Leon Lang - Academia.edu (original) (raw)
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Papers by Leon Lang
In information theory, one major goal is to find useful functions that summarize the amount of in... more In information theory, one major goal is to find useful functions that summarize the amount of information contained in the interaction of several random variables. Specifically, one can ask how the classical Shannon entropy, mutual information, and higher interaction information functions relate to each other. This is formally answered by Hu's theorem, which is widely known in the form of information diagrams: it relates disjoint unions of shapes in a Venn diagram to summation rules of information functions; this establishes a bridge from set theory to information theory. While a proof of this theorem is known, to date it was not analyzed in detail in what generality it could be established. In this work, we view random variables together with the joint operation as a monoid that acts by conditioning on information functions, and entropy as the unique function satisfying the chain rule of information. This allows us to abstract away from Shannon's theory and to prove a gene...
Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with addi... more Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with G-steerable kernels, that is, kernels that satisfy an equivariance constraint themselves. While the G-steerability constraint has been derived, it has to date only been solved for specific use cases - a general characterization of G-steerable kernel spaces is still missing. This work provides such a characterization for the practically relevant case of G being any compact group. Our investigation is motivated by a striking analogy between the constraints underlying steerable kernels on the one hand and spherical tensor operators from quantum mechanics on the other hand. By generalizing the famous Wigner-Eckart theorem for spherical tensor operators, we prove that steerab...
ArXiv, 2021
Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with addi... more Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with G-steerable kernels, that is, kernels that satisfy an equivariance constraint themselves. While the G-steerability constraint has been derived, it has to date only been solved for specific use cases - a general characterization of G-steerable kernel spaces is still missing. This work provides such a characterization for the practically relevant case of G being any compact group. Our investigation is motivated by a striking analogy between the constraints underlying steerable kernels on the one hand and spherical tensor operators from quantum mechanics on the other hand. By generalizing the famous Wigner-Eckart theorem for spherical tensor operators, we prove that steerab...
Proceedings of the Beyond Vision and LANguage: inTEgrating Real-world kNowledge (LANTERN)
In information theory, one major goal is to find useful functions that summarize the amount of in... more In information theory, one major goal is to find useful functions that summarize the amount of information contained in the interaction of several random variables. Specifically, one can ask how the classical Shannon entropy, mutual information, and higher interaction information functions relate to each other. This is formally answered by Hu's theorem, which is widely known in the form of information diagrams: it relates disjoint unions of shapes in a Venn diagram to summation rules of information functions; this establishes a bridge from set theory to information theory. While a proof of this theorem is known, to date it was not analyzed in detail in what generality it could be established. In this work, we view random variables together with the joint operation as a monoid that acts by conditioning on information functions, and entropy as the unique function satisfying the chain rule of information. This allows us to abstract away from Shannon's theory and to prove a gene...
Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with addi... more Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with G-steerable kernels, that is, kernels that satisfy an equivariance constraint themselves. While the G-steerability constraint has been derived, it has to date only been solved for specific use cases - a general characterization of G-steerable kernel spaces is still missing. This work provides such a characterization for the practically relevant case of G being any compact group. Our investigation is motivated by a striking analogy between the constraints underlying steerable kernels on the one hand and spherical tensor operators from quantum mechanics on the other hand. By generalizing the famous Wigner-Eckart theorem for spherical tensor operators, we prove that steerab...
ArXiv, 2021
Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with addi... more Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with G-steerable kernels, that is, kernels that satisfy an equivariance constraint themselves. While the G-steerability constraint has been derived, it has to date only been solved for specific use cases - a general characterization of G-steerable kernel spaces is still missing. This work provides such a characterization for the practically relevant case of G being any compact group. Our investigation is motivated by a striking analogy between the constraints underlying steerable kernels on the one hand and spherical tensor operators from quantum mechanics on the other hand. By generalizing the famous Wigner-Eckart theorem for spherical tensor operators, we prove that steerab...
Proceedings of the Beyond Vision and LANguage: inTEgrating Real-world kNowledge (LANTERN)