14 research outputs found
Rota--Baxter operators and skew left brace structures over Heisenberg Group
No full textRota--Baxter operators over groups have been recently defined in \cite{LHY2021}, and they share a close connection with skew braces, as demonstrated in \cite{VV2022}. In this paper, we classify all Rota--Baxter operators of weight 1 over the Heisenberg Lie algebra of dimension 3 by directly solving the operators defining equations. Using the fact that the exponential map from the Heisenberg Lie algebra to the Heisenberg Group is bijective, we induces these operators to the Heisenberg Group. Finally, we enumerate all skew left brace structures over the Heisenberg Group induced by these Rota--Baxter operators.14 pages. The mistake in the matrix multiplication given to compute skew left brace structures has now been correcte
Extensions and automorphisms of Rota-Baxter groups
No full textThe notion of Rota-Baxter groups was recently introduced by Guo, Lang and Sheng [{\em Adv. Math.} 387 (2021), 107834, 34 pp.] in the geometric study of Rota-Baxter Lie algebras. They are closely related to skew braces as observed by Bardakov and Gubarev. In this paper, we study extensions of Rota-Baxter groups by constructing suitable cohomology theories. Among others, we find relations with the extensions of skew braces. Given an extension of Rota-Baxter groups, we also construct a short exact sequence connecting various automorphism groups, which generalizes the Wells short exact sequence.This is a preliminary version of this research. we will soon upload a final version free from typos and error
Relative Rota-Baxter groups and skew left braces
No full textRelative Rota-Baxter groups are generalisations of Rota-Baxter groups and introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota-Baxter groups with skew left braces, which are well-known to give non-degenerate set-theoretic solutions of the Yang-Baxter equation. We prove that every relative Rota-Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota-Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota-Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota-Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces.Comments and Suggestions are welcom
Representations of skew braces
No full textIn this paper, we explore linear representations of skew left braces, which are known to provide bijective non-degenerate set-theoretical solutions to the Yang--Baxter equation that are not necessarily involutive. A skew left brace induces an action λ^{\op}: (A, \circ) \to \Aut (A, \cdot), which gives rise to the group Λ_{A^{\op}} = (A, \cdot) \rtimes_{λ^{\op}} (A, \circ). We prove that if and are isoclinic skew left braces, then Λ_{A^{\op}} and Λ_{B^{\op}} are also isoclinic under some mild restrictions on the centers of the respective groups. Our key observation is that there is a one-to-one correspondence between the set of equivalence classes of irreducible representations of and that of the group Λ_{A^{\op}}. We obtain a decomposition of the induced representation of the additive group and of the multiplicative group corresponding to the regular representation of the group Λ_{A^{\op}}. As examples, we compute the dimensions of the irreducible representations for several skew left braces with prime power orders.17 pages, added Proposition 3.3: Proof that if is any skew left brace, then and Λ_{A^{\op}} are isomorphic group
Cohomology and Extensions of Relative Rota-Baxter groups
No full textRelative Rota-Baxter groups are generalisations of Rota-Baxter groups and
recently shown to be intimately related to skew left braces, which are
well-known to yield bijective non-degenerate solutions to the Yang-Baxter
equation. In this paper, we develop an extension theory of relative Rota-Baxter
groups and introduce their low dimensional cohomology groups, which are
distinct from the ones known in the context of Rota-Baxter operators on Lie
groups. We establish an explicit bijection between the set of equivalence
classes of extensions of relative Rota-Baxter groups and their second
cohomology. Further, we delve into the connections between this cohomology and
the cohomology of associated skew left braces. We prove that for bijective
relative Rota-Baxter groups, the two cohomologies are isomorphic in dimension
two.Comment: 30 page
Zorro: Valid, sparse, and stable explanations in graph neural networks
No full textWith the ever-increasing popularity and applications of graph neural networks, several proposals have been made to explain and understand the decisions of a graph neural network. Explanations for graph neural networks differ in principle from other input settings. It is important to attribute the decision to input features and other related instances connected by the graph structure. We find that the previous explanation generation approaches that maximize the mutual information between the label distribution produced by the model and the explanation to be restrictive. Specifically, existing approaches do not enforce explanations to be valid, sparse, or robust to input perturbations. In this paper, we lay down some of the fundamental principles that an explanation method for graph neural networks should follow and introduce a metric RDT-Fidelity as a measure of the explanation's effectiveness. We propose a novel approach Zorro based on the principles from rate-distortion theory that uses a simple combinatorial procedure to optimize for RDT-Fidelity. Extensive experiments on real and synthetic datasets reveal that Zorro produces sparser, stable, and more faithful explanations than existing graph neural network explanation approaches.Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Multimedia ComputingWeb Information System
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Narratives of the 1658 War of Succession for the Mughal Throne, 1658-1707
