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We give algorithms with provable guarantees that learn a class of deep nets in the generative model view popularized by Hinton and others. Our generative model is an n node multilayer neural net that has degree at most n^ γ for some γ< 1 and each edge has a random edge weight in [-1, 1]. Our algorithm learns almost all networks in this class with polynomial running time. The sample complexity is quadratic or cubic depending upon the details of the model. The algorithm uses layerwise learning. It is based upon a novel idea of observing correlations among features and using these to infer the underlying edge structure via a global graph recovery procedure. The analysis of the algorithm reveals interesting structure of neural nets with random edge weights.