf{B}$ having the standard $s$-sparse and $\sigma$-sparse RIP their Kronecker product $\mathbf{A}\otimes\mathbf{B}$ has two-level $(s,\sigma)$-hierarchically sparse RIP (HiRIP). This result can be recursively generalized to signals with multiple hierarchical sparsity levels and measurements with multiple Kronecker product factors. As a corollary we establish the efficient reconstruction of hierarchical sparse signals from Kronecker product measurements using the HiHTP algorithm. We argue that Kronecker product measurement matrices allow to design large practical compressed sensing systems that are deterministically certified to reliably recover signals in a stable fashion. We elaborate on their motivation from the perspective of applications.
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Allerton - Hierarchical restricted isometry property for Kronecker product measurements
2018 56th Annual Allerton Conference on Communication Control and Computing (Allerton), 2018Co-Authors: Ingo Roth, Axel Flinth, Richard Kueng, Jens Eisert, Gerhard WunderAbstract:Hierarchically sparse signals and Kronecker product structured measurements naturally arise in a variety of applications. The simplest example of a hierarchical sparsity structure is two-level $(s,~\sigma )$-hierarchical sparsity which features s-block-sparse signals with s-sparse blocks. For a large class of algorithms recovery guarantees can be derived based on the restricted isometry property (RIP) of the measurement matrix and model-based variants thereof. We show that given two matrices A and B having the standard s-sparse and $\sigma $-sparse RIP their Kronecker product $\mathbf {A}\otimes \mathbf {B}$ has two-level $(s,~\sigma )$-hierarchically sparse RIP (HiRIP). This result can be recursively generalized to signals with multiple hierarchical sparsity levels and measurements with multiple Kronecker product factors. As a corollary we establish the efficient reconstruction of hierarchical sparse signals from Kronecker product measurements using the HiHTP algorithm. We argue that Kronecker product measurement matrices allow to design large practical compressed sensing systems that are deterministically certified to reliably recover signals in a stable fashion. We elaborate on their motivation from the perspective of applications.
Simone Severini - One of the best experts on this subject based on the ideXlab platform.
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Quantum walk search on Kronecker graphs
Physical Review A, 2018Co-Authors: Thomas G. Wong, Konstantin Wünscher, Joshua Lockhart, Simone SeveriniAbstract:Kronecker graphs, obtained by repeatedly performing the Kronecker product of the adjacency matrix of an ``initiator'' graph with itself, have risen in popularity in network science due to their ability to generate complex networks with real-world properties. We explore spatial search by continuous-time quantum walk on Kronecker graphs. Specifically, we give analytical proofs for quantum search on first-, second-, and third-order Kronecker graphs with the complete graph as the initiator, showing that search takes Grover's $O(\sqrt{N})$ time. Numerical simulations indicate that higher-order Kronecker graphs with the complete initiator also support optimal quantum search.
Ingo Roth - One of the best experts on this subject based on the ideXlab platform.
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Hierarchical restricted isometry property for Kronecker product measurements
arXiv: Information Theory, 2018Co-Authors: Ingo Roth, Axel Flinth, Richard Kueng, Jens Eisert, Gerhard WunderAbstract:Hierarchically sparse signals and Kronecker product structured measurements arise naturally in a variety of applications. The simplest example of a hierarchical sparsity structure is two-level $(s,\sigma)$-hierarchical sparsity which features $s$-block-sparse signals with $\sigma$-sparse blocks. For a large class of algorithms recovery guarantees can be derived based on the restricted isometry property (RIP) of the measurement matrix and model-based variants thereof. We show that given two matrices $\mathbf{A}$ and $\mathbf{B}$ having the standard $s$-sparse and $\sigma$-sparse RIP their Kronecker product $\mathbf{A}\otimes\mathbf{B}$ has two-level $(s,\sigma)$-hierarchically sparse RIP (HiRIP). This result can be recursively generalized to signals with multiple hierarchical sparsity levels and measurements with multiple Kronecker product factors. As a corollary we establish the efficient reconstruction of hierarchical sparse signals from Kronecker product measurements using the HiHTP algorithm. We argue that Kronecker product measurement matrices allow to design large practical compressed sensing systems that are deterministically certified to reliably recover signals in a stable fashion. We elaborate on their motivation from the perspective of applications.
