# Spectrahedral Regression

arXiv: https://arxiv.org/abs/2110.14779 · HTML: https://mlsanity.com/paper/2110.14779/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 31 references that are in the map.
- First posted on arXiv 27 Oct 2021.

## Builds on (most cited first, 5 of 5)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Fitting Tractable Convex Sets to Support Function Evaluations](https://mlsanity.com/paper/1903.04194/) (2019)
- [Convex Regression in Multidimensions: Suboptimality of Least Squares Estimators](https://mlsanity.com/paper/2006.02044/) (2020)
- [On Suboptimality of Least Squares with Application to Estimation of Convex Bodies](https://mlsanity.com/paper/2006.04046/) (2020)
- [Lifting for Simplicity: Concise Descriptions of Convex Sets](https://mlsanity.com/paper/2002.09788/) (2020)

## Built on by (most cited first, 2 of 2)

- [An Efficient Minimax Optimal Estimator For Multivariate Convex Regression](https://mlsanity.com/paper/2205.03368/) (2022)
- [The Spectral Neuron](https://mlsanity.com/paper/2608.08003/) (2026)

## Abstract (arXiv)

Convex regression is the problem of fitting a convex function to a data set consisting of input-output pairs. We present a new approach to this problem called spectrahedral regression, in which we fit a spectrahedral function to the data, i.e. a function that is the maximum eigenvalue of an affine matrix expression of the input. This method represents a significant generalization of polyhedral (also called max-affine) regression, in which a polyhedral function (a maximum of a fixed number of affine functions) is fit to the data. We prove bounds on how well spectrahedral functions can approximate arbitrary convex functions via statistical risk analysis. We also analyze an alternating minimization algorithm for the non-convex optimization problem of fitting the best spectrahedral function to a given data set. We show that this algorithm converges geometrically with high probability to a small ball around the optimal parameter given a good initialization. Finally, we demonstrate the utility of our approach with experiments on synthetic data sets as well as real data arising in applications such as economics and engineering design.