  Solving geometric tasks involving point clouds by using machine learning is a
challenging problem. Standard feed-forward neural networks combine linear or,
if the bias parameter is included, affine layers and activation functions.
Their geometric modeling is limited, which motivated the prior work introducing
the multilayer hypersphere perceptron (MLHP). Its constituent part, i.e., the
hypersphere neuron, is obtained by applying a conformal embedding of Euclidean
space. By virtue of Clifford algebra, it can be implemented as the Cartesian
dot product of inputs and weights. If the embedding is applied in a manner
consistent with the dimensionality of the input space geometry, the decision
surfaces of the model units become combinations of hyperspheres and make the
decision-making process geometrically interpretable for humans. Our extension
of the MLHP model, the multilayer geometric perceptron (MLGP), and its
respective layer units, i.e., geometric neurons, are consistent with the 3D
geometry and provide a geometric handle of the learned coefficients. In
particular, the geometric neuron activations are isometric in 3D, which is
necessary for rotation and translation equivariance. When classifying the 3D
Tetris shapes, we quantitatively show that our model requires no activation
function in the hidden layers other than the embedding to outperform the
vanilla multilayer perceptron. In the presence of noise in the data, our model
is also superior to the MLHP.
