Local Search Algorithms for Rank-Constrained Convex Optimization
It addresses optimization problems with rank constraints, such as in matrix completion and robust PCA, offering improved theoretical bounds and practical variants, but is incremental in refining existing analyses.
The paper tackles rank-constrained convex optimization by proposing greedy and local search algorithms, achieving a solution with rank O(r*·min{κ log((R(0)-R(A*))/ε), κ^2}) and error R(A) ≤ R(A*) + ε, generalizing prior results for sparse and smooth functions.
We propose greedy and local search algorithms for rank-constrained convex optimization, namely solving $\underset{\mathrm{rank}(A)\leq r^*}{\min}\, R(A)$ given a convex function $R:\mathbb{R}^{m\times n}\rightarrow \mathbb{R}$ and a parameter $r^*$. These algorithms consist of repeating two steps: (a) adding a new rank-1 matrix to $A$ and (b) enforcing the rank constraint on $A$. We refine and improve the theoretical analysis of Shalev-Shwartz et al. (2011), and show that if the rank-restricted condition number of $R$ is $κ$, a solution $A$ with rank $O(r^*\cdot \min\{κ\log \frac{R(\mathbf{0})-R(A^*)}ε, κ^2\})$ and $R(A) \leq R(A^*) + ε$ can be recovered, where $A^*$ is the optimal solution. This significantly generalizes associated results on sparse convex optimization, as well as rank-constrained convex optimization for smooth functions. We then introduce new practical variants of these algorithms that have superior runtime and recover better solutions in practice. We demonstrate the versatility of these methods on a wide range of applications involving matrix completion and robust principal component analysis.