  We give an algorithm for source identification of a mixture of $k$ product
distributions on $n$ bits. This is a fundamental problem in machine learning
with many applications. Our algorithm identifies the source parameters of an
identifiable mixture, given, as input, approximate values of multilinear
moments (derived, for instance, from a sufficiently large sample), using
$2^{O(k^2)} n^{O(k)}$ arithmetic operations. Our result is the first explicit
bound on the computational complexity of source identification of such
mixtures. The running time improves previous results by Feldman, O'Donnell, and
Servedio (FOCS 2005) and Chen and Moitra (STOC 2019) that guaranteed only
learning the mixture (without parametric identification of the source). Our
analysis gives a quantitative version of a qualitative characterization of
identifiable sources that is due to Tahmasebi, Motahari, and Maddah-Ali (ISIT
2018).
