Classical ReLU-based Input Convex Neural Networks (ICNNs) are equivalent to the optimal value functions of Linear Programming (LP). This intrinsic structural equivalence restricts their representational capacity to piecewise-linear polyhedral functions. To overcome this representational bottleneck, we propose the SOC-ICNN, an architecture that generalizes the underlying optimization class from LP to Second-Order Cone Programming (SOCP). By explicitly injecting positive semi-definite curvature and Euclidean norm-based conic primitives, our formulation introduces native smooth curvature into the representation while preserving a rigorous optimization-theoretic interpretation. We formally prove that SOC-ICNNs strictly expand the representational space of ReLU-ICNNs without increasing the asymptotic order of forward-pass complexity. Extensive experiments demonstrate that SOC-ICNN substantially improves function approximation, while delivering competitive downstream decision quality. The code is available at https://github.com/Kanyooo/SOC-ICNN.

翻译：摘要：基于经典ReLU的输入凸神经网络（ICNN）等价于线性规划（LP）的最优值函数。这种内在的结构等价性将其表示能力限制为分段线性多面体函数。为克服这一表示瓶颈，我们提出SOC-ICNN架构，该架构将底层优化类别从LP推广至二阶锥规划（SOCP）。通过显式注入半正定曲率与基于欧几里得范数的锥原语，我们的公式在保持严格优化理论解释的同时，为表示引入了固有的平滑曲率。我们严格证明了SOC-ICNN在不增加前向计算渐近复杂度阶数的前提下，严格扩展了ReLU-ICNN的表示空间。大量实验表明，SOC-ICNN显著提升了函数逼近性能，并提供了具有竞争力的下游决策质量。代码见https://github.com/Kanyooo/SOC-ICNN。