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]:
The paper describes a sparse coding model with complex valued basis functions. For training, it proposes to minimize reconstruction error plus penalty terms that encourage the amplitudes and phases of the basis functions to be smooth.

At first sight the model seems reminiscent of Cadieu, Olshausen (2012) [4]. But in that work, it is the coefficients that are penalized to have smooth amplitudes over time, whereas here, it is the basis functions themselves that are penalized to be smooth.

The model is applied to time-domain speech signals (one-dimensional data). The paper compares the results of complex valued sparse coding with smoothness penalties versus complex valued sparse coding without. The comparison shows that with penalties, basis functions seem to be more localized and filters within a pair tend to have quadrature relations.

Without penalties they do not seem to. I find this somewhat surprising because I would have thought that minimizing reconstruction error (plus orthonormalizing filters within each pair as suggested) would already achieve this, like it does in the case of images. The paper does suggest that sound data is fundamentally different from image data. I am curious what it is about sound data that causes it to require this extra machinery for learning complex basis functions. It would be very good to have actual results on images as a control.  This would also help disentangle two topics that are hard to separate in the paper, which are 1) fundamental  differences in sound data versus image data, and 2) learning complex bases with and without smoothness penalties.

I am wondering in what way the smoothness penalties are related to weight decay, or in what way they may just help find better local optima. It seems like this would be easy to check by initializing model A (no penalties) with model B (with penalties).

The title says 'natural sounds' but as far as I can tell, all experiments were done on a speech dataset. I'm not sure I completely agree with the statement that speech is a good enough proxy for natural sounds in general.

There are a lot of typos (e.g., 'Gramm-Schmidt', 'strucutre', 'analyzis').