Here is a specific tweak to FastFood in order to enable a large set of kernels to better fit group theoretic transformation. It is a little bit a similar path taken in the ScatNet approach.

A la Carte - Learning Fast Kernels by Zichao Yang, Alexander J. Smola, Le Song, Andrew Gordon Wilson

Kernel methods have great promise for learning rich statistical representations of large modern datasets. However, compared to neural networks, kernel methods have been perceived as lacking in scalability and flexibility. We introduce a family of fast, flexible, lightly parametrized and general purpose kernel learning methods, derived from Fastfood basis function expansions. We provide mechanisms to learn the properties of groups of spectral frequencies in these expansions, which require only O(mlogd) time and O(m) memory, for m basis functions and d input dimensions. We show that the proposed methods can learn a wide class of kernels, outperforming the alternatives in accuracy, speed, and memory consumption.

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