Tensor Norm Research Articles

Multi-view Subspace Clustering Based on Weighted Tensor Schatten-P Norm

In the era of advancing technology and continuous societal evolution, an abundance of voluminous data from various domains has become prevalent in the public domain. Managing this wealth of data necessitates a series of preprocessing procedures, encompassing tasks such as data storage, computation, and analysis, with the aim of unveiling intrinsic features and patterns contained within. Among these procedures, clustering stands out as a crucial step, with the primary goal of unveiling the hidden structures within the data and organizing data samples into meaningful clusters based on specific criteria. Early clustering efforts were based on a singleview. Real world data, however, often contains interrelated information, leading to the emergence of multi-view clustering. Multiview subspace clustering (T-SVD) has been widely used in recent years. However, the equal treatment of each singular value in this approach lacks relevance when the data is noisy or contaminated. In order to improve the robustness and clustering performance, this paper presents a T-SVD weighted tensor norm. In order to improve the robustness, a Schatten-P norm was proposed. This paper explores multi-view subspace clustering is based on the Schatten-P norm weighted tensor. Experiments are carried out on 6 benchmark data sets, and the results are compared with 4 existing ones. Experiments show that the proposed approach is better than the most advanced multi-view subspace clustering methods.

Numerical range for weighted Moore-Penrose inverse of tensor

This article first introduces the notion of weighted singular value decomposition (WSVD) of a tensor via the Einstein product. The WSVD is then used to compute the weighted Moore-Penrose inverse of an arbitrary-order tensor. We then define the notions of weighted normal tensor for an even-order square tensor and weighted tensor norm. Finally, we apply these to study the theory of numerical range for the weighted Moore-Penrose inverse of an even-order square tensor and exploit its several properties. We also obtain a few new results in matrix setting.

Improving Generalization via Coupled Tensor Norm Regularization

Improving Generalization via Coupled Tensor Norm Regularization

Strain-based method for fatigue failure prediction of additively manufactured lattice structures

Lattice structures find application in numerous technological domains, including aerospace and automotive industries for structural components, biomedical sector implants, and heat exchangers. In many instances, especially those pertaining to structural applications, fatigue resistance stands as a critical and stringent requirement. The objective of this paper is to advance the analysis of fatigue failure in additively manufactured lattice structures by introducing a predictive fatigue failure model based on the finite element (FE) method and experimentally validating the results. The model utilizes linear homogenization to reduce computational effort in FE simulations. By employing a strain-based parameter, the most critical lattice cell is identified, enabling the prediction of fatigue crack nucleation locations. The Crossland multiaxial fatigue failure criterion is employed to assess the equivalent stress, furnishing the fatigue limit threshold essential for predicting component failure. Inconel 625 specimens are manufactured via the laser-based powder bed fusion of metals additive manufacturing process. In order to validate the model, cantilevers comprising octa-truss lattice cells in both uniform and graded configurations undergo experimental testing subjected to bending loads within the high cycle fatigue regime. The proposed methodology effectively forecasts the location of failure in seventeen out of eighteen samples, establishing itself as a valuable tool for lattice fatigue analysis. Failure consistently manifests in sections of uniform and graded lattice structures characterized by the maximum strain tensor norm. The estimated maximum force required to prevent fatigue failure in the samples is 20 N, based on the computed Crossland equivalent stress.

A multilinear variant of the Saphar theorem for the gauche tensor norm

A multilinear variant of the Saphar theorem for the gauche tensor norm

Approximating Tensor Norms via Sphere Covering: Bridging the Gap between Primal and Dual

Approximating Tensor Norms via Sphere Covering: Bridging the Gap between Primal and Dual

