Robust sparse analysis regularization

ABSTRACT This work studies some properties of 1 -analysis regularization for the resolution of linear inverse problems. Analysis regularization minimizes the 1 norm of the correlations between the signal and the atoms in the dictionary. The corresponding variational problem includes several well-known regularizations such as the discrete total variation and the fused lasso. We give sufficient conditions such that analysis regularization is robust to noise. ANALYSIS VERSUS SYNTHESIS Regularization through variational analysis is a popular way to compute an approximation of x 0 ∈ R N from the measurements y ∈ R Q as defined by an inverse problem y = Φx 0 + w where w is some additive noise and Φ is a linear operator, for instance a super-resolution or an inpainting operator. N which is used to synthesize a signal Common examples in signal processing of dictionary include the wavelet transform or a finite-difference operator. Synthesis regularization corresponds to the following minimization problem where Ψ = ΦD, and x = Dα. Properties of synthesis prior had been studied intensively, see for instance Analysis regularization corresponds to the following minimization problem In the noiseless case, w = 0, one uses the constrained optimization which reads min x∈R N ||D * x|| 1 subject to Φx = y. This prior had been less studied than the synthesis prior, see for instance UNION OF SUBSPACES MODEL It is natural to keep track of the support of this correlation vector, as done in the following definition. A signal x such that D * x is sparse lives in a cospace G J of small dimension where G J is defined as follow. Definition 2. Given a dictionary D, and J a subset of {1 · · · P }, the cospace G J is defined as where D J is the subdictionary whose columns are indexed by J. The signal space can thus be decomposed as a union of subspaces of increasing dimensions For the 1-D total variation prior, Θ k is the set of piecewise constant signals with k − 1 steps

Robust Image Analysis by L1-Norm Semi-supervised Learning

This paper presents a novel L1-norm semi-supervised learning algorithm for robust image analysis by giving new L1-norm formulation of Laplacian regularization which is the key step of graph-based semi-supervised learning. Since our L1-norm Laplacian regularization is defined directly over the eigenvectors of the normalized Laplacian matrix, we successfully formulate semi-supervised learning as an L1-norm linear reconstruction problem which can be effectively solved with sparse coding. By working with only a small subset of eigenvectors, we further develop a fast sparse coding algorithm for our L1-norm semi-supervised learning. Due to the sparsity induced by sparse coding, the proposed algorithm can deal with the noise in the data to some extent and thus has important applications to robust image analysis, such as noise-robust image classification and noise reduction for visual and textual bag-of-words (BOW) models. In particular, this paper is the first attempt to obtain robust image representation by sparse co-refinement of visual and textual BOW models. The experimental results have shown the promising performance of the proposed algorithm.Comment: This is an extension of our long paper in ACM MM 201

Collaborative Representation based Classification for Face Recognition

By coding a query sample as a sparse linear combination of all training samples and then classifying it by evaluating which class leads to the minimal coding residual, sparse representation based classification (SRC) leads to interesting results for robust face recognition. It is widely believed that the l1- norm sparsity constraint on coding coefficients plays a key role in the success of SRC, while its use of all training samples to collaboratively represent the query sample is rather ignored. In this paper we discuss how SRC works, and show that the collaborative representation mechanism used in SRC is much more crucial to its success of face classification. The SRC is a special case of collaborative representation based classification (CRC), which has various instantiations by applying different norms to the coding residual and coding coefficient. More specifically, the l1 or l2 norm characterization of coding residual is related to the robustness of CRC to outlier facial pixels, while the l1 or l2 norm characterization of coding coefficient is related to the degree of discrimination of facial features. Extensive experiments were conducted to verify the face recognition accuracy and efficiency of CRC with different instantiations.Comment: It is a substantial revision of a previous conference paper (L. Zhang, M. Yang, et al. "Sparse Representation or Collaborative Representation: Which Helps Face Recognition?" in ICCV 2011