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- W3065170192 abstract "Let $F: mathbb{R}^n rightarrow mathbb{R}$ be a feedforward ReLU neural network. It is well-known that for any choice of parameters, $F$ is continuous and piecewise affine-linear. We lay some foundations for a systematic investigation of how the architecture of $F$ impacts the geometry and topology of its possible decision regions, $F^{-1}((-infty, t))$ and $F^{-1}((t, infty))$, for binary classification tasks. Following the classical progression for smooth functions in differential topology, we first define the notion of a generic, transversal ReLU neural network and show that almost all ReLU networks are generic and transversal. We then define a partially oriented linear 1-complex in the domain of $F$ and identify properties of this complex that yield an obstruction to the existence of bounded connected components of a decision region. We use this obstruction to prove that a decision region of a generic, transversal ReLU network $F: mathbb{R}^n rightarrow mathbb{R}$ with a single hidden layer of dimension $n+1$ can have no more than one bounded connected component." @default.
- W3065170192 created "2020-08-24" @default.
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- W3065170192 date "2022-04-25" @default.
- W3065170192 modified "2023-09-29" @default.
- W3065170192 title "On Transversality of Bent Hyperplane Arrangements and the Topological Expressiveness of ReLU Neural Networks" @default.
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- W3065170192 doi "https://doi.org/10.1137/20m1368902" @default.
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