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Our goal in the paper is to address the following problem: From an unknown matrix, we are given inner products of that matrix with a set of prescribed matrices, and wish to find the unknown matrix. We shall consider this problem by using the notion of minimal norm interpolation. A fixed-point proximity algorithm for solving this problem will be developed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we will present a generalization for a minimization problem from I. Daubechies, M. Defrise, and C. DeMol (2004) . This generalization is useful for solving many practical problems in which more than one constraint are involved. In this regard, we will conclude the findings of many papers (most of which are on image processing) from this generalization. It is hoped that the approach proposed in this paper will be a suitable reference for some applied works where multi-frames, multi-wavelets, or multi-constraints are present in linear inverse problems. [Copyright &y& Elsevier]

Matrix completion involves recovering a matrix from a subset of its entries by utilizing interdependency between the entries, typically through low rank structure. Despite matrix completion requiring the global solution of a non-convex objective, there are many computationally efficient algorithms which are effective for a broad class of matrices. In this paper, we introduce an alternating steepest descent algorithm (ASD) and a scaled variant, ScaledASD, for the fixed-rank matrix completion problem. Empirical evaluation of ASD and ScaledASD on both image inpainting and random problems show they are competitive with other state-of-the-art matrix completion algorithms in terms of recoverable rank and overall computational time. In particular, their low per iteration computational complexity makes ASD and ScaledASD efficient for large size problems, especially when computing the solutions to moderate accuracy such as in the presence of model misfit, noise, and/or as an initialization strategy for higher order methods. A preliminary convergence analysis is also presented. [ABSTRACT FROM AUTHOR]

Abstract: After a decade of extensive study of the sparse representation synthesis model, we can safely say that this is a mature and stable field, with clear theoretical foundations, and appealing applications. Alongside this approach, there is an analysis counterpart model, which, despite its similarity to the synthesis alternative, is markedly different. Surprisingly, the analysis model did not get a similar attention, and its understanding today is shallow and partial. In this paper we take a closer look at the analysis approach, better define it as a generative model for signals, and contrast it with the synthesis one. This work proposes effective pursuit methods that aim to solve inverse problems regularized with the analysis-model prior, accompanied by a preliminary theoretical study of their performance. We demonstrate the effectiveness of the analysis model in several experiments, and provide a detailed study of the model associated with the 2D finite difference analysis operator, a close cousin of the TV norm. [Copyright &y& Elsevier]