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]:
A sparse coding model of natural sounds (speech) is proposed. The signal is represented by a complex sparse coding problem with smoothness priors on both amplitude and phase. Learning and inference proceeds as in standard sparse coding. The method is analyzed in terms of statistics of complex pairs filters as well as denoising.

The method is not very novel. Complex sparse coding was already introduced in the past and the sparsity priors on the amplitudes and coefficients are a straightforward extension (or simplification compared to the work by Cadieu et al.).

Pros:
- interesting application
- fairly clear written paper

Cons:
- insights may be good but I probably did not fully understood them. Why are sounds inherently different from images? Is it an artifact of how the experimental set up? Without sparsity/smoothness constraints, the problem is clearly underdetermined and therefore filters do not necessarily converge to quadrature pairs.
- what is the contribution of this work compared to Cadieu et al? They had an extra layer, but the basic idea of smoothness of phase and amplitude is present also in that work.
- empirical validation is not sufficient because:
  - more quantitative results would be beneficial to assess the benefits of this model. For instance, the authors may want to compare and cite:
Y. Karklin, C. Ekanadham, and E. P. Simoncelli, Hierarchical spike coding of sound, Adv in Neural Information Processing Systems (NIPS), 2012
- some parts need clarification
  - eq. 9: why is this a good choice? wouldn’t it be better to have it bounded below?
   - sec. 2.2 why rescaling the gradients when there are beta and gamma?
   - in the compression experiment, shouldn’t the reconstruction error be taken into account?

Overall, this is interesting work. However, several clarifications are required in order to better assess novelty and to understand the method. Also, the empirical validation should be strengthened.