# Learning PSD-valued functions using kernel sums-of-squares

arXiv: https://arxiv.org/abs/2111.11306 · HTML: https://mlsanity.com/paper/2111.11306/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 36 references that are in the map.
- First posted on arXiv 22 Nov 2021.

## Builds on (most cited first, 9 of 9)

- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [PSD Representations for Effective Probability Models](https://mlsanity.com/paper/2106.16116/) (2021)
- [A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation](https://mlsanity.com/paper/2101.05380/) (2021)
- [Infinite-Dimensional Sums-of-Squares for Optimal Control](https://mlsanity.com/paper/2110.07396/) (2021)
- [Handling Hard Affine SDP Shape Constraints in RKHSs](https://mlsanity.com/paper/2101.01519/) (2021)
- [A Note on Optimizing Distributions using Kernel Mean Embeddings](https://mlsanity.com/paper/2106.09994/) (2021)

## Built on by (most cited first, 5 of 5)

- [Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence](https://mlsanity.com/paper/2305.15557/) (2023)
- [Approximation of optimization problems with constraints through kernel Sum-Of-Squares](https://mlsanity.com/paper/2301.06339/) (2023)
- [Scalable and adaptive prediction bands with kernel sum-of-squares](https://mlsanity.com/paper/2505.21039/) (2025)
- [The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations](https://mlsanity.com/paper/2505.11622/) (2025)
- [Kernel-Based Nonparametric Tests For Shape Constraints](https://mlsanity.com/paper/2510.16745/) (2025)

## Abstract (arXiv)

Shape constraints such as positive semi-definiteness (PSD) for matrices or convexity for functions play a central role in many applications in machine learning and sciences, including metric learning, optimal transport, and economics. Yet, very few function models exist that enforce PSD-ness or convexity with good empirical performance and theoretical guarantees. In this paper, we introduce a kernel sum-of-squares model for functions that take values in the PSD cone, which extends kernel sums-of-squares models that were recently proposed to encode non-negative scalar functions. We provide a representer theorem for this class of PSD functions, show that it constitutes a universal approximator of PSD functions, and derive eigenvalue bounds in the case of subsampled equality constraints. We then apply our results to modeling convex functions, by enforcing a kernel sum-of-squares representation of their Hessian, and show that any smooth and strongly convex function may be thus represented. Finally, we illustrate our methods on a PSD matrix-valued regression task, and on scalar-valued convex regression.