Tensor Algebra Research Articles

Cosmic decoherence: primordial power spectra and non-Gaussianities

We study the effect of quantum decoherence on the inflationary cosmological perturbations. This process might imprint specific observational signatures revealing the quantum nature of the inflationary mechanism being related to the longstanding issue of the quantum-to-classical transition of inflationary fluctuations. Several works have investigated the effect of quantum decoherence on the statistical properties of primordial fluctuations. In particular, it has been shown that cosmic decoherence leads to corrections to the curvature power spectrum predicted by standard slow-roll inflation. Equally interesting, a non zero curvature trispectrum has been shown to be purely induced by cosmic decoherence but, surprisingly, decoherence seems not to generate any bispectrum. We further develop such an analysis by adopting a generalized form of the pointer observable, showing that decoherence does induce a non vanishing curvature bispectrum and providing a specific underlying concrete physical process. Present constraints on primordial bispectra allow to put an upper bound on the strength of the environment-system interaction. In full generality, the decoherence-induced bispectrum can be scale dependent provided one imposes the corresponding correction to the power spectrum to be scale independent. Such scale dependence on the largest cosmological scales might represent a distinctive imprint of the quantum decoherence process taking place during inflation. We also provide a criterion that allows to understand when cosmic decoherence induces scale independent corrections, independently of the type of environment considered. As a final result, we study the effect of cosmic decoherence on tensor perturbations and we derive the decoherence corrected tensor-to-scalar perturbation ratio. In specific cases, decoherence induces a blue tilted correction to the standard tensor power spectrum.

Hamming Sandwiches

We describe primitive association schemes {mathfrak {X}} of degree n such that textrm{Aut}({mathfrak {X}}) is imprimitive and |textrm{Aut}({mathfrak {X}})| ge exp (n^{1/8}), contradicting a conjecture of Babai. This and other examples we give are the first known examples of nonschurian primitive coherent configurations (PCC) with more than a quasipolynomial number of automorphisms. Our constructions are “Hamming sandwiches”, association schemes sandwiched between the dth tensor power of the trivial scheme and the d-dimensional Hamming scheme. We study Hamming sandwiches in general, and exhaustively for d le 8. We revise Babai’s conjecture by suggesting that any PCC with more than a quasipolynomial number of automorphisms must be an association scheme sandwiched between a tensor power of a Johnson scheme and the corresponding full Cameron scheme. If true, it follows that any nonschurian PCC has at most exp O(n^{1/8} log n) automorphisms.

USING A TENSOR MODEL TO HANDLE UNCERTAINTY IN COMPLEX DYNAMICAL SYSTEMS

The article is devoted to research related to the processing of uncertainty by means of tensor algebra in complex dynamical systems. The “smart electronic hitch” system is considered as an example of a complex dynamic system. The use of such a system is especially important when organizing traffic in difficult conditions. The solution of this problem occurs under conditions of uncertainty that may appear at different levels of the traffic management process. To identify and study in detail the properties of uncertainty, the author suggests using a tensor model. The tensor model makes it possible to identify additional properties of uncertainty, the study of which is not available when using traditional formalisms to represent uncertainty. Using the tensor model allows us to study the spatial model of uncertainty, real and imaginary values of uncertainty, as well as uncertainty invariants with respect to various transformations of the coordinate system. The article proposes a classification of uncertainty in a complex system. Using the example of the organization of interaction of “smart” controllers in an electronic coupling, the author shows the results of applying tensor analysis methods of networks to obtain a computational base of an electronic coupling. Tensor equations provide efficient processing of big data, obtaining information in real time, the stability of the dynamic system to changes in the topology of the connection of controllers and changes in the soft and hard components of the connection. The results obtained in the article show that the tensor uncertainty model can be successfully implemented in a dynamic system of any complexity level.

Real eigenstructure of regular simplex tensors

We are concerned with the real eigenstructure of symmetric tensors. As in the matrix case, normalized tensor eigenvectors are fixed points of the tensor power iteration map. However, unless the given tensor is orthogonally decomposable, some of these fixed points may be repelling and therefore be undetectable by any numerical scheme. In this paper, we consider the case of regular simplex tensors whose symmetric decomposition is induced by an overcomplete, equiangular set of n+1 vectors from Rn. We discuss the full real eigenstructure of such tensors, including the robustness analysis of all normalized eigenvectors. As it turns out, regular simplex tensors have robust as well as non-robust eigenvectors which, moreover, only partly coincide with the generators from the symmetric tensor decomposition.

