A perturbative picture of cubic tensors in dually flat spaces
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Some effects of cubic tensors in divergences, which characterize dually flat spaces, are investigated. Focusing on the small area around a specific point, how tangent space approximation is corrected by the cubic tensor at the point, is described. The cubic tensor in divergence is treated as a perturbation, supposing that the coordinates in tangent space are unperturbed solutions. A first order perturbation solution is presented as a main result. On the other hand, the unperturbed solution is yet another coordinate different from the usual affine coordinate. The divergences are shown to be written in a simple form if it is written in terms of the unperturbed solution. Furthermore, a relation which transforms similarities to the divergences is shown. Consequently, the asymmetry in the divergences between the tangent point and the other point is shown to be attributed to the difference between the squared norms of the two kinds of coordinates.
KeywordsInformation geometry Divergence Dually flat space Multidimensional scaling
Mathematics Subject Classification53A15 91C15 94A17
The author greatly appreciates the valuable comments of the reviewers.
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