The Project SMALL
The main goal of the project is to develop a new foundational framework for processing signals, using adaptive sparse structured representations.
While sparsity provides an enormous dimensionality reduction, it does not fully capture all the structure of natural data that is readily available to us. For example, in images, the size of wavelet coefficients decays down the wavelet tree and there are strong dependencies between coefficients due to the underlying geometry of images.
Suitable sparse models have yet to be discovered for many "exotic" or heterogeneous types of data such as multi-channel or multi-modal data, as found in audiovisual (typically two sound tracks and a video track), bio-medical, or climate monitoring applications.
Traditionally, the idea of using learned dictionaries implies computational cost. This is because the complexity issue is left aside at the learning stage. To deploy learned dictionaries on large-scale data, learning algorithms should encompass this aspect.
The state-of-the-art in dictionary learning assumes fully unstructured dictionaries.
Instead, we believe dictionaries should reflect the natural structures present in signals, such as shift-invariance, rotation-invariance, and/or scale-invariance.
The dictionary design problem has been essentially addressed through empirical
algorithms. This is in contrast to the extensive body of theoretical work providing solid foundations to sparse decomposition algorithms, and calls for a solid theoretical underpinning of dictionary design algorithms.