A Full-Body Relative Orbital Motion of Spacecraft Using Dual Tensor Algebra and Dual Quaternions

This paper proposes a new non-linear differential equation for the six degrees of freedom (6-DOF) relative rigid bodies motion. A representation theorem is provided for the 6-DOF differential equation of motion in the arbitrary non-inertial reference frame. The problem of the 6-DOF relative motion of two spacecraft in the specific case of Keplerian confocal orbits is proposed. The result is an analytical method without secular terms and singularities. Tensors dual algebra and dual quaternions play a fundamental role, with the solution representation being the relative problem. Furthermore, the representation theorems for the rotation and translation parts of the 6-DOF relative orbital motion problems are obtained.

Primordial black holes from Higgs inflation with a Gauss-Bonnet coupling

Primordial black holes (PBHs) can be the source for all or a part of today's dark matter density. Inflation provides a mechanism for generating the seeds of PBHs in the presence of a temporal period where the velocity of an inflaton field $\phi$ rapidly decreases toward 0. We compute the primordial power spectra of curvature perturbations generated during Gauss-Bonnet (GB) corrected Higgs inflation in which the inflaton field has not only a nonminimal coupling to gravity but also a GB coupling. For a scalar-GB coupling exhibiting a rapid change during inflation, we show that curvature perturbations are sufficiently enhanced by the appearance of an effective potential $V_{\rm eff}(\phi)$ containing the structures of plateau-type, bump-type, and their intermediate type. We find that there are parameter spaces in which PBHs can constitute all dark matter for these three types of $V_{\rm eff}(\phi)$. In particular, models with bump- and intermediate-types give rise to the primordial scalar and tensor power spectra consistent with the recent Planck data on scales relevant to the observations of cosmic microwave background. This property is attributed to the fact that the number of e-foldings $\Delta N_c$ acquired around the bump region of $V_{\rm eff}(\phi)$ can be as small as a few, in contrast to the plateau-type where $\Delta N_c$ typically exceeds the order of 10.

Limit shape for infinite rank limit of tensor power decomposition for Lie algebras of series *

We consider the Plancherel measure on irreducible components of tensor powers of the spinor representation of . The irreducible representations correspond to the generalized Young diagrams. With respect to this measure the probability of an irreducible representation is the product of its multiplicity and dimension, divided by the total dimension of the tensor product. We study the limit shape of the generalized Young diagram when the tensor power N and the rank n of the algebra tend to infinity with fixed. We derive an explicit formula for the limit shape and prove convergence to it in probability. We prove central limit theorem for global fluctuations around the limit shape.

Invariant Theory and wheeled PROPs

We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra V=T(V)⊗T(V⋆), where T(V) is the tensor algebra on an n-dimensional vector space over a field K of characteristic 0. First we classify all the ideals of the initial object Z in the category of wheeled PROPs. We show that non-degenerate sub-wheeled PROPs of V are exactly subalgebras of the form VG where G is a closed, reductive subgroup of the general linear group GL(V). When V is a finite dimensional Hilbert space, a similar description of invariant tensors for an action of a compact group was given by Schrijver. We also generalize the theorem of Procesi that says that trace rings satisfying the n-th Cayley-Hamilton identity can be embedded in an n×n matrix ring over a commutative algebra R. Namely, we prove that a wheeled PROP can be embedded in R⊗V for a commutative K-algebra R if and only if it satisfies certain relations.

Hyperspectral image mixed noise removal via tensor robust principal component analysis with tensor-ring decomposition

ABSTRACT Due to the instability of sensors and other factors, hyperspectral images (HSIs) are inevitably polluted by various types of mixed noise. To explore a better denoising method based on the existing research, combining the denoising advantages of tensor-ring (TR) decomposition and tensor robust principal component analysis (TRPCA), a mixed-noise removal method for HSIs is proposed in this paper. First, TRPCA maintains the tensor structure of the image itself, accurately recovers the low-rank part and the sparse part from their sum and separates the sparse noise in the form of sparse tensors. Then, TR decomposition is introduced to denoise the low-rank tensors. To verify the effectiveness and superiority of this method, experiments are carried out on two simulated data sets and two real data sets. Compared with the traditional denoising methods and several existing improved denoising methods from both visual and quantitative aspects, the proposed TRPCA-TR method provides better denoising results.

Exact solutions for Janssen effect of lattice disk packings

The rhombic lattice packing model proposed here clearly shows the micro causes of the Janssen effect and is solved analytically. Based on the analysis of force equilibrium equations with an exterior product method, a recurrence relation with a force transmission matrix Q is obtained which shows that a contact force will be transmitted and accumulated along the lattice vector direction and then be reflected by lateral walls, and eventually, reach a saturation value if the packing is frictional. A method based on tensor algebra and difference equation is proposed for the diagonalization of Q. Using roundup and remainder functions, we give the exact and approximate expressions which can describe the piecewise smoothness of the pressure function data given by numerical calculations. Numerical results also show the non-uniformity of pressure distribution. These quantitative results have certain deviations with Janssen model except for very small top load or large force reflectivity.

