Convergence Rate of Block-Coordinate Maximization Burer-Monteiro Method for Solving Large SDPs

Semidefinite programming (SDP) with equality constraints arise in many optimization and machine learning problems, such as Max-Cut, community detection and robust PCA. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to address this issue, Burer and Monteiro \cite{burer2003nonlinear} proposed to reduce the dimension of the problem by appealing to a low-rank factorization, and solve the subsequent non-convex problem instead. It is well-understood that the resulting non-convex problem acts as a reliable surrogate to the original SDP, and can be efficiently solved using the block-coordinate maximization method. Despite its simplicity, remarkable success, and wide use in practice, the theoretical understanding of the convergence of this method is limited. We prove that the block-coordinate maximization algorithm applied to the non-convex Burer-Monteiro approach enjoys a global sublinear rate without any assumptions on the problem, and a local linear convergence rate despite no local maxima is locally strongly concave. We illustrate our results through examples and numerical experiments.

Adaptive and Efficient Nonlinear Channel Equalization for Underwater Acoustic Communication

We investigate underwater acoustic (UWA) channel equalization and introduce hierarchical and adaptive nonlinear channel equalization algorithms that are highly efficient and provide significantly improved bit error rate (BER) performance. Due to the high complexity of nonlinear equalizers and poor performance of linear ones, to equalize highly difficult underwater acoustic channels, we employ piecewise linear equalizers. However, in order to achieve the performance of the best piecewise linear model, we use a tree structure to hierarchically partition the space of the received signal. Furthermore, the equalization algorithm should be completely adaptive, since due to the highly non-stationary nature of the underwater medium, the optimal MSE equalizer as well as the best piecewise linear equalizer changes in time. To this end, we introduce an adaptive piecewise linear equalization algorithm that not only adapts the linear equalizer at each region but also learns the complete hierarchical structure with a computational complexity only polynomial in the number of nodes of the tree. Furthermore, our algorithm is constructed to directly minimize the final squared error without introducing any ad-hoc parameters. We demonstrate the performance of our algorithms through highly realistic experiments performed on accurately simulated underwater acoustic channels.