On Multi-Layer Basis Pursuit, Efficient Algorithms and Convolutional Neural Networks
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Parsimonious representations in data modeling are ubiquitous and central for processing information. Motivated by the recent Multi-Layer Convolutional Sparse Coding (ML-CSC) model, we herein generalize the traditional Basis Pursuit regression problem to a multi-layer setting, introducing similar sparse enforcing penalties at different representation layers in a symbiotic relation between synthesis and analysis sparse priors. We propose and analyze different iterative algorithms to solve this new problem in practice. We prove that the presented multi-layer Iterative Soft Thresholding (ML-ISTA) and multi-layer Fast ISTA (ML-FISTA) converge to the global optimum of our multi-layer formulation at a rate of O(1/k) and O(1/k^2), respectively. We further show how these algorithms effectively implement particular recurrent neural networks that generalize feed-forward architectures without any increase in the number of parameters. We demonstrate the different architectures resulting from unfolding the iterations of the proposed multi-layer pursuit algorithms, providing a principled way to construct deep recurrent CNNs from feed-forward ones. We demonstrate the emerging constructions by training them in an end-to-end manner, consistently improving the performance of classical networks without introducing extra filters or parameters.
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