Remarks on symplectic capacities of p-products

In this paper, we study the behavior of symplectic capacities of convex domains in the classical phase space with respect to symplectic [Formula: see text]-products. As an application, by using a “tensor power trick”, we show that it is enough to prove the weak version of Viterbo’s volume-capacity conjecture in the asymptotic regime, i.e. when the dimension is sent to infinity. In addition, we introduce a conjecture about higher-order capacities of [Formula: see text]-products, and show that if it holds, then there are no nontrivial [Formula: see text]-decompositions of the symplectic ball.

Application of Tensor Decomposition to Reduce the Complexity of Neural Min-Sum Channel Decoding Algorithm

Channel neural decoding is very promising as it outperforms the traditional channel decoding algorithms. Unfortunately, it still faces the disadvantage of high computational complexity and storage complexity compared with the traditional decoding algorithms. In this paper, we propose that low rank decomposition techniques based on tensor train decomposition and tensor ring decomposition can be utilized in neural offset min-sum (NOMS) and neural scale min-sim (NSMS) decoding algorithms. The experiment results show that the proposed two algorithms achieve near state-of-the-art performance with low complexity.

Guaranteed Functional Tensor Singular Value Decomposition

This article introduces the functional tensor singular value decomposition (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. The problem is motivated by high-order longitudinal data analysis. Our model assumes the observed data to be a random realization of an approximate CP low-rank functional tensor measured on a discrete time grid. Incorporating tensor algebra and the theory of reproducing kernel Hilbert space (RKHS), we propose a novel RKHS-based constrained power iteration with spectral initialization. Our method can successfully estimate both singular vectors and functions of the low-rank structure in the observed data. With mild assumptions, we establish the non-asymptotic contractive error bounds for the proposed algorithm. The superiority of the proposed framework is demonstrated via extensive experiments on both simulated and real data. Supplementary materials for this article are available online.

Monic modules and semi-Gorenstein-projective modules

The category gp ( Λ ) of Gorenstein-projective modules over tensor algebra Λ = A ⊗ k B can be described as the monomorphism category mon ( B , gp ( A ) ) of B over gp ( A ) . In particular, Gorenstein-projective Λ-modules are monic. In this paper, we find the similar relation between semi-Gorenstein-projective Λ-modules and A -modules, via monic modules, namely, mon ( B , A ⊥ ) = mon ( B , A ) ∩ Λ ⊥ . Using this, it is proved that if A is weakly Gorenstein, then Λ is weakly Gorenstein if and only each semi-Gorenstein-projective Λ-modules are monic; and that if B = k Q with Q a finite acyclic quiver, then Λ is weakly Gorenstein if and only if A is weakly Gorenstein. However, this relation itself does not answer the question whether there exist double semi-Gorenstein-projective Λ-modules which are not monic. Using the recent discovered examples of double semi-Gorenstein-projective A -modules which are not torsionless, we positively answer this question, by explicitly constructing a class of double semi-Gorenstein-projective T 2 ( A ) -modules with one parameter such that they are not monic, and hence not torsionless. The corresponding results are obtained also for the monic modules and semi-Gorenstein-projective modules over the triangular matrix algebras given by bimodules .

Gravitational wave probes of massive gauge bosons at the cosmological collider

We extend the reach of the “cosmological collider” for massive gauge boson production during inflation from the CMB scales to the interferometer scales. Considering a Chern-Simons coupling between the gauge bosons and the pseudoscalar inflaton, one of the transverse gauge modes is efficiently produced and its inverse decay leaves an imprint in the primordial scalar and tensor perturbations. We study the correlation functions of these perturbations and derive the updated constraints on the parameter space from CMB observables. We then extrapolate the tensor power spectrum to smaller scales consistently taking into account the impact of the gauge field on inflationary dynamics. Our results show that the presence of massive gauge fields during inflation can be detected from characteristic gravitational wave signals encompassing the whole range of current and planned interferometers.

Tensor Ring decomposition for context-aware recommendation

As a powerful tool for processing high-dimensional data, tensor decomposition has been widely used in context-aware recommendation. Most of the current tensor decomposition-based recommendation models use CP decomposition or Tucker decomposition. In practical recommendation applications, fewer parameters limit the performance of CP decomposition, and the high-order tensor kernel makes the computational complexity of Tucker decomposition exponential. In order to improve the performance of recommendation while maintaining high computational efficiency, we propose a bias Tensor Ring decomposition framework for context-aware recommendation, which employs Tensor Ring decomposition to extract user-item-context latent factors. Comparing to current tensor decomposition-based recommendation models, our framework achieves a better balance between recommendation performance and computational complexity by Tensor Ring decomposition. This is a highly scalable recommendation framework that can easily further integrate additional information such as social trust and implicit feedback, thus used for more complex recommendation tasks. To the best of our knowledge, this work is the first to formulate context-aware recommendation as Tensor Ring decomposition process. Our in-depth analyses confirm that Tensor Ring decomposition has certain advantages over CP decomposition and Tucker decomposition. Multiple experiments on three real datasets also show that the proposed framework helps to improve the recommendation performance.