Low-Rank tensor completion based on nonconvex regularization

In this paper, we consider the low-rank tensor completion which aims to exactly recover incomplete high-dimensional visual data. Existing studies utilize widely tensor nuclear norm minimization (TNNM), a convex relaxation to tensor-rank minimization (TRM), to solve tensor completion tasks. Nevertheless, TNNM ignores the difference between different tensor singular values induced by the tensor singular value decompositions (t-SVD) and then the obtained solution may be suboptimal. In this paper, we propose a nonconvex minimization approach to solve the tensor completion problem more effectively by adopting a nonconvex regularization to further approximate the tensor-rank. Moreover, alternating direction method of multipliers (ADMM) and iteratively reweighted nuclear norm (IRNN) are adopted to solve the constructed nonconvex models efficiently, and the convergence can also be guaranteed. Finally, we present that the proposed nonconvex optimization methods are suitable for solving other TRM problems induced by any invertible linear transform, such as subspace clustering based on low-rank representation. Extensive experiments on real images and videos validate the superiority of our approach over the state-of-the-art algorithms.

Prediction Skill and Practical Predictability Depending on the Initial Atmospheric States in S2S Forecasts

Abstract The hypothesis that predictability depends on the atmospheric state in the planetary-scale low-frequency variability in boreal winter was examined. We first computed six typical weather patterns from 500-hPa geopotential height anomalies in the Northern Hemisphere using self-organizing map (SOM) and k-clustering analysis. Next, using 11 models from the subseasonal-to-seasonal (S2S) operational and reforecast archive, we computed each model’s climatology as a function of lead time to evaluate model bias. Although the forecast bias depends on the model, it is consistently the largest when the forecast begins from the atmospheric state with a blocking-like pattern in the eastern North Pacific. Moreover, the ensemble-forecast spread based on S2S multimodel forecast data was compared with empirically estimated Fokker–Planck equation (FPE) parameters based on reanalysis data. The multimodel mean ensemble-forecast spread was correlated with the diffusion tensor norm; they are large for the cases when the atmospheric state started from a cluster with a blocking-like pattern. As the multimodel mean is expected to substantially reduce model biases and may approximate the predictability inherent in nature, we can summarize that the atmospheric state corresponding to the cluster was less predictable than others. Significance Statement The purpose of this study is to examine the performance of week-to-month forecasts by analyzing multimodel forecast results. We established the hypothesis proposed by the previous studies that the accuracy of forecasts depended on the atmospheric state. Together with the data-based method on predictability, an atmospheric state with the anticyclone anomaly in the eastern North Pacific exhibited low predictability. Our results provide a method to foresee the ability of week-to-month forecasts.

Improvement of Incomplete Multiview Clustering by the Tensor Reconstruction of the Connectivity Graph

With the development of data collection technologies, a significant volume of multiview data has appeared, and their clustering has become topical. Most methods of multiview clustering assume that all views are fully observable. However, in many cases this is not the case. Several tensor methods have been proposed to deal with incomplete multiview data. However, the traditional tensor norm is computationally expensive, and such methods generally cannot handle undersampling and imbalances of various views. A new method for clustering incomplete multiview data is proposed. A new tensor norm is defined to reconstruct the connectivity graph, and the graphs are regularized to a consistent low-dimensional representation of patterns. The weights are then iteratively updated for each view. Compared to the existing ones, the proposed method not only determines the consistency between views but also obtains a low-dimensional representation of the samples using the resulting projection matrix. An efficient optimization algorithm based on the method of indefinite Lagrange multipliers is developed for the solution. The experimental results on four data sets demonstrate the effectiveness of the method.

Multilinear absolutely summing operators associated to a tensor norm

Multilinear absolutely summing operators associated to a tensor norm

Sketching Low-Rank Matrices With a Shared Column Space by Convex Programming

In many practical applications including remote sensing, multi-task learning, and multi-spectrum imaging, data are described as a set of matrices sharing a common column space. We consider the joint estimation of such matrices from their noisy linear measurements. We study a convex estimator regularized by a pair of matrix norms. The measurement model corresponds to block-wise sensing and the reconstruction is possible only when the total energy is well distributed over blocks. The first norm, which is the maximum-block-Frobenius norm, favors such a solution. This condition is analogous to the notion of low-spikiness in matrix completion or column-wise sensing. The second norm, which is a tensor norm on a pair of suitable Banach spaces, induces low-rankness in the solution together with the first norm. We demonstrate that the joint estimation provides a significant gain over the individual recovery of each matrix when the number of matrices sharing a column space and the ambient dimension of the shared column space are large relative to the number of columns in each matrix. The convex estimator is cast as a semidefinite program and an efficient ADMM algorithm is derived. The empirical behavior of the convex estimator is illustrated using Monte Carlo simulations and recovery performance is compared to existing methods in the literature.

