On the Topology of Neural Network Superlevel Sets
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Abstract (arXiv)
We show that neural networks with activations satisfying a Riccati-type ordinary differential equation condition, an assumption arising in recent universal approximation results in the uniform topology, produce Pfaffian outputs on analytic domains with format controlled only by the architecture. Consequently, superlevel sets, as well as Lie bracket rank drop loci for neural network parameterized vector fields, admit architecture-only bounds on topological complexity, in particular on total Betti numbers, uniformly over all weights.
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References · 7
- 2014On the Number of Linear Regions of Deep Neural Networkscited by 381
- 2016On the Expressive Power of Deep Neural Networkscited by 306
- 2016Benefits of depth in neural networkscited by 250
- 2020Topology of deep neural networkscited by 112
- 2018On Characterizing the Capacity of Neural Networks using Algebraic Topologycited by 38
- 2020Universal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theorycited by 18
- 2023Topological Expressivity of ReLU Neural Networkscited by 10
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