We generalize quasi-arithmetic means beyond scalars by considering the gradient map of a Legendre type real-valued function. The gradient map of a Legendre type function is proven strictly comonotone with a global inverse. It thus yields a generalization of strictly mononotone and differentiable functions generating scalar quasi-arithmetic means. Furthermore, the Legendre transformation gives rise to pairs of dual quasi-arithmetic averages via the convex duality. We study the invariance and equivariance properties under affine transformations of quasi-arithmetic averages via the lens of dually flat spaces of information geometry. We show how these quasi-arithmetic averages are used to express points on dual geodesics and sided barycenters in the dual affine coordinate systems. We then consider quasi-arithmetic mixtures and describe several parametric and non-parametric statistical models which are closed under the quasi-arithmetic mixture operation.

翻译：我们通过考虑勒让德型实值函数的梯度映射，将拟算术均值推广至标量之外。证明表明，勒让德型函数的梯度映射具有严格共单调性且存在全局逆映射，从而实现了生成标量拟算术均值的严格单调可微函数的一般化。进一步地，勒让德变换通过凸对偶性导出了成对的对偶拟算术平均。我们通过信息几何中对偶平坦空间的视角，研究了拟算术平均在仿射变换下的不变性与等变性性质。揭示了这些拟算术平均如何用于表达对偶测地线上的点以及对偶仿射坐标系中的有向重心。随后我们探讨了拟算术混合，并描述了若干在拟算术混合运算下封闭的参数与非参数统计模型。