Nonsmooth rank-one symmetric matrix factorization landscape

TL;DR

The paper analyzes the landscape of the nonsmooth rank-one symmetric matrix factorization objective $f(x) = 1/2 \sum_{i,j=1}^n |x_i x_j - u_i u_j|$ for $u \in R^n$. It employs Rockafellar–Wets variational analysis to derive the subdifferential and conduct a detailed first- and second-order optimality analysis, including a case split on coordinates where $u_i$ vanishes or not. The main result is that there are no spurious second-order stationary points: any such point with a nonnegative second-order directional derivative must satisfy $f(x) = 0$, hence $x = \pm u$ are the only second-order stationary points. This clarifies the optimization landscape for this nonsmooth factorization problem and connects to the broader ell1-rank-one factorization literature, informing potential algorithmic approaches.

Abstract

We consider nonsmooth rank-one symmetric matrix factorization. It has no spurious second-order stationary points.

Figures (2)

• Figure 1: Subdifferential of $\alpha$
• Figure 2: Visualization of the cases where $[0,\epsilon] \subset \partial \alpha(x_{i_0}/u_{i_0})$.

Theorems & Definitions (7)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof