# Cubic-Regularized Newton for Spectral Constrained Matrix Optimization and its Application to Fairness

arXiv: https://arxiv.org/abs/2209.01229 · HTML: https://mlsanity.com/paper/2209.01229/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 38 references that are in the map.
- First posted on arXiv 2 Sep 2022.

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

- [A Survey on Bias and Fairness in Machine Learning](https://mlsanity.com/paper/1908.09635/) (2019)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [The Price of Fair PCA: One Extra Dimension](https://mlsanity.com/paper/1811.00103/) (2018)
- [Convex Formulations for Fair Principal Component Analysis](https://mlsanity.com/paper/1802.03765/) (2018)
- [Gradient Descent Finds the Cubic-Regularized Non-Convex Newton Step](https://mlsanity.com/paper/1612.00547/) (2016)
- [Adaptive regularization with cubics on manifolds](https://mlsanity.com/paper/1806.00065/) (2018)
- [Rigorous Guarantees for Tyler's M-estimator via quantum expansion](https://mlsanity.com/paper/2002.00071/) (2020)
- [Distributionally Robust Fair Principal Components via Geodesic Descents](https://mlsanity.com/paper/2202.03071/) (2022)
- [A Cubic Regularized Newton's Method over Riemannian Manifolds](https://mlsanity.com/paper/1805.05565/) (2018)
- [An accelerated first-order method with complexity analysis for solving cubic regularization subproblems](https://mlsanity.com/paper/1911.12545/) (2019)
- [Frank-Wolfe-based Algorithms for Approximating Tyler's M-estimator](https://mlsanity.com/paper/2206.09370/) (2022)
- [Cubic regularization methods with second-order complexity guarantee based on a new subproblem reformulation](https://mlsanity.com/paper/2112.09291/) (2021)
- [On global minimizers of quadratic functions with cubic regularization](https://mlsanity.com/paper/1711.01303/) (2017)

## Abstract (arXiv)

Matrix functions are utilized to rewrite smooth spectral constrained matrix optimization problems as smooth unconstrained problems over the set of symmetric matrices which are then solved via the cubic-regularized Newton method. A second-order chain rule identity for matrix functions is proven to compute the higher-order derivatives to implement cubic-regularized Newton, and a new convergence analysis is provided for cubic-regularized Newton for matrix vector spaces. We demonstrate the applicability of our approach by conducting numerical experiments on both synthetic and real datasets. In our experiments, we formulate a new model for estimating fair and robust covariance matrices in the spirit of the Tyler's M-estimator (TME) model and demonstrate its advantage.