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Improved Bounds on Neural Complexity for Representing Piecewise Linear Functions
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Scale depth with the number of pieces using the max-min representation instead of constraining depth by input dimension | ReLU networks representing CPWL functions · CPWL functions with q pieces | hidden neurons (upper bound): \mathcal{O}\left(n2^{q^2}\right) -> \mathcal{O}(q^2) | Helped |
| Use the known number of linear components k as well as the number of pieces q | Proposed ReLU network · CPWL functions with k linear components and q pieces | hidden neurons (upper bound): \mathcal{O}(q^2) -> \mathcal{O}(kq) | Helped |
| Use the input dimension to tighten the number-of-pieces bound when only k is known | Proposed ReLU network · CPWL functions with k linear components and low-dimensional inputs | hidden neurons (upper bound): \mathcal{O}\left(k\cdot k!\right) -> \mathcal{O}\left(k^{2n+1}\right) | Helped |
| Use the proposed bound instead of the prior bound inferred from [cite] | ReLU networks representing CPWL functions · CPWL functions with k linear components | hidden neurons (upper bound): \mathcal{O}\left(k^{2n^2+3n+1}\log_2\left(n+1\right)\right) -> \mathcal{O}\left(k\cdot k!\right) | Helped |
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- Decomposition Polyhedra of Piecewise Linear Functions (ICLR 2025 · similarity 0.86)
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