# Certifying the absence of spurious local minima at infinity

arXiv: https://arxiv.org/abs/2303.03536 · HTML: https://mlsanity.com/paper/2303.03536/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Mar 2023.

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

- [SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structures](https://mlsanity.com/paper/2505.09572/) (2025)
- [Flat Channels to Infinity in Neural Loss Landscapes](https://mlsanity.com/paper/2506.14951/) (2025)

## Abstract (arXiv)

When searching for global optima of nonconvex unconstrained optimization problems, it is desirable that every local minimum be a global minimum. This property of having no spurious local minima is true in various problems of interest nowadays, including principal component analysis, matrix sensing, and linear neural networks. However, since these problems are non-coercive, they may yet have spurious local minima at infinity. The classical tools used to analyze the optimization landscape, namely the gradient and the Hessian, are incapable of detecting spurious local minima at infinity. In this paper, we identify conditions that certify the absence of spurious local minima at infinity, one of which is having bounded subgradient trajectories. We check that they hold in several applications of interest.