  This paper proposes a novel method for deep learning based on the analytical
convolution of multidimensional Gaussian mixtures. In contrast to tensors,
these do not suffer from the curse of dimensionality and allow for a compact
representation, as data is only stored where details exist. Convolution kernels
and data are Gaussian mixtures with unconstrained weights, positions, and
covariance matrices. Similar to discrete convolutional networks, each
convolution step produces several feature channels, represented by independent
Gaussian mixtures. Since traditional transfer functions like ReLUs do not
produce Gaussian mixtures, we propose using a fitting of these functions
instead. This fitting step also acts as a pooling layer if the number of
Gaussian components is reduced appropriately. We demonstrate that networks
based on this architecture reach competitive accuracy on Gaussian mixtures
fitted to the MNIST and ModelNet data sets.
