# SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterization

arXiv: https://arxiv.org/abs/2609.33515 · HTML: https://mlsanity.com/paper/2609.33515/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 86 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 30 of 34)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification](https://mlsanity.com/paper/1602.02068/) (2016)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Differentiable Ranks and Sorting using Optimal Transport](https://mlsanity.com/paper/1905.11885/) (2019)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [Implicit Reparameterization Gradients](https://mlsanity.com/paper/1805.08498/) (2018)
- [Deep Lattice Networks and Partial Monotonic Functions](https://mlsanity.com/paper/1709.06680/) (2017)
- [Certified Monotonic Neural Networks](https://mlsanity.com/paper/2011.10219/) (2020)
- [Pathwise Derivatives Beyond the Reparameterization Trick](https://mlsanity.com/paper/1806.01851/) (2018)
- [Constrained Monotonic Neural Networks](https://mlsanity.com/paper/2205.11775/) (2022)
- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [Physics-Informed Neural Networks with Hard Linear Equality Constraints](https://mlsanity.com/paper/2402.07251/) (2024)
- [FSNet: Feasibility-Seeking Neural Network for Constrained Optimization with Guarantees](https://mlsanity.com/paper/2506.00362/) (2025)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [An introduction to sampling via measure transport](https://mlsanity.com/paper/1602.05023/) (2016)
- [Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraints](https://mlsanity.com/paper/2507.08124/) (2025)
- [End-to-End Probabilistic Framework for Learning with Hard Constraints](https://mlsanity.com/paper/2506.07003/) (2025)
- [Point and probabilistic forecast reconciliation for general linearly constrained multiple time series](https://mlsanity.com/paper/2305.05330/) (2023)
- [Smooth Min-Max Monotonic Networks](https://mlsanity.com/paper/2306.01147/) (2023)
- [Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues](https://mlsanity.com/paper/2303.17277/) (2023)
- [LMI-Net: Linear Matrix Inequality--Constrained Neural Networks via Differentiable Projection Layers](https://mlsanity.com/paper/2604.05374/) (2026)
- [Soft-Radial Projection for Constrained End-to-End Learning](https://mlsanity.com/paper/2602.03461/) (2026)
- [Hard Constraint Projection in a Physics Informed Neural Network](https://mlsanity.com/paper/2601.06244/) (2026)

## Abstract (arXiv)

Many probabilistic predictors must satisfy exact structure in every stochastic realization, yet common hard-constraint approaches form predictions in ambient coordinates and then correct or project them. We introduce SLP-ProbHard, a cross-family, representation-centered framework for probabilistic hard-constrained learning when explicit structural parameterizations are available. Its core object, a Structural Feasible Latent Parameterization (SFLP), combines a structural latent law Z ~ PZθ(·mid x) with a feasible map Y=hφ(x,Z) that satisfies the constraint for every latent realization. Together these components define the predictive law itself, including its support and boundary probabilities, rather than serving as a final feasibility wrapper. We study how feasible coordinates and maps affect stochastic dimension, dependence, calibration, expressiveness, and computation. Experiments use Gaussian latent laws and fixed geometry-derived maps across affine equalities, ordering and simplex constraints, nonlinear manifolds, and three structural representations of seven-basin hydrological flow-duration-curve (FDC) data. In an official-source affine comparison with ProbHardE2E/DPPL, both methods achieve zero practical constraint violations. SLP-ProbHard uses 8 instead of 11 stochastic coordinates and improves MSE/MAE, while DPPL yields better marginal CRPS and closer-to-nominal coverage; a paired test detects no Energy Score difference across ten seeds. Real-world affine and nonlinear FDC representations reduce 13 to 7 and 14 to 8 ambient versus computational coordinates, respectively. Exact feasibility alone thus does not determine a predictive law, motivating direct structural generation when meaningful feasible coordinates are available.