The Experts below are selected from a list of 45246 Experts worldwide ranked by ideXlab platform
Sanjaye Ramgoolam - One of the best experts on this subject based on the ideXlab platform.
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Tensor models, Kronecker coefficients and permutation centralizer algebras
Journal of High Energy Physics, 2017Co-Authors: Joseph Ben Geloun, Sanjaye RamgoolamAbstract:A bstractWe show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras. These algebras are shown to be semi-simple and their Wedderburn-Artin decompositions into matrix blocks are given in terms of Clebsch-Gordan coefficients of symmetric Groups. The matrix basis for the algebras also gives an orthogonal basis for the tensor observables which diagonalizes the Gaussian two-point functions. The centres of the algebras are associated with correlators which are expressible in terms of Kronecker coefficients (Clebsch-Gordan multiplicities of symmetric Groups). The color-exchange symmetry present in the Gaussian model, as well as a large class of interacting models, is used to refine the description of the permutation centralizer algebras. This discussion is extended to a general number of colors d: it is used to prove the integrality of an infinite family of number sequences related to color-symmetrizations of colored graphs, and expressible in terms of symmetric Group Representation Theory data. Generalizing a connection between matrix models and Belyi maps, correlators in Gaussian tensor models are interpreted in terms of covers of singular 2-complexes. There is an intriguing difference, between matrix and higher rank tensor models, in the computational complexity of superficially comparable correlators of observables parametrized by Young diagrams.
Shubhendu Trivedi - One of the best experts on this subject based on the ideXlab platform.
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clebsch gordan nets a fully fourier space spherical convolutional neural network
arXiv: Machine Learning, 2018Co-Authors: Risi Kondor, Zhen Lin, Shubhendu TrivediAbstract:Recent work by Cohen \emph{et al.} has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from Group Representation Theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but from an implementation point of view is actually simpler. An unusual feature of the proposed architecture is that it uses the Clebsch--Gordan transform as its only source of nonlinearity, thus avoiding repeated forward and backward Fourier transforms. The underlying ideas of the paper generalize to constructing neural networks that are invariant to the action of other compact Groups.
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clebsch gordan nets a fully fourier space spherical convolutional neural network
Neural Information Processing Systems, 2018Co-Authors: Risi Kondor, Zhen Lin, Shubhendu TrivediAbstract:Recent work by Cohen et al. has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from Group Representation Theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but from an implementation point of view is actually simpler. An unusual feature of the proposed architecture is that it uses the Clebsch--Gordan transform as its only source of nonlinearity, thus avoiding repeated forward and backward Fourier transforms. The underlying ideas of the paper generalize to constructing neural networks that are invariant to the action of other compact Groups.
Joseph Ben Geloun - One of the best experts on this subject based on the ideXlab platform.
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Tensor models, Kronecker coefficients and permutation centralizer algebras
Journal of High Energy Physics, 2017Co-Authors: Joseph Ben Geloun, Sanjaye RamgoolamAbstract:A bstractWe show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras. These algebras are shown to be semi-simple and their Wedderburn-Artin decompositions into matrix blocks are given in terms of Clebsch-Gordan coefficients of symmetric Groups. The matrix basis for the algebras also gives an orthogonal basis for the tensor observables which diagonalizes the Gaussian two-point functions. The centres of the algebras are associated with correlators which are expressible in terms of Kronecker coefficients (Clebsch-Gordan multiplicities of symmetric Groups). The color-exchange symmetry present in the Gaussian model, as well as a large class of interacting models, is used to refine the description of the permutation centralizer algebras. This discussion is extended to a general number of colors d: it is used to prove the integrality of an infinite family of number sequences related to color-symmetrizations of colored graphs, and expressible in terms of symmetric Group Representation Theory data. Generalizing a connection between matrix models and Belyi maps, correlators in Gaussian tensor models are interpreted in terms of covers of singular 2-complexes. There is an intriguing difference, between matrix and higher rank tensor models, in the computational complexity of superficially comparable correlators of observables parametrized by Young diagrams.
Risi Kondor - One of the best experts on this subject based on the ideXlab platform.
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clebsch gordan nets a fully fourier space spherical convolutional neural network
arXiv: Machine Learning, 2018Co-Authors: Risi Kondor, Zhen Lin, Shubhendu TrivediAbstract:Recent work by Cohen \emph{et al.} has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from Group Representation Theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but from an implementation point of view is actually simpler. An unusual feature of the proposed architecture is that it uses the Clebsch--Gordan transform as its only source of nonlinearity, thus avoiding repeated forward and backward Fourier transforms. The underlying ideas of the paper generalize to constructing neural networks that are invariant to the action of other compact Groups.
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clebsch gordan nets a fully fourier space spherical convolutional neural network
Neural Information Processing Systems, 2018Co-Authors: Risi Kondor, Zhen Lin, Shubhendu TrivediAbstract:Recent work by Cohen et al. has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from Group Representation Theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but from an implementation point of view is actually simpler. An unusual feature of the proposed architecture is that it uses the Clebsch--Gordan transform as its only source of nonlinearity, thus avoiding repeated forward and backward Fourier transforms. The underlying ideas of the paper generalize to constructing neural networks that are invariant to the action of other compact Groups.
Franz Luef - One of the best experts on this subject based on the ideXlab platform.
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A Duality Principle for Groups II: Multi-frames Meet Super-Frames
Journal of Fourier Analysis and Applications, 2020Co-Authors: Radu Balan, Dorin Ervin Dutkay, Deguang Han, David R. Larson, Franz LuefAbstract:AbstractThe duality principle for Group Representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of Group Representation Theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary Group Representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of Group Representations. Applying it to the Gabor Representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor Representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d .
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A duality principle for Groups II: Multi-frames meet super-frames
arXiv: Functional Analysis, 2018Co-Authors: Radu Balan, Dorin Ervin Dutkay, Deguang Han, David R. Larson, Franz LuefAbstract:The duality principle for Group Representations developed in \cite{DHL-JFA, HL_BLM} exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of Group Representation Theory. There are two other well-known fundamental properties in Gabor analysis: The Wexler-Raz biorthogonality and the Fundamental Identity of Gabor analysis. In this paper we will show that these fundamental properties remain to be true for general projective unitary Group Representations. The main purpose of this paper is present a more general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pairs. In particular, for the Gabor Representations $\pi_{\Lambda}$ and $\pi_{\Lambda^{o}}$ with respect to a pair of dual time-frequency lattices $\Lambda$ and $\Lambda^{o}$ in $\R^{d}\times \R^{d}$ we have that $\{\pi_{\Lambda}(m, n)g_{1} \oplus ... \oplus \pi_{\Lambda}(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a frame for $L^{2}(\R^{d})\oplus ... \oplus L^{2}(\R^{d})$ if and only if $\cup_{i=1}^{k}\{\pi_{\Lambda^{o}}(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a Riesz sequence, and $\cup_{i=1}^{k}\{\pi_{\Lambda}(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a frame for $L^{2}(\R^{d})$ if and only if $\{\pi_{\Lambda^{o}}(m, n)g_{1} \oplus ... \oplus \pi_{\Lambda^{o}}(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a Riesz sequence. This appears to be new even in the context of Gabor analysis.
