Abstract: A problem by Feichtinger, Heil, and Larson asks whether a positive-definite integral operator with M1 kernel admits a symmetric rank-one decomposition which is strongly square-summable with respect to the M1 norm. In conjunction with a concurrent result by Bandeira-Mixon-Steinerberger, we provide a negative answer to the problem and study special cases where such factorization is possible. We also discuss an application of the results to the semidefinite relaxation of a quadratic program. Algorithms that solve the first problem can be used to solve special cases of the Blind Source Separation problem. Joint work with Fushuai (Black) Jiang: https://arxiv.org/abs/2409.20372