We propose a novel method to sparsify attention in the Transformer model by learning to select the most-informative token representations during the training process, thus focusing on the task-specific parts of an input. A reduction of quadratic time and memory complexity to sublinear was achieved due to a robust trainable top-$k$k operator. Our experiments on a challenging long document summarization task show that even our simple baseline performs comparably to the current SOTA, and with trainable pooling, we can retain its top quality, while being $1.8\times$1.8× faster during training, $4.5\times$4.5× faster during inference, and up to $13\times$13× more computationally efficient in the decoder.