Recovering images from very few measurements is an important task in imaging problems. Doing so requires assuming a model of what makes some images natural. Such a model is called an image prior. Classical priors such as sparsity have led to the speedup of Magnetic Resonance Imaging in certain cases. With the recent developments in machine learning, neural networks have been shown to provide efficient and effective priors for inverse problems arising in imaging. In this talk, we will discuss the use of neural network generative models for inverse problems in imaging. We will present a rigorous recovery guarantee at optimal sample complexity for compressed sensing and other inverse problems under a suitable random model. We will see that generative modelsenable an efficient algorithms for phase retrieval and spiked matrix recovery from generic measurements with optimal sample complexity. In contrast, no efficient algorithm is known for this problem in the case of sparsity priors. We will discuss strengths, weaknesses, and future opportunities of neural networks and generative models as image priors. This talk is intended to be pedagogical in nature and will involve technical mathematical results using nonasymptotic random matrix theory.
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