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Relevant bibliographies by topics / Lie transformation groups

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Journal articles

Academic literature on the topic 'Lie transformation groups'

Author: Grafiati

Published: 10 March 2023

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Journal articles on the topic "Lie transformation groups"

1

Miao, Xu, and Rajesh P. N. Rao. "Learning the Lie Groups of Visual Invariance." Neural Computation 19, no. 10 (2007): 2665–93. http://dx.doi.org/10.1162/neco.2007.19.10.2665.

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A fundamental problem in biological and machine vision is visual invariance: How are objects perceived to be the same despite transformations such as translations, rotations, and scaling? In this letter, we describe a new, unsupervised approach to learning invariances based on Lie group theory. Unlike traditional approaches that sacrifice information about transformations to achieve invariance, the Lie group approach explicitly models the effects of transformations in images. As a result, estimates of transformations are available for other purposes, such as pose estimation and visuomotor control. Previous approaches based on first-order Taylor series expansions of images can be regarded as special cases of the Lie group approach, which utilizes a matrix-exponential-based generative model of images and can handle arbitrarily large transformations. We present an unsupervised expectation-maximization algorithm for learning Lie transformation operators directly from image data containing examples of transformations. Our experimental results show that the Lie operators learned by the algorithm from an artificial data set containing six types of affine transformations closely match the analytically predicted affine operators. We then demonstrate that the algorithm can also recover novel transformation operators from natural image sequences. We conclude by showing that the learned operators can be used to both generate and estimate transformations in images, thereby providing a basis for achieving visual invariance.

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2

Al-Shomrani, M. M. "Lie Groups Analysis and Contact Transformations for Ito System." Abstract and Applied Analysis 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/342680.

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Generalized Ito systems of four coupled nonlinear evaluation equations are proposed. New classes of exact invariant solutions by using Lie group analysis are obtained. Moreover, we investigate the existence of a one-parameter group of contact transformations for a generalized Ito system. Consequently, we study the relationship between one-parameter group of a contact transformation and a one-parameter Lie point transformation for a generalized Ito system.

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3

FIGUEROA-O’FARRILL, JOSÉ M., and SONIA STANCIU. "POISSON LIE GROUPS AND THE MIURA TRANSFORMATION." Modern Physics Letters A 10, no. 36 (1995): 2767–73. http://dx.doi.org/10.1142/s0217732395002908.

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We point out that the recent proof of the Kupershmidt-Wilson theorem by Cheng and Mas-Ramos is underpinned by the Poisson Lie property of the second Gel’fand-Dickey bracket. The supersymmetric Kupershmidt-Wilson theorem is also proved along the same lines.

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4

Cariñena, Jose F., Mariano A. Del Olmo, and Mariano Santander. "Locally operating realizations of transformation Lie groups." Journal of Mathematical Physics 26, no. 9 (1985): 2096–106. http://dx.doi.org/10.1063/1.526974.

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5

Lin, Feng, Haohang Xu, Houqiang Li, Hongkai Xiong, and Guo-Jun Qi. "Auto-Encoding Transformations in Reparameterized Lie Groups for Unsupervised Learning." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 10 (2021): 8610–17. http://dx.doi.org/10.1609/aaai.v35i10.17044.

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Unsupervised training of deep representations has demonstrated remarkable potentials in mitigating the prohibitive expenses on annotating labeled data recently. Among them is predicting transformations as a pretext task to self-train representations, which has shown great potentials for unsupervised learning. However, existing approaches in this category learn representations by either treating a discrete set of transformations as separate classes, or using the Euclidean distance as the metric to minimize the errors between transformations. None of them has been dedicated to revealing the vital role of the geometry of transformation groups in learning representations. Indeed, an image must continuously transform along the curved manifold of a transformation group rather than through a straight line in the forbidden ambient Euclidean space. This suggests the use of geodesic distance to minimize the errors between the estimated and groundtruth transformations. Particularly, we focus on homographies, a general group of planar transformations containing the Euclidean, similarity and affine transformations as its special cases. To avoid an explicit computing of intractable Riemannian logarithm, we project homographies onto an alternative group of rotation transformations SR(3) with a tractable form of geodesic distance. Experiments demonstrate the proposed approach to Auto-Encoding Transformations exhibits superior performances on a variety of recognition problems.

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6

García‐Prada, Oscar, Mariano A. del Olmo, and Mariano Santander. "Locally operating realizations of nonconnected transformation Lie groups." Journal of Mathematical Physics 29, no. 5 (1988): 1083–90. http://dx.doi.org/10.1063/1.527946.

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7

Nesterenko, Maryna O. "Transformation groups on real plane and their differential invariants." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–17. http://dx.doi.org/10.1155/ijmms/2006/17410.

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Complete sets of bases of differential invariants, operators of invariant differentiation, and Lie determinants of continuous transformation groups acting on the real plane are constructed. As a necessary preliminary, realizations of finite-dimensional Lie algebras on the real plane are revisited.

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8

Nikonov, V. I. "The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 20, no. 3 (2018): 295–303. http://dx.doi.org/10.15507/2079-6900.20.201802.295-303.

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The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.

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9

Nikonov, Vladimir I. "The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 20, no. 3 (2018): 295–303. http://dx.doi.org/10.15507/2079-6900.20.201803.295-303.

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The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.

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10

Russell, Thomas. "Gorman demand systems and lie transformation groups: A reply." Economics Letters 51, no. 2 (1996): 201–4. http://dx.doi.org/10.1016/0165-1765(96)00806-3.

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