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

From Rank Estimation to Rank Approximation: Rank Residual Constraint for Image Restoration

In this paper, we propose a novel approach to the rank minimization problem, termed rank residual constraint (RRC) model. Different from existing low-rank based approaches, such as the well-known nuclear norm minimization (NNM) and the weighted nuclear norm minimization (WNNM), which estimate the underlying low-rank matrix directly from the corrupted observations, we progressively approximate the underlying low-rank matrix via minimizing the rank residual. Through integrating the image nonlocal self-similarity (NSS) prior with the proposed RRC model, we apply it to image restoration tasks, including image denoising and image compression artifacts reduction. Towards this end, we first obtain a good reference of the original image groups by using the image NSS prior, and then the rank residual of the image groups between this reference and the degraded image is minimized to achieve a better estimate to the desired image. In this manner, both the reference and the estimated image are updated gradually and jointly in each iteration. Based on the group-based sparse representation model, we further provide a theoretical analysis on the feasibility of the proposed RRC model. Experimental results demonstrate that the proposed RRC model outperforms many state-of-the-art schemes in both the objective and perceptual quality

An MDL framework for sparse coding and dictionary learning

The power of sparse signal modeling with learned over-complete dictionaries has been demonstrated in a variety of applications and fields, from signal processing to statistical inference and machine learning. However, the statistical properties of these models, such as under-fitting or over-fitting given sets of data, are still not well characterized in the literature. As a result, the success of sparse modeling depends on hand-tuning critical parameters for each data and application. This work aims at addressing this by providing a practical and objective characterization of sparse models by means of the Minimum Description Length (MDL) principle -- a well established information-theoretic approach to model selection in statistical inference. The resulting framework derives a family of efficient sparse coding and dictionary learning algorithms which, by virtue of the MDL principle, are completely parameter free. Furthermore, such framework allows to incorporate additional prior information to existing models, such as Markovian dependencies, or to define completely new problem formulations, including in the matrix analysis area, in a natural way. These virtues will be demonstrated with parameter-free algorithms for the classic image denoising and classification problems, and for low-rank matrix recovery in video applications