Diffusion echoes are a fundamental concept for understanding the behaviour of nonlinear diffusion filters. They describe the accumulated data exchange during a diffusion process and are given by the columns or rows of the corresponding state transition matrix. Unfortunately, they involve a prohibitively large amount of data. Therefore, we propose the first compression strategy to efficiently represent and reconstruct diffusion echoes. Using a truncated singular value decomposition (SVD) we can reduce the storage requirements substantially, while still obtaining very accurate reconstructions. The SVD works on all echoes jointly and captures the redundancy between them. To approximate the singular value decomposition, we use a powerful probabilistic approach: the randomised subspace iteration. We show on a test case with a difficult, rapidly decaying diffusivity that we can reduce the storage requirements by a factor of 20, without creating visually noticeable errors. Furthermore, our compressed data representation enables an efficient reconstruction, which allows a fast and detailed echo investigation. This paves the way for various future applications of the diffusion echo that have been prevented by its huge amount of data so far.