Soft-Radial Projection for Constrained End-to-End Learning
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Abstract (arXiv)
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.
Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 19 of its 38 references that are in the map. First posted on arXiv 3 Feb 2026.
Builds on
19 of 38 references.
References · 19
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7 more
- 2024End-to-end Conditional Robust Optimizationcited by 17
- 2025Pinet: Optimizing hard-constrained neural networks with orthogonal projection layerscited by 17
- 2023A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraintscited by 12
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- 2025HoP: Homeomorphic Polar Learning for Hard Constrained Optimizationcited by 1
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Citing papers per year: 2026: 3.
3 papers.