We present novel analysis and algorithms for solving sparse phase retrieval and sparse principal component analysis (PCA) with convex lifted matrix formulations. The key innovation is a new mixed atomic matrix norm that, when used as regularization, promotes low-rank matrices with sparse factors. We show that convex programs with this atomic norm as a regularizer provide near-optimal sample complexity and error rate guarantees for sparse phase retrieval and sparse PCA. While we do not know how to solve the convex programs exactly with an efficient algorithm, for the phase retrieval case we carefully analyze the program and its dual and thereby derive a practical heuristic algorithm. We show empirically that this practical algorithm performs similarly to existing state-of-the-art algorithms.
翻译:我们提出新的分析和算法,以解决微小的阶段检索和稀少的主要组成部分分析(PCA),并配有松散的矩阵配方。关键创新是一个新的混合原子矩阵规范,在用作正规化时,会促进低级矩阵,并带有稀有因素。我们表明,这种原子规范的细微程序为稀少的阶段检索和稀少的五氯苯甲醚提供了近乎最佳的样本复杂性和错误率保障。虽然我们不知道如何用高效的算法来完全解决松动程序,但对于阶段检索案例,我们仔细分析程序及其双重性,从而得出一种实用的超自然算法。我们从经验中表明,这种实际算法与现有最先进的算法类似。