Deep neural networks have been successfully applied to a broad range of problems where overparametrization yields weight matrices which are partially random. A comparison of weight matrixsingular vectors to the Porter-Thomas distribution suggests that there is a boundary between randomness and learned information in the singular value spectrum. Inspired by this finding, weintroduce an algorithm for noise filtering, which both removes small singular values and reducesthe magnitude of large singular values to counteract the effect of level repulsion between the noiseand the information part of the spectrum. For networks trained in the presence of label noise, wefind that the generalization performance improves significantly due to noise filtering.