A Novel Approach to Canonical Divergences within Information Geometry
A Novel Approach to Canonical Divergences within Information Geometry
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DOI：
10.3390/e17127866
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发表时间：
2015-12-01
期刊：
影响因子：
2.7
通讯作者：
Amari, Shun-ichi
中科院分区：
文献类型：
--
作者：
Ay, Nihat;Amari, Shun-ichi
流形M上的散度函数定义了黎曼度量g和M上的对偶耦合仿射联络δ和δ *。当M是对偶平坦的，即关于δ和δ * 是平坦的，则已知正则散度，其由（M，g，δ，δ *）唯一确定。我们提出了一个一般的，不一定是平坦的，M通过使用测地积分的逆指数映射的典范发散的自然定义。在对偶平坦性的情况下，正则发散的新定义简化为已知的正则发散。最后，我们证明了逆指数映射的可积性蕴涵测地投影性质。
A divergence function on a manifold M defines a Riemannian metric g and dually coupled affine connections delta anddelta *on M. When M is dually flat, that is flat with respect to delta anddelta *, a canonical divergence is known, which is uniquely determined from( M , g , delta ,delta *). We propose a natural definition of a canonical divergence for a general, not necessarily flat, M by using the geodesic integration of the inverse exponential map. The new definition of a canonical divergence reduces to the known canonical divergence in the case of dual flatness. Finally, we show that the integrability of the inverse exponential map implies the geodesic projection property.