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M-Representation of Polytopes
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We introduce the M-representation of polytopes, which makes it possible to compute linear transformations, convex hulls, and Minkowski sums with linear complexity in the dimension of the polytopes. When the polytope is a convex hull of a zonotope and a polytope, the representation size can be smaller than any of the known representations (V-representation, H-representation, and Z-representation). We also provide a variant of the M-representation: The chain representation is more compact and we can directly use it to compute linear transformations and convex hulls -- for all other operations on the chain representation, one requires a conversion to the M-representation.
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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.
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