A Relative Homology Theory of Representation in Neural Networks

Previous research has proven that the set of maps implemented by neural networks with a ReLU activation function is identical to the set of piecewise linear continuous maps. Furthermore, such networks induce a hyperplane arrangement splitting the input domain into convex polyhedra $G_J$ over which the network $\Phi$ operates in an affine manner. In this work, we leverage these properties to define the equivalence class of inputs $\sim_\Phi$, which can be split into two sets related to the local rank of $\Phi_J$ and the intersections $\cap \text{Im}\Phi_{J_i}$. We refer to the latter as the overlap decomposition $O_\Phi$ and prove that if the intersections between each polyhedron and the input manifold are convex, the homology groups of neural representations are isomorphic to relative homology groups $H_k(\Phi(M)) \simeq H_k(M,O_\Phi)$. This lets us compute Betti numbers without the choice of an external metric. We develop methods to numerically compute the overlap decomposition through linear programming and a union-find algorithm. Using this framework, we perform several experiments on toy datasets showing that, compared to standard persistent homology, our relative homology-based computation of Betti numbers tracks purely topological rather than geometric features. Finally, we study the evolution of the overlap decomposition during training on various classification problems while varying network width and depth and discuss some shortcomings of our method.

A rank decomposition for the topological classification of neural representations

Neural networks can be thought of as applying a transformation to an input dataset. The way in which they change the topology of such a dataset often holds practical significance for many tasks, particularly those demanding non-homeomorphic mappings for optimal solutions, such as classification problems. In this work, we leverage the fact that neural networks are equivalent to continuous piecewise-affine maps, whose rank can be used to pinpoint regions in the input space that undergo non-homeomorphic transformations, leading to alterations in the topological structure of the input dataset. Our approach enables us to make use of the relative homology sequence, with which one can study the homology groups of the quotient of a manifold $\mathcal{M}$ and a subset $A$, assuming some minimal properties on these spaces. As a proof of principle, we empirically investigate the presence of low-rank (topology-changing) affine maps as a function of network width and mean weight. We show that in randomly initialized narrow networks, there will be regions in which the (co)homology groups of a data manifold can change. As the width increases, the homology groups of the input manifold become more likely to be preserved. We end this part of our work by constructing highly non-random wide networks that do not have this property and relating this non-random regime to Dale's principle, which is a defining characteristic of biological neural networks. Finally, we study simple feedforward networks trained on MNIST, as well as on toy classification and regression tasks, and show that networks manipulate the topology of data differently depending on the continuity of the task they are trained on.

Isometric Representations in Neural Networks Improve Robustness

Artificial and biological agents cannon learn given completely random and unstructured data. The structure of data is encoded in the metric relationships between data points. In the context of neural networks, neuronal activity within a layer forms a representation reflecting the transformation that the layer implements on its inputs. In order to utilize the structure in the data in a truthful manner, such representations should reflect the input distances and thus be continuous and isometric. Supporting this statement, recent findings in neuroscience propose that generalization and robustness are tied to neural representations being continuously differentiable. In machine learning, most algorithms lack robustness and are generally thought to rely on aspects of the data that differ from those that humans use, as is commonly seen in adversarial attacks. During cross-entropy classification, the metric and structural properties of network representations are usually broken both between and within classes. This side effect from training can lead to instabilities under perturbations near locations where such structure is not preserved. One of the standard solutions to obtain robustness is to add ad hoc regularization terms, but to our knowledge, forcing representations to preserve the metric structure of the input data as a stabilising mechanism has not yet been studied. In this work, we train neural networks to perform classification while simultaneously maintaining within-class metric structure, leading to isometric within-class representations. Such network representations turn out to be beneficial for accurate and robust inference. By stacking layers with this property we create a network architecture that facilitates hierarchical manipulation of internal neural representations. Finally, we verify that isometric regularization improves the robustness to adversarial attacks on MNIST.