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Consider an information processing algorithm that is designed to process an input data object onto an output data object via a number of successive internal {\it layers} and mappings between them. The possible activation state within each layer can be represented as a cube within Euclidean space of a high dimension (e.g. equal to the number of artificial neurons at that level). Multiple instances of such input objects produce a point cloud within each layer’s cube: this is the “representation of the reality” at that layer, as sampled by the set of input objects.

Most neural networks reduce the dimension of each layer’s cube from layer to successive layer. This gives the false impression of refining the inner representations of reality, distilling it down to fewer dimensions from which to discriminate or to infer outcomes (whatever is the aim). However, the representation of reality realised within each layer’s cube is a manifold, a curved subset embedded within it and of much lower dimension. Investigations show that such manifolds may not always be reducing in their local dimension. Instead, the manifold may become folded over and over, filling up further dimensions and creating non-realistic (unforeseeable) proximities.

We discuss some of the likely consequences of these relatively unforeseen characteristics and, in particular, the possible vulnerability of such algorithms to non-realistic perturbations. We consider a possible response to this issue.

New forms of calibration are necessary, using geometric/topological loss functions, as opposed to simple (variation-limiting) regularisation terms.

We apply persistent homology methods to understand how the images of the point cloud (representing the sampled reality) change as they pass from layer to layer.