Calcium imaging is an essential tool to study the activity of neuronal populations. However, the high level of background fluorescence in images hinders the accurate identification of neurons and the extraction of neuronal activities. While robust principal component analysis (RPCA) is a promising method that can decompose the foreground and background in such images, its computational complexity and memory requirement are prohibitively high to process large-scale calcium imaging data. Here, we propose BEAR, a simple bilinear neural network for the efficient approximation of RPCA which achieves an order of magnitude speed improvement with GPU acceleration compared to the conventional RPCA algorithms. In addition, we show that BEAR can perform foreground-background separation of calcium imaging data as large as tens of gigabytes. We also demonstrate that two BEARs can be cascaded to perform simultaneous RPCA and non-negative matrix factorization for the automated extraction of spatial and temporal footprints from calcium imaging data. The source code used in the paper is available at https://github.com/NICALab/BEAR.