In the process of recording, storage and transmission of time-domain audio signals, errors may be introduced that are difficult to correct in an unsupervised way. Here, we train a convolutional deep neural network to re-synthesize input time-domain speech signals at its output layer. We then use this abstract transformation, which we call a deep transform (DT), to perform probabilistic re-synthesis on further speech (of the same speaker) which has been degraded. Using the convolutive DT, we demonstrate the recovery of speech audio that has been subject to extreme degradation. This approach may be useful for correction of errors in communications devices.