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，将拟算术均值推广到标量之外。勒让德型函数的梯度映射被证明是严格共单调的且具有全局逆，从而推广了生成标量拟算术均值的严格单调可微函数。进一步地，勒让德变换通过凸对偶性产生对偶拟算术平均对。借助信息几何的对偶平坦空间视角，我们研究了拟算术平均在仿射变换下的不变性与等变性。我们展示了如何利用这些拟算术平均在对偶仿射坐标系中表示对偶测地线上的点与旁侧重心。随后，我们考虑拟算术混合，并描述了几类在拟算术混合运算下封闭的参数化与非参数化统计模型。