# The Landscape of Non-convex Empirical Risk with Degenerate Population Risk

arXiv: https://arxiv.org/abs/1907.05520 · HTML: https://mlsanity.com/paper/1907.05520/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2019.

## Built on by (most cited first, 1 of 1)

- [Direction-Magnitude Decomposition for Low-Rank Matrix Optimization: Faster Convergence and Saddle-to-saddle Dynamics](https://mlsanity.com/paper/2606.31390/) (2026)

## Abstract (arXiv)

The landscape of empirical risk has been widely studied in a series of machine learning problems, including low-rank matrix factorization, matrix sensing, matrix completion, and phase retrieval. In this work, we focus on the situation where the corresponding population risk is a degenerate non-convex loss function, namely, the Hessian of the population risk can have zero eigenvalues. Instead of analyzing the non-convex empirical risk directly, we first study the landscape of the corresponding population risk, which is usually easier to characterize, and then build a connection between the landscape of the empirical risk and its population risk. In particular, we establish a correspondence between the critical points of the empirical risk and its population risk without the strongly Morse assumption, which is required in existing literature but not satisfied in degenerate scenarios. We also apply the theory to matrix sensing and phase retrieval to demonstrate how to infer the landscape of empirical risk from that of the corresponding population risk.