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We present a general framework for designing efficient algorithms forunsupervised learning problems, such as mixtures of Gaussians and subspaceclustering. Our framework is based on a meta algorithm that learns arithmeticcircuits in the presence of noise, using lower bounds. This builds upon therecent work of Garg, Kayal and Saha (FOCS 20), who designed such a frameworkfor learning arithmetic circuits without any noise. A key ingredient of ourmeta algorithm is an efficient algorithm for a novel problem called RobustVector Space Decomposition. We show that our meta algorithm works well whencertain matrices have sufficiently large smallest non-zero singular values. Weconjecture that this condition holds for smoothed instances of our problems,and thus our framework would yield efficient algorithms for these problems inthe smoothed setting.

Learning Arithmetic Formulas in the Presence of Noise: A General Framework and Applications to Unsupervised Learning