Integrated Spatio-Spectral–Temporal Fusion via Anisotropic Sparsity Constrained Low-Rank Tensor Approximation

Although spatio-spectral and spatio-temporal fusion has been well explored, few efforts are made on integrating spatio-spectral-temporal features. As an intrinsic prior, low tensor-rank has been successfully taken into effect by current fusion models, most of which, however, resort to establishing an overall low-rank norm without performing factorization techniques thus have trouble capturing the latent high-order structure of hyperspectral data cube. To address that, a novel Anisotropicly Sparse (AS) tensor norm is developed to make the rank minimization a learnable process under Tucker decomposition. The AS norm enables the model to minimize the multi-linear tensor ranks if imposed on the core tensor after factorization, hence it significantly improves the model’s fusion performance. In the temporal domain, a Hadamard-product based variability descriptor is incorporated into the fusion model to map the former information to current time. Additionally, piece-wise smooth prior of the Tucker factors is employed by extra regularizers as supplement to the loss spatial information. With the Proximal Differential Matrix being developed for optimization, the proposed method reaches state-of-the-art results on both spatio-spectral and spatio-spectral-temporal fusion at low computational cost.

Transformed singular values and spectral norms of third-order tensors

The tensor-tensor product (T-product) and tensor singular value decomposition (T-SVD) for third-order tensors, which are based upon the general revertible linear transformation, are widely used in engineering and data science. This paper focuses on studying singular values and spectral norms of third-order tensors under unitary transformation. After presenting the decomposition for the third-order Hermitian tensors and the QR factorization of third-order tensors, with the help of the involved invertible linear transformation, several basic properties of singular values for third-order tensors are studied, which include the Lipschitz continuity that characterizes the singular values perturbation. Based upon these, two kinds of spectral norms for third-order tensors are introduced, one of which is the norm induced by Frobenius norm of tensors in the sense of the T-product.

Measurement Incompatibility versus Bell Nonlocality: An Approach via Tensor Norms

Measurement incompatibility and quantum non-locality are two key features of quantum theory. Violations of Bell inequalities require quantum entanglement and incompatibility of the measurements used by the two parties involved in the protocol. We analyze the converse question: for which Bell inequalities is the incompatibility of measurements enough to ensure a quantum violation? We relate the two questions by comparing two tensor norms on the space of dichotomic quantum measurements: one characterizing measurement compatibility and the second one characterizing violations of a given Bell inequality. We provide sufficient conditions for the equivalence of the two notions in terms of the matrix describing the correlation Bell inequality. We show that the CHSH inequality and its variants are the only ones satisfying it.

Robust tensor recovery with nonconvex and nonsmooth regularization

This paper considers the least squares loss minimization problem regularized by tensor average rank and zero norm, to decompose the noisy observation into a low-rank tensor, a sparse tensor and a noise tensor. Due to the nonconvex and nonsmooth of the average rank and the zero norm of tensors, it is not easy to solve the problem directly. We construct the variational characterizations of the tensor average rank and the tensor zero norm, get an equivalent mathematical program with an equilibrium constraint, and then propose an equivalent Lipschitz surrogate for the regularization problem by adding the equality constraint into the objective. Moreover, we design a convex algorithm with multi-stage to verify the efficiency of the proposed method. Numerical experiments are reported for simulation data and actual data to show the performance of the new method.

