Convolutions are fundamental elements in deep learning architectures. Here, we present a theoretical framework for combining extrinsic and intrinsic approaches to manifold convolution through isometric embeddings into tori. In this way, we define a convolution operator for a manifold of arbitrary topology and dimension. We also explain geometric and topological conditions that make some local definitions of convolutions which rely on translating filters along geodesic paths on a manifold, computationally intractable. A result of Alan Turing from 1938 underscores the need for such a toric isometric embedding approach to achieve a global definition of convolution on computable, finite metric space approximations to a smooth manifold.

The emerging subject of deep learning mathematical analysis [1] has been tasked with answering some "mysterious" facts that appear to be inexplicable using traditional mathematical methodologies. They are attempting to comprehend what a neural network actually does. Deep Neural Networks (DNN) transform data at each layer, producing a new representation as output. DNN attempts to divide data in a classification problem, enhancing this action layer by layer until it reaches an output layer when DNN provides its best possible result. Under the manifold hypothesis (natural data creates lower-dimensional manifolds in its embedding space), this task can be viewed as the separation of lower-dimensional manifolds in a data space. DNN layers are linked by a realization function, Φ (an affine transformation) and a component-wise activation function, ρ. Consider the fully connected feedforward neural network depicted in Figure 2. The network architecture can be described by defining the number of layers N, L, the number of neurons, and the activation function.