Hyperspectral image restoration via hybrid smoothness regularized auto-weighted low-rank tensor ring factorization

Restoration of hyperspectral images (HSI) is a crucial step in many potential applications as a preprocessing step. Recently, low-rank tensor ring factorization was applied for HSI reconstruction, which has high-order tensors’ powerful and generalized representation ability. Although low-rank TR-based approaches with nuclear norm regularization achieved successful results for restoring hyperspectral images, there is still room for improved tensor low-rank approximation. In this article, we propose a novel Auto-weighted low-rank Tensor Ring Factorization with Hybrid Smoothness regularization (ATRFHS) for mixed noise removal in HSI. Nonlocal Cuboid Tensorization (NCT) is leveraged to transform HSI data into high-order tensors. TR factorization using latent factors rank minimization removes the mixed noise in HSI data. To highlight nuclear norms of factor tensors differently effective, an auto-weighted strategy is employed to reduce the more prominent factors while shrinking the smaller ones. A hybrid regularization combining total variation (TV) and phase congruency (PC) is incorporated into a low-rank tensor ring factorization model for the HSI noise removal problem. This efficient combination yields sharper edge preservation and resolves this weakness of existing pure TV regularization. Moreover, we develop an efficient algorithm for solving the resulting optimization problem using the framework of alternating minimization. Extensive experimental results demonstrate that our proposed method can significantly outperform existing approaches for mixed noise removal in HSI. The proposed algorithm is validated on synthetic and natural HSI data.

An $$\mathfrak {sl}_2$$-type tensor category for the Virasoro algebra at central charge 25 and applications

Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge $1$. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge $25$ as a $PSL_2(\mathbb{C})$-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of $SL_2$ at level $-1$, showing that it has a symmetric tensor category of representations equivalent to $\mathrm{Rep}\,PSL_2(\mathbb{C})$. This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges $1$ and $25$, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.

Spatial-Temporal Tensor Ring Norm Regularization for Infrared Small Target Detection

Infrared small target detection (ISTD) technique is widely used in infrared searching and tracking (IRST) and military surveillance. Existing detection methods must sufficiently address the challenges of the heterogeneous background with high concealment targets. In this letter, we propose a novel spatial-temporal tensor ring norm regularization (STT-TRNR) to detect infrared small targets. First, to utilize the spatial and temporal context information in a sequence, nonrepetitive spatial-temporal patches are formed by sliding windows in the consecutive frames. The patches are stacked into a tensor structure with spatial and temporal information. Second, the tensor ring nuclear norm is introduced to approximate the rank of the background tensor. The tensor ring regularization improves the correlation between dimensions, protects the internal structure of the tensor, and avoids the dimension disaster caused by train decomposition. Third, the local contrastive feature is used as a priori information to suppress the false alarms caused by the corner edges and other noises and avoid the distortion caused by target movement. Finally, the alternating direction multiplier method (ADMM) is employed to reconstruct the sequence images and retrieve the targets. The experimental results reveal that the suggested model provides improved detection performance and higher robustness across various complicated scene types.

Infrared Small Target Detection Using Spatiotemporal 4-D Tensor Train and Ring Unfolding

Infrared small target detection (ISTD) is vital for civil and military applications. However, existing methods often face challenges in coping with complex scenes, discriminating targets from similar objects, or leveraging temporal information effectively. To tackle these limitations, we offer an innovative approach that exploits the spatio-temporal structure of infrared images. A four-dimensional (4D) infrared tensor is initially constructed from a sequence of infrared images, and decomposed into lower-dimensional tensors using the tensor train (TT) and its extension – tensor ring (TR) techniques. The ISTD problem is then formulated as a sparse plus low-rank decomposition problem, where the sparse part is the target and the low-rank part is the background. We factorize the composed tensors into matrices via TT and TR unfolding approaches, which mitigates the imbalance between different modes containing spatial and temporal information. By constraining the balanced unfolded components with the weighted sum of nuclear norm, we solve the problem using the alternating direction multiplier method (ADMM). Furthermore, we validate models on several datasets and benchmark them with state-of-the-art techniques in detection accuracy and background suppression. Comparison results demonstrate the superiority of our approach over the existing methods. Moreover, the results of an ablation study with three-dimensional (3D) tensor structures show the effectiveness and feasibility of the dimension expansion to 4D.