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We give an algorithm for source identification of a mixture of $k$ productdistributions on $n$ bits. This is a fundamental problem in machine learningwith many applications. Our algorithm identifies the source parameters of anidentifiable mixture, given, as input, approximate values of multilinearmoments (derived, for instance, from a sufficiently large sample), using$2^{O(k^2)} n^{O(k)}$ arithmetic operations. Our result is the first explicitbound on the computational complexity of source identification of suchmixtures. The running time improves previous results by Feldman, O'Donnell, andServedio (FOCS 2005) and Chen and Moitra (STOC 2019) that guaranteed onlylearning the mixture (without parametric identification of the source). Ouranalysis gives a quantitative version of a qualitative characterization ofidentifiable sources that is due to Tahmasebi, Motahari, and Maddah-Ali (ISIT2018).

Source Identification for Mixtures of Product Distributions