  We present an alternative layer to convolution layers in convolutional neural
networks (CNNs). Our approach reduces the complexity of convolutions by
replacing it with binary decisions. Those binary decisions are used as indexes
to conditional distributions where each weight represents a leaf in a decision
tree. This means that only the indices to the weights need to be determined
once, thus reducing the complexity of convolutions by the depth of the output
tensor. Index computation is performed by simple binary decisions that require
fewer cycles compared to conventionally used multiplications. In addition, we
show how convolutions can be replaced by binary decisions. These binary
decisions form indices in the conditional distributions and we show how they
are used to replace 2D weight matrices as well as 3D weight tensors. These new
layers can be trained like convolution layers in CNNs based on the
backpropagation algorithm, for which we provide a formalization.
  Our results on multiple publicly available data sets show that our approach
performs similar to conventional neuronal networks. Beyond the formalized
reduction of complexity and the improved qualitative performance, we show the
runtime improvement empirically compared to convolution layers.
