For ReLU networks, the homology of the output representation is isomorphic to the homology of the input manifold quotiented by the network's overlap decomposition, when polyhedron-manifold intersections are convex.
Representation of Polytopes as Polynomial Zonotopes
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abstract
We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes. Previous representations are the vertex representation (V-representation) and the halfspace representation (H-representation). Depending on the polytope, the Z-representation can be more compact than the V-representation and the H-representation. In addition, the Z-representation enables the computation of linear maps, Minkowski addition, and convex hull with a computational complexity that is polynomial in the representation size. The usefulness of the new representation is demonstrated by range bounding within polytopes.
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A Quotient Homology Theory of Representation in Neural Networks
For ReLU networks, the homology of the output representation is isomorphic to the homology of the input manifold quotiented by the network's overlap decomposition, when polyhedron-manifold intersections are convex.