Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. In particular, these techniques offer a route toward principal component analysis (PCA) for petascale data. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider

Abstract. Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition

New image denoising models based on non-local image information have been recently introduced in the literature. These so-called ”non-local” denoising models provide excellent results because these models can denoise smooth regions or/and textured regions simultaneously, unlike standard denoising models. Standard variational models s.a. Total Variation-based models are defined to work in a small local neighborhood, which is enough to denoise smooth regions. However, textures are not local in nature and requires semi-local/non-local information to be denoised efficiently. Several papers have introduced non-local filters and non-local variational models for image denoising. Yet, few studies have been done to develop unsupervised image segmentation models based on non-local information. This will be the goal of this paper. We define and study three unsupervised non-local segmentation models. These models will be based on the continuous global minimization approach for image segmentation recently introduced in [10, 6]. The energy of [10, 6] is a first order energy composed of the weighted Total Variation norm and a linear term. The first proposed non-local segmentation model will extend

Modeling natural and artificial systems has a key role in various applications, and has long been a task that drew enormous efforts. In this work, instead of exploring predefined models, we aim at implicitly identifying the system degrees of freedom. This approach circumvents the dependency of a specific predefined model for a specific task or system, and enables a generic data-driven method to characterize a system based solely on its output observations. We claim that each system can be viewed as a black box controlled by several independent parameters. Moreover, we assume that the perceptual characterization of the system output is determined by these independent parameters. Consequently, by recovering the independent controlling parameters, we find in fact a generic modeling for the system. In this work, we propose a supervised algorithm to recover the controlling parameters of natural and artificial linear systems. The proposed algorithm relies on nonlinear independent component analysis using diffusion kernels and spectral analysis. Employment of the proposed algorithm on both synthetic and real examples has shown accurate recovery of parameters.

Abstract. There are currently several communities working on algorithms for classification of high dimensional data. This work develops a class of variational algorithms that combine recent ideas from spectral methods on graphs with nonlinear edge/region detection methods traditionally used in in the PDE-based imaging community. The algorithms are based on the Ginzburg-Landau functional which has classical PDE connections to total variation minimization. Convex-splitting algorithms allow us to quickly find minimizers of the proposed model and take advantage of fast spectral solvers of linear graph-theoretic problems. We present diverse computational examples involving both basic clustering and semi-supervised learning for different applications. Case studies include feature identification in images, segmentation in social networks, and segmentation of shapes in high dimensional datasets. Key words. Nyström extension, diffuse interfaces, image processing, high dimensional data AMS subject classifications. Insert AMS subject classifications. This work brings together ideas from different communities and for this reason we review various components of the algorithms in order to make the paper accessible to readers familiar with either the PDE-based or graph-theoretic approaches. In

In [5], a general framework for adaptive function regularization was introduced, and this framework was demonstrated in several applications, including image denoising. The basic idea of the method applied to image denoising is to choose

Abstract. We propose a semi-supervised image segmentation method that relies on a non-local continuous version of the min-cut algorithm and labels or seeds provided by a user. The segmentation process is performed via energy minimization. The proposed energy is composed of three terms. The first term defines labels or seed points assigned to objects that the user wants to identify and the background. The second term carries out the diffusion of object and background labels and stops the diffusion when the interface between the object and the background is reached. The diffusion process is performed on a graph defined from image intensity patches. The graph of intensity patches is known to better deal with textures because this graph uses semi-local and non-local image information. The last term is the standard TV term that regularizes the geometry of the interface. We introduce an iterative scheme that provides a unique minimizer. Promising results are presented on synthetic textures and real-world images. 1

Editor: The problems of dimension reduction and inference of statistical dependence are addressed by the modeling framework of learning gradients. The models we propose hold for Euclidean spaces as well as the manifold setting. The central quantity in this approach is an estimate of the gradient and the gradient outer product. We relate the gradient outer product to standard statistical quantities such covariances and provide a simple and precise comparison of a variety of simultaneous regression and dimensionality reduction methods. We provide rates of convergence for both inference of informative directions as well as inference of a graphical model of variable dependencies. We illustrate the efficacy of the method of simulated and real data. Keywords: Gradient estimates, manifold learning, graphical models, inverse regression, dimension reduction 1.

Patch-based de-noising algorithms and patch manifold smoothing have emerged as efficient de-noising methods. This paper provides a new insight on these methods, such as the Non Local Means or the image graph de-noising, by showing its use for filtering a selected pattern.