The recently proposed multilayer convolutional sparse coding (ML-CSC) model, consisting of a cascade of convolutional sparse layers, provides a new interpretation of convolutional neural networks (CNNs). Under this framework, the forward pass in a CNN is equivalent to a pursuit algorithm aiming to estimate the nested sparse representation vectors from a given input signal. Despite having served as a pivotal connection between CNNs and sparse modeling, a deeper understanding of the ML-CSC is still lacking. In this paper, we propose a sound pursuit algorithm for the ML-CSC model by adopting a projection approach. We provide new and improved bounds on the stability of the solution of such pursuit and we analyze different practical alternatives to implement this in practice. We show that the training of the filters is essential to allow for nontrivial signals in the model, and we derive an online algorithm to learn the dictionaries from real data, effectively resulting in cascaded sparse convolutional layers. Last, but not least, we demonstrate the applicability of the ML-CSC model for several applications in an unsupervised setting, providing competitive results. Our work represents a bridge between matrix factorization, sparse dictionary learning, and sparse autoencoders, and we analyze these connections in detail.
Finally, and due to the fact that our formulation effectively provides a convolutional network with sparse kernels, our approach is reminiscent of works attempting to sparsify the filters in deep learning models. For instance, the work in @cite_30 showed that the weights of learned deep convolutional networks can be sparsified without considerable degradation of classification accuracy. Nevertheless, one should perpend the fact that these works are motivated merely by cheaper and faster implementations, whereas our model is intrinsically built by theoretically justified sparse kernels. We do not attempt to compare our approach to such sparsifying methods at this stage, and we defer this to future work.
@cite_30
1935978687
In recent years, sparse signal modeling, especially using the synthesis model has been popular. Sparse coding in the synthesis model is however, NP-hard. Recently, interest has turned to the sparsifying transform model, for which sparse coding is cheap. However, natural images typically contain diverse textures that cannot be sparsified well by a single transform. Hence, in this work, we propose a union of sparsifying transforms model. Sparse coding in this model reduces to a form of clustering. The proposed model is also equivalent to a structured overcomplete sparsifying transform model with block cosparsity, dubbed OCTOBOS. The alternating algorithm introduced for learning such transforms involves simple closed-form solutions. A theoretical analysis provides a convergence guarantee for this algorithm. It is shown to be globally convergent to the set of partial minimizers of the non-convex learning problem. We also show that under certain conditions, the algorithm converges to the set of stationary points of the overall objective. When applied to images, the algorithm learns a collection of well-conditioned square transforms, and a good clustering of patches or textures. The resulting sparse representations for the images are much better than those obtained with a single learned transform, or with analytical transforms. We show the promising performance of the proposed approach in image denoising, which compares quite favorably with approaches involving a single learned square transform or an overcomplete synthesis dictionary, or gaussian mixture models. The proposed denoising method is also faster than the synthesis dictionary based approach. We address the challenging open problem of blindly separating time position varying mixtures, and attempt to separate the sources from such mixtures without having prior information about the sources or the mixing system. Unlike studies concerning instantaneous or convolutive mixtures, we assume that the mixing system (medium) is varying in time position. Attempts to solve this problem have mostly utilized, so far, online algorithms based on tracking the mixing system by methods previously developed for the instantaneous or convolutive mixtures. In contrast with these attempts, we develop a unified approach in the form of staged sparse component analysis (SSCA). Accordingly, we assume that the sources are either sparse or can be “sparsified.” In the first stage, we estimate the filters of the mixing system, based on the scatter plot of the sparse mixtures' data, using a proper clustering and curve surface fitting. In the second stage, the mixing system is inverted, yielding the estimated sources. We use the SSCA approach for solving three types of mixtures: time position varying instantaneous mixtures, single-path mixtures, and multipath mixtures. Real-life scenarios and simulated mixtures are used to demonstrate the performance of our approach. Adaptive sparse representations have been very popular in numerous applications in recent years. The learning of synthesis sparsifying dictionaries has particularly received much attention, and such adaptive dictionaries have been shown to be useful in applications such as image denoising, and magnetic resonance image reconstruction. In this work, we focus on the alternative sparsifying transform model, for which sparse coding is cheap and exact, and study the learning of tall or overcomplete sparsifying transforms from data. We propose various penalties that control the sparsifying ability, condition number, and incoherence of the learnt transforms. Our alternating algorithm for transform learning converges empirically, and significantly improves the quality of the learnt transform over the iterations. We present examples demonstrating the promising performance of adaptive overcomplete transforms over adaptive overcomplete synthesis dictionaries learnt using K-SVD, in the application of image denoising. This paper presents methods to reduce the complexity of convolutional neural networks (CNN). These include: (1) A method to quickly and easily sparsify a given network. (2) Fine tune the sparse network to obtain the lost accuracy back (3) Quantize the network to be able to implement it using 8-bit fixed point multiplications efficiently. (4) We then show how an inference engine can be designed to take advantage of the sparsity. These techniques were applied to full frame semantic segmentation and the degradation due to the sparsity and quantization is found to be negligible. We show by analysis that the complexity reduction achieved is significant. Results of implementation on Texas Instruments TDA2x SoC [17] are presented. We have modified Caffe CNN framework to do the sparse, quantized training described in this paper. The source code for the training is made available at https: github.com tidsp caffe-jacinto