Inverse problems and variational auto-encoders

This post is going to trace a line from inverse probability problems to variational auto-encoders, where nothing really changes except for the symbols and terminology. An underlying concept that will remain intact is that we have a deterministic forward function, and from among the set of input parameters for this function, it makes sense to optimize over some of them and to integrate over the rest. Integration in the inverse setting relies on the stability of Gaussian distributions through linear maps in order to convert this integration to a linear transformation involving covariance matrices. Integration in the neural network setting arises as latent variable sampling that is amortized during training.

I'll start from a tangible problem: estimate the reflectance of terrestrial surfaces from satellite images. I'll follow the approach of . The reflectance of a surface is the fraction of incident light that is reflected as a function of wavelength. There is camera taking hyperspectral images, measuring radiance, and the images will depend on the illumination from the sun, the reflectance properties of the surface and on various atmospheric conditions affecting light.