Tensor and its tucker core: The invariance relationships

Jiang, Bo; Yang, Fan; Zhang, Shuzhong

doi:10.1002/nla.2086

Cited by 30 publications

(38 citation statements)

References 38 publications

(102 reference statements)

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“…. , d. In fact, the "low-rank" tensor in the above model corresponds to the tensor with a small core; however a recent work [36] demonstrates that the CP-rank of the core regardless of its size could be as large as the original tensor. Therefore, if one wants to find the low CP-rank decomposition, then the following model is preferred:…”

Section: Introductionmentioning

confidence: 95%

Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis

et al. 2018

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Nonconvex and nonsmooth optimization problems are frequently encountered in much of statistics, business, science and engineering, but they are not yet widely recognized as a technology in the sense of scalability. A reason for this relatively low degree of popularity is the lack of a well developed system of theory and algorithms to support the applications, as is the case for its convex counterpart. This paper aims to take one step in the direction of disciplined nonconvex and nonsmooth optimization. In particular, we consider in this paper some constrained nonconvex optimization models in block decision variables, with or without coupled affine constraints. In the absence of coupled constraints, we show a sublinear rate of convergence to an ǫ-stationary solution in the form of variational inequality for a generalized conditional gradient method, where the convergence rate is dependent on the Hölderian continuity of the gradient of the smooth part of the objective. For the model with coupled affine constraints, we introduce corresponding ǫ-stationarity conditions, and apply two proximal-type variants of the ADMM to solve such a model, assuming the proximal ADMM updates can be implemented for all the block variables except for the last block, for which either a gradient step or a majorization-minimization step is implemented. We show an iteration complexity bound of O(1/ǫ 2 ) to reach an ǫ-stationary solution for both algorithms. Moreover, we show that the same iteration complexity of a proximal BCD method follows immediately. Numerical results are provided to illustrate the efficacy of the proposed algorithms for tensor robust PCA.

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Section: Introductionmentioning

confidence: 95%

Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis

et al. 2018

Self Cite

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“…The independent Tucker format has the following important properties if the equality in (3.10) holds exactly (see, e.g., [105] and references therein):…”

mentioning

confidence: 99%

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Cichocki

Lee

Oseledets

et al. 2016

FNT in Machine Learning

371

342

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Modern applications in engineering and data science are increasingly based on multidimensional data of exceedingly high volume, variety, and structural richness. However, standard machine learning algorithms typically scale exponentially with data volume and complexity of cross-modal couplings - the so called curse of dimensionality - which is prohibitive to the analysis of large-scale, multi-modal and multi-relational datasets. Given that such data are often efficiently represented as multiway arrays or tensors, it is therefore timely and valuable for the multidisciplinary machine learning and data analytic communities to review low-rank tensor decompositions and tensor networks as emerging tools for dimensionality reduction and large scale optimization problems. Our particular emphasis is on elucidating that, by virtue of the underlying low-rank approximations, tensor networks have the ability to alleviate the curse of dimensionality in a number of applied areas. In Part 1 of this monograph we provide innovative solutions to low-rank tensor network decompositions and easy to interpret graphical representations of the mathematical operations on tensor networks. Such a conceptual insight allows for seamless migration of ideas from the flat-view matrices to tensor network operations and vice versa, and provides a platform for further developments, practical applications, and non-Euclidean extensions. It also permits the introduction of various tensor network operations without an explicit notion of mathematical expressions, which may be beneficial for many research communities that do not directly rely on multilinear algebra. Our focus is on the Tucker and tensor train (TT) decompositions and their extensions, and on demonstrating the ability of tensor networks to provide linearly or even super-linearly (e.g., logarithmically) scalable solutions, as illustrated in detail in Part 2 of this monograph

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“…Then, the three‐factor matrices,

U^{(n)} \in R^{d_{n} \times R_{n}} ((), n = 1, 2, 3)

, which represent the feature structure of the data in each dimension, can be obtained. Projecting the original data

A \in R^{d_{1} \times d_{2} \times d_{3}}

onto its factor matrices obtains the core tensor

S \in R^{R_{1} \times R_{2} \times R_{3}}

, which denotes the coupling relationship among the data in each dimension …”

Section: Basic Ideas and Preliminariesmentioning

confidence: 99%

“…Projecting the original data A ∈ R d 1 ×d 2 ×d 3 onto its factor matrices obtains the core tensor S ∈ R R 1 ×R 2 ×R 3 , which denotes the coupling relationship among the data in each dimension. 33,34 FIGURE 2 Tucker decomposition of the three-order tensor A ∈ R d1×d2×d3…”

Section: Tensor Tucker Decompositionmentioning

confidence: 99%

Digital watermarking scheme for colour remote sensing image based on quaternion wavelet transform and tensor decomposition

Che

Luo

et al. 2019

Math Methods in App Sciences

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Digital watermarking is important for protecting the intellectual property of remote sensing images. Unlike watermarking in ordinary colour images, in colour remote sensing images, watermarking has an important requirement: robustness. In this paper, a robust nonblind watermarking scheme for colour remote sensing images, which considers both frequency and statistical pattern features, is constructed based on the quaternion wavelet transform (QWT) and tensor decomposition. Using the QWT, not only the abundant phase information can be used to preserve detailed host image features to improve the imperceptibility of the watermark, but also the frequency coefficients of the host image can provide a stable position to embed the watermark. To further strengthen the robustness, the global statistical feature structure acquired through the tensor Tucker decomposition is employed to distribute the watermark's energy among different colour bands. Because both the QWT frequency coefficients and the tensor decomposition global statistical feature structure are highly stable against external distortion, their integration yields the proposed scheme, which is robust to many image manipulations. A simulation experiment shows that our method can balance the trade‐off between imperceptibility and robustness and that it is more robust than the traditional QWT and discrete wavelet transform (DWT) methods under many different types of image manipulations.

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Tensor and its tucker core: The invariance relationships

Cited by 30 publications

References 38 publications

Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis

Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Digital watermarking scheme for colour remote sensing image based on quaternion wavelet transform and tensor decomposition

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