  Learning features from data has shown to be more successful than using
hand-crafted features for many machine learning tasks. In music information
retrieval (MIR), features learned from windowed spectrograms are highly variant
to transformations like transposition or time-shift. Such variances are
undesirable when they are irrelevant for the respective MIR task. We propose an
architecture called Complex Autoencoder (CAE) which learns features invariant
to orthogonal transformations. Mapping signals onto complex basis functions
learned by the CAE results in a transformation-invariant "magnitude space" and
a transformation-variant "phase space". The phase space is useful to infer
transformations between data pairs. When exploiting the invariance-property of
the magnitude space, we achieve state-of-the-art results in audio-to-score
alignment and repeated section discovery for audio. A PyTorch implementation of
the CAE, including the repeated section discovery method, is available online.
