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Bounding and Counting Linear Regions of Deep Neural Networks
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the dimension-aware upper bound instead of the previous bound | ReLU networks · Networks with arbitrary layer widths and input dimension | upper bound on linear regions | Helped |
| Construct the one-dimensional zigzag with n+1 regions rather than n | ReLU networks · Input dimension one | maximal number of linear regions: n+1 regions instead of n | Helped |
| Use the improved one-dimensional construction in the lower-bound construction | ReLU networks · n_l \geq 3n_0 | lower bound on maximal number of linear regions: \left\lfloor n_l/n_0 \right\rfloor + 1 | Helped |
| Use the one-dimensional construction in the remaining layers | ReLU networks · Networks of size 2m + w(L-1) | linear regions: w+1 replaces w | Helped |
| Use a shallow network with Ln neurons instead of L layers of width n | ReLU networks · n_0 \geq Ln | maximal number of linear regions: R(n_0, \underbrace{n, \ldots, n}_{\text{$L$ times}}) < R(n_0, Ln) | Helped |
| Use the dimension-aware upper bound to count exact regions with a mixed-integer linear formulation | ReLU networks · Bounded polyhedral input domains | exact region counting | Helped |
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