On the Wavelet Scattering Transform and its Generalizations

In this thesis, we look into generalizations of Mallat's wavelet scattering transform. In the second chapter, we generalize finite depth wavelet scattering transforms, which we formulate as L^q norms of a cascade of continuous wavelet transforms (or dyadic wavelet transforms) and contractive nonlinearities. We then provide norms for these operators, prove that these operators are well-defined, and are Lipschitz continuous to the action of C^2 diffeomorphisms in specific cases; additionally, we extend our results to formulate an operator invariant to the action of rotations and an operator that is equivariant to the action of rotations. In the third and fourth chapters, we generalize our results to stochastic process and signals on compact manifolds, respectively.