  In this work, we introduce 6D Convolutional Neural Network (6DCNN) designed
to tackle the problem of detecting relative positions and orientations of local
patterns when processing three-dimensional volumetric data. 6DCNN also includes
SE(3)-equivariant message-passing and nonlinear activation operations
constructed in the Fourier space. Working in the Fourier space allows
significantly reducing the computational complexity of our operations. We
demonstrate the properties of the 6D convolution and its efficiency in the
recognition of spatial patterns. We also assess the 6DCNN model on several
datasets from the recent CASP protein structure prediction challenges. Here,
6DCNN improves over the baseline architecture and also outperforms the state of
the art.