Lipschitz 𝑝-summing multilinear operators correspond to Lipschitz 𝑝-summing operators

We give conditions that ensure that an operator satisfying a Piestch domination in a given setting also satisfies a Piestch domination in a different setting. From this we derive that a bounded multilinear operator T T is Lipschitz p p -summing if and only if the mapping f T ( x 1 ⊗ ⋯ ⊗ x n ) ≔ T ( x 1 , … , x n ) f_T(x_1\otimes \cdots \otimes x_n)≔T(x_1,\ldots , x_n) is Lipschitz p p -summing. The results are based on the projective tensor norm. An example with the Hilbert tensor norm is provided to show that the statement may not hold when a reasonable cross-norm other than the projective tensor norm is considered.

The Gauge Group and Perturbation Semigroup of an Operator System

The perturbation semigroup was first defined in the case of $*$-algebras by Chamseddine, Connes and van Suijlekom. In this paper, we take $\mathcal{E}$ as a concrete operator system with unit. We first give a definition of gauge group $\mathcal{G}(\mathcal{E})$ of $\mathcal{E}$, after that we give the definition of perturbation semigroup of $\mathcal{E}$, and the closed perturbation semigroup of $\mathcal{E}$ with respect to the Haagerup tensor norm. We also show that there is a continuous semigroup homomorphism from the closed perturbation semigroup to the collection of unital completely bounded Hermitian maps over $\mathcal{E}$. Finally we compute the gauge group and perturbation semigroup of the Toeplitz system as an example.

Some Generalized Versions of Chevet–Saphar Tensor Norms

The paper is concerned with some generalized versions gE and wE of classical tensor norms. We find a Banach space E for which gE and wE are finitely generated tensor norms, and show that gE and wE are associated with the ideals of some E-nuclear operators. We also initiate the study of some theories of our tensor norms.

A Hybrid Norm for Guaranteed Tensor Recovery

Benefiting from the superiority of tensor Singular Value Decomposition (t-SVD) in excavating low-rankness in the spectral domain over other tensor decompositions (like Tucker decomposition), t-SVD-based tensor learning has shown promising performance and become an emerging research topic in computer vision and machine learning very recently. However, focusing on modeling spectral low-rankness, the t-SVD-based models may be insufficient to exploit low-rankness in the original domain, leading to limited performance while learning from tensor data (like videos) that are low-rank in both original and spectral domains. To this point, we define a hybrid tensor norm dubbed the “Tubal + Tucker” Nuclear Norm (T2NN) as the sum of two tensor norms, respectively, induced by t-SVD and Tucker decomposition to simultaneously impose low-rankness in both spectral and original domains. We further utilize the new norm for tensor recovery from linear observations by formulating a penalized least squares estimator. The statistical performance of the proposed estimator is then analyzed by establishing upper bounds on the estimation error in both deterministic and non-asymptotic manners. We also develop an efficient algorithm within the framework of Alternating Direction Method of Multipliers (ADMM). Experimental results on both synthetic and real datasets show the effectiveness of the proposed model.

Incompatibility in General Probabilistic Theories, Generalized Spectrahedra, and Tensor Norms

In this work, we investigate measurement incompatibility in general probabilistic theories (GPTs). We show several equivalent characterizations of compatible measurements. The first is in terms of the positivity of associated maps. The second relates compatibility to the inclusion of certain generalized spectrahedra. For this, we extend the theory of free spectrahedra to ordered vector spaces. The third characterization connects the compatibility of dichotomic measurements to the ratio of tensor crossnorms of Banach spaces. We use these characterizations to study the amount of incompatibility present in different GPTs, i.e. their compatibility regions. For centrally symmetric GPTs, we show that the compatibility degree is given as the ratio of the injective and the projective norm of the tensor product of associated Banach spaces. This allows us to completely characterize the compatibility regions of several GPTs, and to obtain optimal universal bounds on the compatibility degree in terms of the 1-summing constants of the associated Banach spaces. Moreover, we find new bounds on the maximal incompatibility present in more than three qubit measurements.