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Allerton - Hierarchical restricted isometry property for Kronecker product measurements
2018 56th Annual Allerton Conference on Communication Control and Computing (Allerton), 2018Co-Authors: Ingo Roth, Axel Flinth, Richard Kueng, Jens Eisert, Gerhard WunderAbstract:Hierarchically sparse signals and Kronecker product structured measurements naturally arise in a variety of applications. The simplest example of a hierarchical sparsity structure is two-level $(s,~\sigma )$-hierarchical sparsity which features s-block-sparse signals with s-sparse blocks. For a large class of algorithms recovery guarantees can be derived based on the restricted isometry property (RIP) of the measurement matrix and model-based variants thereof. We show that given two matrices A and B having the standard s-sparse and $\sigma $-sparse RIP their Kronecker product $\mathbf {A}\otimes \mathbf {B}$ has two-level $(s,~\sigma )$-hierarchically sparse RIP (HiRIP). This result can be recursively generalized to signals with multiple hierarchical sparsity levels and measurements with multiple Kronecker product factors. As a corollary we establish the efficient reconstruction of hierarchical sparse signals from Kronecker product measurements using the HiHTP algorithm. We argue that Kronecker product measurement matrices allow to design large practical compressed sensing systems that are deterministically certified to reliably recover signals in a stable fashion. We elaborate on their motivation from the perspective of applications.
Sandip Roy - One of the best experts on this subject based on the ideXlab platform.
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ACC - Kronecker products of defective matrices: Some spectral properties and their implications on observability
2012 American Control Conference (ACC), 2012Co-Authors: Mengran Xue, Sandip RoyAbstract:The eigenvector analysis of the Kronecker product of defective matrices is revisited, and is brought to bear in characterizing observability of linear systems with Kronecker-product state matrices. Specifically, a new proof is given for a classical result on the eigenspace dimension for the Kronecker product of two Jordan chains, which highlights that the dimension is essentially dependent on the off-diagonal entries' zero pattern. Based on the spectral characterization of the Kronecker product, observability conditions are then developed for linear systems with Kronecker-product state matrices. Simple examples from two application domains are also included, to illustrate how the spectral and observability results can inform stochastic-systems analysis.
Christian Ikenmeyer - One of the best experts on this subject based on the ideXlab platform.
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On vanishing of Kronecker coefficients
computational complexity, 2017Co-Authors: Christian Ikenmeyer, Ketan Mulmuley, Michael WalterAbstract:We show that the problem of deciding positivity of Kronecker coefficients is NP-hard. Previously, this problem was conjectured to be in P, just as for the Littlewood–Richardson coefficients. Our result establishes in a formal way that Kronecker coefficients are more difficult than Littlewood–Richardson coefficients, unless P = NP. We also show that there exists a #P-formula for a particular subclass of Kronecker coefficients whose positivity is NP-hard to decide. This is an evidence that, despite the hardness of the positivity problem, there may well exist a positive combinatorial formula for the Kronecker coefficients. Finding such a formula is a major open problem in representation theory and algebraic combinatorics. Finally, we consider the existence of the partition triples $${(\lambda, \mu, \pi)}$$ such that the Kronecker coefficient $${k^\lambda_{\mu, \pi} = 0}$$ but the Kronecker coefficient $${k^{l\lambda}_{l \mu, l \pi} > 0}$$ for some integer l > 1. Such “holes” are of great interest as they witness the failure of the saturation property for the Kronecker coefficients, which is still poorly understood. Using insight from computational complexity theory, we turn our hardness proof into a positive result: We show that not only do there exist many such triples, but they can also be found efficiently. Specifically, we show that, for any $${0 < \epsilon \leq 1}$$ , there exists $${0 < a < 1}$$ such that, for all m, there exist $${\Omega(2^{m^a})}$$ partition triples $${(\lambda,\mu,\mu)}$$ in the Kronecker cone such that: (a) the Kronecker coefficient $${k^\lambda_{\mu,\mu}}$$ is zero, (b) the height of $${\mu}$$ is m, (c) the height of $${\lambda}$$ is $${\leq m^\epsilon}$$ , and (d) $${|\lambda|=|\mu| \le m^3}$$ . The proof of the last result illustrates the effectiveness of the explicit proof strategy of GCT.
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Rectangular Kronecker coefficients and plethysms in geometric complexity theory
arXiv: Computational Complexity, 2015Co-Authors: Christian Ikenmeyer, Greta PanovaAbstract:We prove that in the geometric complexity theory program the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial. Moreover, we prove the positivity of rectangular Kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition. We also compare rectangular Kronecker coefficients with their corresponding plethysm coefficients, which leads to a new lower bound for rectangular Kronecker coefficients. Moreover, we prove that the saturation of the rectangular Kronecker semigroup is trivial, we show that the rectangular Kronecker positivity stretching factor is 2 for a long first row, and we completely classify the positivity of rectangular limit Kronecker coefficients that were introduced by Manivel in 2011.
