Abstract

We investigate the effectiveness of convex relaxation and nonconvex optimization in solving bilinear systems of equations under two different designs (i.e., a sort of random Fourier design and Gaussian design). Despite the wide applicability, the theoretical understanding about these two paradigms remains largely inadequate in the presence of random noise. The current article makes two contributions by demonstrating that (i) a two-stage nonconvex algorithm attains minimax-optimal accuracy within a logarithmic number of iterations, and (ii) convex relaxation also achieves minimax-optimal statistical accuracy vis-à-vis random noise. Both results significantly improve upon the state-of-the-art theoretical guarantees. Supplementary materials for this article are available online.

摘要

我们研究了凸松弛和非凸优化在两种不同设计（即一种随机傅立叶设计和高斯设计）下求解双线性方程组的有效性。尽管适用性广泛，但在随机噪声存在的情况下，对这两种范式的理论理解仍然很大程度上不足。当前的文章通过证明 (i) 两阶段非凸算法在对数迭代次数内实现极小极大最优精度，以及 (ii) 凸松弛也实现了相对于随机噪声的极小极大最优统计精度，做出了两项贡献。这两个结果都显着改进了最先进的理论保证。本文的补充材料可在线获取。