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In mathematics, a Bregman divergence or Bregman distance is similar to a metric, but does not satisfy the triangle inequality nor symmetry. There are two ways in which Bregman divergences are important. Firstly, they generalize squared Euclidean distance to a class of distances that all share similar properties. Secondly, they bear a strong connection to exponential families of distributions; as has been shown by (Banerjee et al. 2005), there is a bijection between regular exponential families and regular Bregman divergences. Bregman divergences are named after Lev M. Bregman, who introduced the concept in 1967. More recently researchers in geometric algorithms have shown that many important algorithms can be generalized from Euclidean metrics to distances defined by Bregman divergence (Banerjee et al. 2005; Nielsen and Nock 2006; Boissonnat et al. 2010). == Definition == Let be a continuously-differentiable real-valued and strictly convex function defined on a closed convex set . The Bregman distance associated with ''F'' for points is the difference between the value of ''F'' at point ''p'' and the value of the first-order Taylor expansion of ''F'' around point ''q'' evaluated at point ''p'': : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bregman divergence」の詳細全文を読む スポンサード リンク
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