The Gaussian Radon Transform for Banach Spaces

The classical Radon transform can be thought of as a way to obtain the density of an n-dimensional object from its (n-1)-dimensional sections in diff_x001B_erent directions. A generalization of this transform to infi_x001C_nite-dimensional spaces has the potential to allow one to obtain a function de_x001C_fined on an infi_x001C_nite-dimensional space from its conditional expectations. We work within a standard framework in in_x001C_finite-dimensional analysis, that of abstract Wiener spaces, developed by L. Gross. The main obstacle in infinite dimensions is the absence of a useful version of Lebesgue measure. To overcome this, we work with Gaussian measures. Specifically, we construct Gaussian measures concentrated on closed affine subspaces of infinite-dimensional Banach spaces, and use these measures to define the Gaussian Radon transform. We provide for this transform a disintegration theorem, an inversion procedure and explore possible applications to machine learning