Single-molecule localization microscopy techniques, like stochastic optical reconstruction microscopy (STORM), visualize biological specimens by stochastically exciting sparse blinking emitters. The raw images suffer from unwanted background fluorescence, which must be removed to achieve super-resolution. We introduce a sparsity-based background removal method by adapting a neural network (SLNet) from a different microscopy domain. The SLNet computes a low-rank representation of the images, and then, by subtracting it from the raw images, the sparse component is computed, representing the frames without the background. We compared our approach with widely used background removal methods, such as the median background removal or the rolling ball algorithm, on two commonly used STORM datasets, one glial cell, and one microtubule dataset. The SLNet delivers STORM frames with less background, leading to higher emitters' localization precision and higher-resolution reconstructed images than commonly used methods. Notably, the SLNet is lightweight and easily trainable (<5 min). Since it is trained in an unsupervised manner, no prior information is required and can be applied to any STORM dataset. We uploaded a pre-trained SLNet to the Bioimage model zoo, easily accessible through ImageJ. Our results show that our sparse decomposition method could be an essential and efficient STORM pre-processing tool.