# Handling Hard Affine SDP Shape Constraints in RKHSs

arXiv: https://arxiv.org/abs/2101.01519 · HTML: https://mlsanity.com/paper/2101.01519/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 72 references that are in the map.
- First posted on arXiv 5 Jan 2021.

## Builds on (most cited first, 13 of 13)

- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Finding Global Minima via Kernel Approximations](https://mlsanity.com/paper/2012.11978/) (2020)
- [A Computational Framework for Multivariate Convex Regression and its Variants](https://mlsanity.com/paper/1509.08165/) (2015)
- [Gaussian processes with linear operator inequality constraints](https://mlsanity.com/paper/1901.03134/) (2019)
- [Hard Shape-Constrained Kernel Machines](https://mlsanity.com/paper/2005.12636/) (2020)
- [Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression](https://mlsanity.com/paper/1903.05315/) (2019)
- [Nonparametric Shape-restricted Regression](https://mlsanity.com/paper/1709.05707/) (2017)
- [Infinite-Task Learning with RKHSs](https://mlsanity.com/paper/1805.08809/) (2018)
- [Optimization over Nonnegative and Convex Polynomials With and Without Semidefinite Programming](https://mlsanity.com/paper/1806.06996/) (2018)
- [Nonlinear Functional Output Regression: a Dictionary Approach](https://mlsanity.com/paper/2003.01432/) (2020)
- [Linearly-constrained Linear Quadratic Regulator from the viewpoint of kernel methods](https://mlsanity.com/paper/2011.02196/) (2020)

## Built on by (most cited first, 4 of 4)

- [Learning PSD-valued functions using kernel sums-of-squares](https://mlsanity.com/paper/2111.11306/) (2021)
- [The Minimax Lower Bound of Kernel Stein Discrepancy Estimation](https://mlsanity.com/paper/2510.15058/) (2025)
- [Approximation of optimization problems with constraints through kernel Sum-Of-Squares](https://mlsanity.com/paper/2301.06339/) (2023)
- [Kernel-Based Nonparametric Tests For Shape Constraints](https://mlsanity.com/paper/2510.16745/) (2025)

## Abstract (arXiv)

Shape constraints, such as non-negativity, monotonicity, convexity or supermodularity, play a key role in various applications of machine learning and statistics. However, incorporating this side information into predictive models in a hard way (for example at all points of an interval) for rich function classes is a notoriously challenging problem. We propose a unified and modular convex optimization framework, relying on second-order cone (SOC) tightening, to encode hard affine SDP constraints on function derivatives, for models belonging to vector-valued reproducing kernel Hilbert spaces (vRKHSs). The modular nature of the proposed approach allows to simultaneously handle multiple shape constraints, and to tighten an infinite number of constraints into finitely many. We prove the convergence of the proposed scheme and that of its adaptive variant, leveraging geometric properties of vRKHSs. Due to the covering-based construction of the tightening, the method is particularly well-suited to tasks with small to moderate input dimensions. The efficiency of the approach is illustrated in the context of shape optimization, safety-critical control, robotics and econometrics.