Simultaneous Codeword Optimization (SimCO) for Dictionary Update and Learning

We consider the data-driven dictionary learning problem. The goal is to seek an over-complete dictionary from which every training signal can be best approximated by a linear combination of only a few codewords. This task is often achieved by iteratively executing two operations: sparse coding and dictionary update. In the literature, there are two benchmark mechanisms to update a dictionary. The first approach, such as the MOD algorithm, is characterized by searching for the optimal codewords while fixing the sparse coefficients. In the second approach, represented by the K-SVD method, one codeword and the related sparse coefficients are simultaneously updated while all other codewords and coefficients remain unchanged. We propose a novel framework that generalizes the aforementioned two methods. The unique feature of our approach is that one can update an arbitrary set of codewords and the corresponding sparse coefficients simultaneously: when sparse coefficients are fixed, the underlying optimization problem is similar to that in the MOD algorithm; when only one codeword is selected for update, it can be proved that the proposed algorithm is equivalent to the K-SVD method; and more importantly, our method allows us to update all codewords and all sparse coefficients simultaneously, hence the term simultaneous codeword optimization (SimCO). Under the proposed framework, we design two algorithms, namely, primitive and regularized SimCO. We implement these two algorithms based on a simple gradient descent mechanism. Simulations are provided to demonstrate the performance of the proposed algorithms, as compared with two baseline algorithms MOD and K-SVD. Results show that regularized SimCO is particularly appealing in terms of both learning performance and running speed.Comment: 13 page

Robust Orthogonal Complement Principal Component Analysis

Recently, the robustification of principal component analysis has attracted lots of attention from statisticians, engineers and computer scientists. In this work we study the type of outliers that are not necessarily apparent in the original observation space but can seriously affect the principal subspace estimation. Based on a mathematical formulation of such transformed outliers, a novel robust orthogonal complement principal component analysis (ROC-PCA) is proposed. The framework combines the popular sparsity-enforcing and low rank regularization techniques to deal with row-wise outliers as well as element-wise outliers. A non-asymptotic oracle inequality guarantees the accuracy and high breakdown performance of ROC-PCA in finite samples. To tackle the computational challenges, an efficient algorithm is developed on the basis of Stiefel manifold optimization and iterative thresholding. Furthermore, a batch variant is proposed to significantly reduce the cost in ultra high dimensions. The paper also points out a pitfall of a common practice of SVD reduction in robust PCA. Experiments show the effectiveness and efficiency of ROC-PCA in both synthetic and real data

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Robust Kronecker-decomposable component analysis for low-rank modeling

Dictionary learning and component analysis are part of one of the most well-studied and active research fields, at the intersection of signal and image processing, computer vision, and statistical machine learning. In dictionary learning, the current methods of choice are arguably K-SVD and its variants, which learn a dictionary (i.e., a decomposition) for sparse coding via Singular Value Decomposition. In robust component analysis, leading methods derive from Principal Component Pursuit (PCP), which recovers a low-rank matrix from sparse corruptions of unknown magnitude and support. However, K-SVD is sensitive to the presence of noise and outliers in the training set. Additionally, PCP does not provide a dictionary that respects the structure of the data (e.g., images), and requires expensive SVD computations when solved by convex relaxation. In this paper, we introduce a new robust decomposition of images by combining ideas from sparse dictionary learning and PCP. We propose a novel Kronecker-decomposable component analysis which is robust to gross corruption, can be used for low-rank modeling, and leverages separability to solve significantly smaller problems. We design an efficient learning algorithm by drawing links with a restricted form of tensor factorization. The effectiveness of the proposed approach is demonstrated on real-world applications, namely background subtraction and image denoising, by performing a thorough comparison with the current state of the art

Robust Kronecker-decomposable component analysis for low-rank modeling

Dictionary learning and component analysis are part of one of the most well-studied and active research fields, at the intersection of signal and image processing, computer vision, and statistical machine learning. In dictionary learning, the current methods of choice are arguably K-SVD and its variants, which learn a dictionary (i.e., a decomposition) for sparse coding via Singular Value Decomposition. In robust component analysis, leading methods derive from Principal Component Pursuit (PCP), which recovers a low-rank matrix from sparse corruptions of unknown magnitude and support. However, K-SVD is sensitive to the presence of noise and outliers in the training set. Additionally, PCP does not provide a dictionary that respects the structure of the data (e.g., images), and requires expensive SVD computations when solved by convex relaxation. In this paper, we introduce a new robust decomposition of images by combining ideas from sparse dictionary learning and PCP. We propose a novel Kronecker-decomposable component analysis which is robust to gross corruption, can be used for low-rank modeling, and leverages separability to solve significantly smaller problems. We design an efficient learning algorithm by drawing links with a restricted form of tensor factorization. The effectiveness of the proposed approach is demonstrated on real-world applications, namely background subtraction and image denoising, by performing a thorough comparison with the current state of the art

A non-adapted sparse approximation of PDEs with stochastic inputs

We propose a method for the approximation of solutions of PDEs with stochastic coefficients based on the direct, i.e., non-adapted, sampling of solutions. This sampling can be done by using any legacy code for the deterministic problem as a black box. The method converges in probability (with probabilistic error bounds) as a consequence of sparsity and a concentration of measure phenomenon on the empirical correlation between samples. We show that the method is well suited for truly high-dimensional problems (with slow decay in the spectrum)