Multi-view low-rank sparse subspace clustering

With the increasing availability of various sensor technologies, we now have access to large amounts of multi-block (also called multi-set, multi-relational, or multi-view) data that need to be jointly analyzed to explore their latent connections. Various component analysis methods have played an increasingly important role for the analysis of such coupled data. In this paper, we first provide a brief review of existing matrix-based (two-way) component analysis methods for the joint analysis of such data with a focus on biomedical applications. Then, we discuss their important extensions and generalization to multi-block multiway (tensor) data. We show how constrained multi-block tensor decomposition methods are able to extract similar or statistically dependent common features that are shared by all blocks, by incorporating the multiway nature of data. Special emphasis is given to the flexible common and individual feature analysis of multi-block data with the aim to simultaneously extract common and individual latent components with desired properties and types of diversity. Illustrative examples are given to demonstrate their effectiveness for biomedical data analysis.

Linked Component Analysis from Matrices to High Order Tensors: Applications to Biomedical Data

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Facility Layout And Location An Analytical Approach

Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications have received wide attention and achievement in areas like data mining, computer vision, signal processing, and neuroscience. In this survey, we provide a modern overview of recent advances in tensor completion algorithms from the perspective of big data analytics characterized by diverse variety, large volume, and high velocity. We characterize these advances from the following four perspectives: general tensor completion algorithms, tensor completion with auxiliary information (variety), scalable tensor completion algorithms (volume), and dynamic tensor completion algorithms (velocity). Further, we identify several tensor completion applications on real-world data-driven problems and present some common experimental frameworks popularized in the literature along with several available software repositories. Our goal is to summarize these popular methods and introduce them to researchers and practitioners for promoting future research and applications. We conclude with a discussion of key challenges and promising research directions in this community for future exploration.

https://dl.acm.org/doi/pdf/10.1145/3278607

Tensor Completion Algorithms in Big Data Analytics

With the increasing availability of various sensor technologies, we now have access to large amounts of multiblock (also called multiset, multirelational, or multiview) data that need to be jointly analyzed to explore their latent connections. Various component analysis methods have played an increasingly important role for the analysis of such coupled data. In this article, we first provide a brief review of existing matrix-based (two-way) component analysis methods for the joint analysis of such data with a focus on biomedical applications. Then, we discuss their important extensions and generalization to multiblock multiway (tensor) data. We show how constrained multiblock tensor decomposition methods are able to extract similar or statistically dependent common features that are shared by all blocks, by incorporating the multiway nature of data. Special emphasis is given to the flexible common and individual feature analysis of multiblock data with the aim to simultaneously extract common and individual latent components with desired properties and types of diversity. Illustrative examples are given to demonstrate their effectiveness for biomedical data analysis.

Linked Component Analysis From Matrices to High-Order Tensors: Applications to Biomedical Data

The problem of tensor completion arises often in signal processing and machine learning. It consists of recovering a tensor from a subset of its entries. The usual structural assumption on a tensor that makes the problem well posed is that the tensor has low rank in every mode. Several tensor completion methods based on minimization of nuclear norm, which is the closest convex approximation of rank, have been proposed recently, with applications mostly in image inpainting problems. It is often stated in these papers that methods based on Tucker factorization perform poorly when the true ranks are unknown. In this paper, we propose a simple algorithm for Tucker factorization of a tensor with missing data and its application to low-$$n$$n-rank tensor completion. The algorithm is similar to previously proposed method for PARAFAC decomposition with missing data. We demonstrate in several numerical experiments that the proposed algorithm performs well even when the ranks are significantly overestimated. Approximate reconstruction can be obtained when the ranks are underestimated. The algorithm outperforms nuclear norm minimization methods when the fraction of known elements of a tensor is low.

/pdf/tucker-factorization-with-missing-data-with-application-to-1o6kbzz1pr.pdf

Tucker factorization with missing data with application to low-$$n$$n-rank tensor completion

The first contribution of this paper is the comparison of learned dictionary based approaches to inpainting and denoising of images in natural scenes, where emphasis is given on the use of complete and overcomplete dictionary learned by independent component analysis. The second contribution of the paper relates to the formulation of a problem of denoising an image corrupted by a salt and pepper type of noise (this problem is equivalent to estimating saturated pixel values), as a noiseless inpainting problem, whereupon noise corrupted pixels are treated as missing pixels. A maximum a posteriori (MAP) approach to image denoising is not applicable in such a case due to the fact that variance of the impulsive noise is infinite and the MAP based estimation relies on solving an optimization problem with an inequality constraint that depends on the variance of the additive noise. Through extensive comparative performance analysis of the inpainting task, it is demonstrated that ICA-learned basis outperforms K-SVD and morphological component analysis approaches in terms of visual quality. It yielded similar performance as a field of experts method but with more than two orders of magnitude lower computational complexity. On the same problems, Fourier and wavelet bases as representatives of fixed bases, exhibited the poorest performance. It is also demonstrated that noiseless inpainting-based approach to image denoising (estimation of the saturated pixel values) greatly outperforms denoising based on two-dimensional myriad filtering that is a theoretically optimal solution for this class of additive impulsive noise.

A comparison of dictionary based approaches to inpainting and denoising with an emphasis to independent component analysis learned dictionaries

We address the problem of restoration of images which have been affected by impulse or a combination of impulse and Gaussian noise. We propose a patch-based approach that exploits approximate sparse representations of image patches in learned dictionaries. For every patch, sparse representation in a dictionary is enforced by l 1 -norm penalty, and sparsity of the residual is enforced by l 0 -quasi-norm penalty. The obtained non-convex problem is solved iteratively by a combination of soft and hard thresholding, and a proof of convergence to a local minimum is given. Experimental evaluation suggests that the proposed approach can produce state-of-the-art results for some types of images, especially in terms of the structural similarity (SSIM) measure. In addition, the proposed iterative thresholding algorithm could possibly be applied to general inverse problems.

Restoration of images corrupted by mixed Gaussian-impulse noise by iterative soft-hard thresholding

Sparse representation of natural images over redundant dictionary enables solution of the inpainting problem. A major challenge, in this regard, is learning of a dictionary that is well adapted to the image. Efficient methods are developed for grayscale images represented in patch space by using, for example, K-SVD or independent component analysis algorithms. Here, we address the problem of patch space-based dictionary learning for color images. To this end, an image in RGB color space is represented as a collection of vectorized 3D patch tensors. This leads to the state-of-the-art results in inpainting random and structured patterns of missing values as it is demonstrated in the paper.

Inpainting color images in learned dictionary

https://bib.irb.hr/datoteka/633391.Jukic_Filipovic_2013_PRLetters.pdf

Marko Filipović

Papers

Tucker factorization with missing data with application to low-$$n$$n-rank tensor completion

A comparison of dictionary based approaches to inpainting and denoising with an emphasis to independent component analysis learned dictionaries

Restoration of images corrupted by mixed Gaussian-impulse noise by iterative soft-hard thresholding

Inpainting color images in learned dictionary

Supervised feature extraction for tensor objects based on maximization of mutual information