Sparse, complex-valued representations of natural sounds learned with phase and amplitude continuity priors
Authors: Wiktor Mlynarski
ICLR 2014

Abstract: Complex-valued sparse coding is a data representation which employs a dictionary of two-dimensional subspaces, while imposing a sparse, factorial prior on complex amplitudes. When trained on a dataset of natural image patches, it learns phase invariant features which closely resemble receptive fields of complex cells in the visual cortex. Features trained on natural sounds however, rarely reveal phase invariance and capture other aspects of the data. This observation is a starting point of the present work. As its first contribution, it provides an analysis of natural sound statistics by means of learning sparse, complex representations of short speech intervals. Secondly, it proposes priors over the basis function set, which bias them towards phase-invariant solutions. In this way, a dictionary of complex basis functions can be learned from the data statistics, while preserving the phase invariance property. Finally, representations trained on speech sounds with and without priors are compared. Prior-based basis functions reveal performance comparable to unconstrained sparse coding, while explicitly representing phase as a temporal shift. Such representations can find applications in many perceptual and machine learning tasks.


Peer review:
[Review]:
This paper shows that imposing a prior over the basis functions in a complex representation of sound results in bases that are closer to hilbert pairs, with smooth amplitude envelope and linear phase precession.  

It is not clear why imposing the prior directly on the basis functions is necessary.  If you  think of the complex pair as a phase-shiftable basis function, then it would make sense for the real and imaginary parts to be related by hilbert transform.  It makes me wonder whether the optimization was done correctly in inferring the sparse amplitudes - I.e., the phase must be allowed to steer to the optimal position, yielding a sparse representation.  It appears the gradients were computed with respect to the real and imaginary parts of the coefficients, rather than the amplitude and phase, which may be why the phase is not being properly  inferred.

The slowness prior on the phase doesn't make sense  - this would bias the bases toward low frequencies, no?   Some comment seems warranted.

The learned tiling in time-frequency doesn't make much sense.  What is causing the arching pattern?  It's not clear.

Most of all, it's not clear what we gain from this representation beyond previous attempts to learn a sparse representation of sound (Smith & Lewicki).  It would have been nice to compare coding efficiency and so forth against a purely real (vs. complex) representation.