提出软径向投影层，解决约束深度学习中梯度消失问题。
Soft-Radial Projection for Constrained End-to-End Learning
- 用径向映射将输出映射到可行域内部，避免边界投影的梯度饱和
- 理论证明可保留通用逼近能力，实验显示收敛更快、解更优
- 适合安全关键系统中的端到端约束学习，如机器人控制
将硬性约束融入深度学习对安全关键系统至关重要。现有基于构造层的投影方法在将预测投影到约束边界时存在根本瓶颈：梯度饱和。标准正交投影会将外部点压缩到低维流形上，导致雅可比矩阵秩不足，使与活动约束正交的梯度归零，阻碍优化。本文提出软径向投影（Soft-Radial Projection），一种可微重构层，通过从欧氏空间到可行域内部的径向映射规避此问题。该设计保证严格可行性的同时，在几乎处处保持满秩雅可比，防止边界方法常见的优化停滞。理论上证明该架构保持通用逼近能力，实验证明其在收敛速度和解质量上优于当前最先进的优化与投影基基线方法。
原文摘要 · Abstract (English)
Integrating hard constraints into deep learning is essential for safety-critical systems. Yet existing constructive layers that project predictions onto constraint boundaries face a fundamental bottleneck: gradient saturation. By collapsing exterior points onto lower-dimensional surfaces, standard orthogonal projections induce rank-deficient Jacobians, which nullify gradients orthogonal to active constraints and hinder optimization. We introduce Soft-Radial Projection, a differentiable reparameterization layer that circumvents this issue through a radial mapping from Euclidean space into the interior of the feasible set. This construction guarantees strict feasibility while preserving a full-rank Jacobian almost everywhere, thereby preventing the optimization stalls typical of boundary-based methods. We theoretically prove that the architecture retains the universal approximation property and empirically show improved convergence behavior and solution quality over state-of-the-art optimization- and projection-based baselines.
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