Neural Control Barrier Functions (NCBFs) have shown significant promise in enforcing safety constraints on nonlinear autonomous systems. State-of-the-art exact approaches to verifying safety of NCBF-based controllers exploit the piecewise-linear structure of ReLU neural networks, however, such approaches still rely on enumerating all of the activation regions of the network near the safety boundary, thus incurring high computation cost. In this paper, we propose a framework for Synthesis with Efficient Exact Verification (SEEV). Our framework consists of two components, namely (i) an NCBF synthesis algorithm that introduces a novel regularizer to reduce the number of activation regions at the safety boundary, and (ii) a verification algorithm that exploits tight over-approximations of the safety conditions to reduce the cost of verifying each piecewise-linear segment. Our simulations show that SEEV significantly improves verification efficiency while maintaining the CBF quality across various benchmark systems and neural network structures.

In this article, we attempt to capture the difference between the maximum complexity of deep networks and the complexity of functions that are actually learned (see Figure 1).

The appendix is organized as follows. Appendix A Proofs related to activation patterns and activation regions. Appendix B Proofs related to the numbers of regions attained with positive probability. Appendix D Proofs related to the expected volume of activation regions. Appendix E Proofs related to the expected number of activation regions.

Learning with neural networks relies on the complexity of the representable functions, but more importantly, the particular assignment of typical parameters to functions of different complexity.

This assists in analysis and provides a foundation for further research.

In this article, we attempt to capture the difference between the maximum complexity of deep networks and the complexity of functions that are actually learned (see Figure 1).

We extend the ReLU Transition Graph (RTG) framework into a comprehensive graph-theoretic model for understanding deep ReLU networks. In this model, each node represents a linear activation region, and edges connect regions that differ by a single ReLU activation flip, forming a discrete geometric structure over the network's functional behavior. We prove that RTGs at random initialization exhibit strong expansion, binomial degree distributions, and spectral properties that tightly govern generalization. These structural insights enable new bounds on capacity via region entropy and on generalization via spectral gap and edge-wise KL divergence. Empirically, we construct RTGs for small networks, measure their smoothness and connectivity properties, and validate theoretical predictions. Our results show that region entropy saturates under overparameterization, spectral gap correlates with generalization, and KL divergence across adjacent regions reflects functional smoothness. This work provides a unified framework for analyzing ReLU networks through the lens of discrete functional geometry, offering new tools to understand, diagnose, and improve generalization.

The appendix is organized as follows. Appendix A Proofs related to activation patterns and activation regions. Appendix B Proofs related to the numbers of regions attained with positive probability. Appendix D Proofs related to the expected volume of activation regions. Appendix E Proofs related to the expected number of activation regions.

This assists in analysis and provides a foundation for further research.