#### Frank-Wolfe Optimization for Symmetric-NMF under Simplicial Constraint

##### Han Zhao, Geoff Gordon

Symmetric nonnegative matrix factorization has found abundant applications in various domains by providing a symmetric low-rank decomposition of nonnegative matrices. In this paper we propose a Frank-Wolfe (FW) solver to optimize the symmetric nonnegative matrix factorization problem under a simplicial constraint, which has recently been proposed for probabilistic clustering. Compared with existing solutions, this algorithm is simple to implement, and has no hyperparameters to be tuned. Building on the recent advances of FW algorithms in nonconvex optimization, we prove an $O(1/\varepsilon^2)$ convergence rate to $\varepsilon$-approximate KKT points, via a tight bound $\Theta(n^2)$ on the curvature constant, which matches the best known result in unconstrained nonconvex setting using gradient methods. Numerical results demonstrate the effectiveness of our algorithm. As a side contribution, we construct a simple nonsmooth convex problem where the FW algorithm fails to converge to the optimum. This result raises an interesting question about necessary conditions of the success of the FW algorithm on convex problems.

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