Abstract
Lie group machine learning is recognized as the theoretical basis of brain intelligence, brain learning, higher machine learning, and higher artificial intelligence. Sample sets of Lie group matrices are widely available in practical applications. Lie group learning is a vibrant field of increasing importance and extraordinary potential and thus needs to be developed further. This study aims to provide a comprehensive survey on recent advances in Lie group machine learning. We introduce Lie group machine learning techniques in three major categories: supervised Lie group machine learning, semisupervised Lie group machine learning, and unsupervised Lie group machine learning. In addition, we introduce the special application of Lie group machine learning in image processing. This work covers the following techniques: Lie group machine learning model, Lie group subspace orbit generation learning, symplectic group learning, quantum group learning, Lie group fiber bundle learning, Lie group cover learning, Lie group deep structure learning, Lie group semisupervised learning, Lie group kernel learning, tensor learning, frame bundle connection learning, spectral estimation learning, Finsler geometric learning, homology boundary learning, category representation learning, and neuromorphic synergy learning. Overall, this survey aims to provide an insightful overview of state-of-the-art development in the field of Lie group machine learning. It will enable researchers to comprehensively understand the state of the field, identify the most appropriate tools for particular applications, and identify directions for future research.
Highlights
Machine learning, as a branch of artificial intelligence, has been playing an increasingly important role in scientific research in recent years[1,2,3,4]
The relationship between Lie groups and Lie algebras is that the tangent space of group G at identity e, Te, is called the Lie algebra
The exponential map exp is a mapping from the Lie algebra elements to the Lie group elements
Summary
As a branch of artificial intelligence, has been playing an increasingly important role in scientific research in recent years[1,2,3,4]. LML realizes the cognitive view of “solving complex problems with simple models” It has a unique advantage with regard to the use of model continuity theory to solve realistic discrete data. A simple conclusion exists about the general form of Lie brackets for Lie algebra, including closure, bilinearity, alternating, and Jacobi identity. To establish the definition of a Lie group, we first define a smooth map. Definition 3 Let G be a smooth manifold which is a topological group with multiplication map mult: G G ! G is a Lie group if mult and inv are smooth maps. The Lie group distance between two points is defined as d.x1; x2/ D log.x1 1x2/ , where k k is the Frobenius norm of the resulting algebra element[8,9]. Orthogonal matrices are known to have a determinant of either
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