Rodríguez Miguel Á, Romaniega Álvaro, Tempesta Piergiulio
Departamento de Física Teórica, Facultad de Físicas, Universidad Complutense de Madrid, 28040 Madrid, Spain.
Instituto de Ciencias Matemáticas, c/ Nicolás Cabrera, No. 13-15, 28049 Madrid, Spain.
Proc Math Phys Eng Sci. 2019 Feb;475(2222):20180633. doi: 10.1098/rspa.2018.0633. Epub 2019 Feb 6.
In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for a large class of entropies. In addition, a method for defining new entropies, using previously known ones with some desired group-theoretical properties is proposed. In the second part of this work, the information geometrical counterpart of the previous construction is examined and a general class of divergences are proposed and studied. Finally, a method of constructing new divergences from known ones is discussed; in particular, some results concerning the Riemannian structure associated with the class of divergences under investigation are formulated.
在这项工作中，我们在群论框架下研究广义熵和信息几何。我们探索确保一大类熵存在某些自然性质以及同时存在群论结构的条件。此外，提出了一种使用具有某些期望群论性质的已知熵来定义新熵的方法。在这项工作的第二部分，研究了先前构造的信息几何对应物，提出并研究了一类通用的散度。最后，讨论了一种从已知散度构造新散度的方法；特别地，阐述了一些与所研究的散度类相关的黎曼结构的结果。
