Title: Maximum a Posteriori on a Submanifold: a General Image Restoration Method with GAN. Abstract: We propose a general method for various image restoration problems, such as denoising, deblurring, super-resolution and inpainting. The problem is formulated as a constrained optimization problem. Its objective is to maximize a posteriori probability of latent variables, and its constraint is that the image generated by these latent variables must be the same as the degraded image. We use a Generative Adversarial Network (GAN) as our density estimation model. Convincing results are obtained on MNIST dataset.