# Variational analysis of spectral functions simplified

arXiv: https://arxiv.org/abs/1506.05170 · HTML: https://mlsanity.com/paper/1506.05170/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2015.

## Built on by (most cited first, 3 of 3)

- [Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization](https://mlsanity.com/paper/2108.11832/) (2021)
- [On the Partial Convexification for Low-Rank Spectral Optimization: Rank Bounds and Algorithms](https://mlsanity.com/paper/2305.07638/) (2023)
- [The radius of statistical efficiency](https://mlsanity.com/paper/2405.09676/) (2024)

## Abstract (arXiv)

Spectral functions of symmetric matrices -- those depending on matrices only through their eigenvalues -- appear often in optimization. A cornerstone variational analytic tool for studying such functions is a formula relating their subdifferentials to the subdifferentials of their diagonal restrictions. This paper presents a new, short, and revealing derivation of this result. We then round off the paper with an illuminating derivation of the second derivative of twice differentiable spectral functions, highlighting the underlying geometry. All of our arguments have direct analogues for spectral functions of Hermitian matrices, and for singular value functions of rectangular matrices.