Lower Bounds for Compressed Sensing with Generative Models

The goal of compressed sensing is to learn a structured signal x from a limited number of noisy linear measurements y ≈ Ax. In traditional compressed sensing, "structure" is represented by sparsity in some known basis. Inspired by the success of deep learning in modeling images, recent work starting with~citeBJPD17 has instead considered structure to come from a generative model G: ℝk → ℝn. We present two results establishing the difficulty of this latter task, showing that existing bounds are tight. First, we provide a lower bound matching the~citeBJPD17 upper bound for compressed sensing from L-Lipschitz generative models G. In particular, there exists such a function that requires roughly Omega(k log L) linear measurements for sparse recovery to be possible. This holds even for the more relaxed goal of nonuniform recovery. Second, we show that generative models generalize sparsity as a representation of structure. In particular, we construct a ReLU-based neural network G: ℝ2k → ℝn with O(1) layers and O(kn) activations per layer, such that the range of G contains all k-sparse vectors.