We define a broader family of corruption processes that generalizes previously known diffusion models. To reverse these general diffusions, we propose a new objective called Soft Score Matching. Soft Score Matching incorporates the degradation process in the network and provably learns the score function for any linear corruption process. Our new loss trains the model to predict a clean image, that after corruption, matches the diffused observation. This objective learns the gradient of the likelihood under suitable regularity conditions for the family of linear corruption processes. We further develop an algorithm to select the corruption levels for general diffusion processes and a novel sampling method that we call Momentum Sampler. We show experimentally that our framework works for general linear corruption processes, such as Gaussian blur and masking. Our method outperforms all linear diffusion models on CelebA-64 achieving FID score 1.85. We also show computational benefits compared to vanilla denoising diffusion.