In this letter, we propose a learning-based compressive sensing (CS) algorithm for denoising side scan sonar (SSS) images. The proposed method is a deep learning-based CS method with enhanced nonlinearity based on an iterative shrinkage and thresholding algorithm (ISTA). Since noise intensity varies depending on the position within SSS images, the proposed method also incorporates CoordConv, which provides coordinate information to the network to help remove nonhomogeneous noise. Through end-to-end training, both the deep learning module and the CS characteristics can be jointly optimized. Representative experimental results show that the proposed method is better than state-of-art methods in terms of both noise removal and memory requirements.