Abstract
This paper develops Sobolev variants of the non-parametric statistical manifolds appearing in [10] and [11]. The manifolds are modelled on a particular class of weighted, mixed-norm Sobolev spaces, including a Hilbert-Sobolev space. Densities are expressed in terms of a deformed exponential function having linear growth, which lifts to a continuous nonlinear superposition (Nemytskii) operator. This property is used in the construction of finite-dimensional mixture and exponential submanifolds, on which approximations can be based. The manifolds of probability measures are developed in their natural setting, as embedded submanifolds of those of finite measures.
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