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We present novel analysis and algorithms for solving sparse phase retrievaland sparse principal component analysis (PCA) with convex lifted matrixformulations. The key innovation is a new mixed atomic matrix norm that, whenused as regularization, promotes low-rank matrices with sparse factors. We showthat convex programs with this atomic norm as a regularizer providenear-optimal sample complexity and error rate guarantees for sparse phaseretrieval and sparse PCA. While we do not know how to solve the convex programsexactly with an efficient algorithm, for the phase retrieval case we carefullyanalyze the program and its dual and thereby derive a practical heuristicalgorithm. We show empirically that this practical algorithm performs similarlyto existing state-of-the-art algorithms.

Optimal convex lifted sparse phase retrieval and PCA with an atomic matrix norm regularizer