No full textThis dissertation studies certain Hindi and Persian narratives of the War of Succession (1658) to succeed Shah Jahan (r.1627-1658). All the narratives under study were written during the reign of Aurangzeb (r.1658-1707), the successor of Shah Jahan. The study evaluates the significance of the War as a landmark moment in the social history of India, especially in the formation and inter-relationships between religious communities. The dissertation demarcates the larger epistemological and ontological canvas on which these communities took shape and interacted with each other. The research outlines the ways and the contexts in which terms such as Hindu, momin, musalman, Islam, din and Rajput were deployed in literary texts. It asks whether Hinduism and Islam were two disparate traditions, as previous histories of the War and Mughal India had contended. The dissertation argues that social communities of Hindus and Muslims were mutually and similarly circumscribed within an Islamic worldview and concept of din. Hindu traditions could portray Muslims in concepts and terms borrowed from Indian epics but within an over-arching Islamic cultural dispensation. The War was not a moment of evolution between two independent Hindu and Muslim traditions. Rather, the War was a moment that saw the evolution, even if it be of an antagonistic kind, of Hindu and Muslim traditions within a larger Islamic framework. Besides the above primary focus, the dissertation provides the reader with important insights and overviews regarding allied subjects such as the literary histories of Persian and of Hindi/Urdu, especially in the Dingal and Khari Boli dialects, the political culture of Hindu India, Rajput political culture, Mughal political culture, patronage networks in Mughal India, notions of soldierly duty in seventeenth century India, language and status, preaching in the Hindu and Islamic traditions, the sociological ideas of acculturation and Islamisation, and twentieth century history-writing.Release 24-Aug-2030Originally embargoed through 11-Aug-2017; updated embargo through 24-Aug-2021 per author request, 10-Aug-2017, Kimberly; updated embargo through 24-Aug-2030 per author request, 17-Aug-2021, Kimberl
CrypTFlow2: Practical 2-Party Secure Inference
No full textWe present CrypTFlow2, a cryptographic framework for secure inference over realistic Deep Neural Networks (DNNs) using
secure 2-party computation. CrypTFlow2 protocols are both correct -- i.e., their outputs are bitwise equivalent to the cleartext execution -- and efficient -- they outperform the state-of-the-art protocols in both latency and scale. At the core of CrypTFlow2, we have new 2PC protocols for secure comparison and division, designed carefully to balance round and communication complexity for secure inference tasks. Using CrypTFlow2, we present the first secure inference over ImageNet-scale DNNs like ResNet50 and DenseNet121. These DNNs are at least an order of magnitude larger than those considered in the prior
work of 2-party DNN inference. Even on the benchmarks considered by prior work, CrypTFlow2 requires an order of magnitude less communication and 20x-30x less time than the state-of-the-art
Private Graph Extraction via Feature Explanations
No full textPrivacy and interpretability are two important ingredients for achieving trustworthy machine learning. We study the interplay of these two aspects in graph machine learning through graph reconstruction attacks. The goal of the adversary here is to reconstruct the graph structure of the training data given access to model explanations. Based on the different kinds of auxiliary information available to the adversary, we propose several graph reconstruction attacks. We show that additional knowledge of post-hoc feature explanations substantially increases the success rate of these attacks. Further, we investigate in detail the differences between attack performance with respect to three different classes of explanation methods for graph neural networks: gradient-based, perturbationbased, and surrogate model-based methods. While gradient-based explanations reveal the most in terms of the graph structure, we find that these explanations do not always score high in utility. For the other two classes of explanations, privacy leakage increases with an increase in explanation utility. Finally, we propose a defense based on a randomized response mechanism for releasing the explanations, which substantially reduces the attack success rate. Our code is available at https://github.com/iyempissy/graphstealing- attacks-with-explanation.Multimedia Computin
CrypTFlow: Secure TensorFlow Inference
No full textWe present CrypTFlow, a first of its kind system that converts TensorFlow inference code into Secure Multi-party Computation (MPC) protocols at the push of a button. To do this, we build three components. Our first component, Athos, is an end-to-end compiler from TensorFlow to a variety of semi-honest MPC protocols. The second component, Porthos, is an improved semi-honest 3-party protocol that provides significant speedups for TensorFlow like applications. Finally, to provide malicious
secure MPC protocols, our third component, Aramis, is a novel technique that uses hardware with integrity guarantees to convert any semi-honest MPC protocol into an MPC protocol that provides malicious security. The malicious security of the
protocols output by Aramis relies on integrity of the hardware and semi-honest security of MPC. Moreover, our system matches the inference accuracy of plaintext TensorFlow.
We experimentally demonstrate the power of our system by showing the secure inference of real-world neural networks such as ResNet50 and DenseNet121 over the ImageNet dataset with running times of about 30 seconds for semi-honest security and under two minutes for malicious security. Prior work in the area of secure inference has been limited to semi-honest security of small networks over tiny datasets such as MNIST or CIFAR. Even on MNIST/CIFAR, CrypTFlow outperforms prior work
