jpeg 2000 swapped the discrete cosine transform for wavelets and got visibly better images at the same bitrate, with none of the block artefacts that give away ordinary jpeg. the obvious question is why the same move is not standard for audio.
part of the answer is that nobody agrees on which wavelet to use. unlike the dct, which is one fixed transform, "wavelet" is a family — and the right member depends on the signal you are compressing. in practice people pick one that has worked before and tune around it.
this research was an attempt at a procedure for finding the optimal basis for a given class of audio rather than choosing by taste: define what optimal means for a signal class, then search the space of admissible bases against a real corpus. the searching is the part that needs the iacs supercomputers. the criterion is the part that needs the mathematics.
read the full article — it is inside the november 2016 newsletter pdf.