Sum Of Tensors Research Articles

Low-rank sparse fully-connected tensor network for tensor completion

Fully-connected tensor network (FCTN) has recently drawn lots of attention in tensor completion due to its full description of all correlations between any two modes. However, the FCTN model has multiple ranks, and existing methods often ignore the difficulty brought about by rank selection, especially when the model is rank-sensitive. To overcome this drawback, this paper proposed a low-rank sparse FCTN for tensor completion. Specifically, we theoretically show that, for a tensor with FCTN structure, its subtensor can be represented as a coefficient sum of basic tensors, where the coefficients are remained in the FCTN’s factors. This means that factors’ sparsity can improve the robustness of FCTN to larger rank selection. Moreover, according to the relation between the target tensor and its FCTN’s factors, low-rank constrain is used to enhance rank robustness. Lastly, we optimize the proposed model by the alternating direction method of multipliers (ADMM) algorithm. Experimental results show that the low-rank sparse constraint effectively improves the rank robustness of the FCTN model, and achieves excellent results compared with other state-of-the-art completion methods.

Polynomial Representations of the Witt Lie Algebra

Abstract The Witt algebra ${\mathfrak{W}}_{n}$ is the Lie algebra of all derivations of the $n$-variable polynomial ring $\textbf{V}_{n}=\textbf{C}[x_{1}, \ldots , x_{n}]$ (or of algebraic vector fields on $\textbf{A}^{n}$). A representation of ${\mathfrak{W}}_{n}$ is polynomial if it arises as a subquotient of a sum of tensor powers of $\textbf{V}_{n}$. Our main theorems assert that finitely generated polynomial representations of ${\mathfrak{W}}_{n}$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $\textbf{Fin}^{\textrm{op}}$, where $\textbf{Fin}$ is the category of finite sets. We also show that polynomial representations of ${\mathfrak{W}}_{n}$ are equivalent to polynomial representations of the endomorphism monoid of $\textbf{A}^{n}$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.

Strassen's rank additivity for small tensors, including tensors of rank less or equal 7

The article is concerned with the problem of additivity of the tensor rank. That is, for two independent tensors, we study when the rank of their direct sum is equal to the sum of their individual ranks. The statement saying that additivity always holds was previously known as Strassen's conjecture (1969) until Shitov proposed counterexamples (2019). They are not explicit and only known to exist asymptotically for very large tensor spaces. In this article, we present families of pairs of small three-way tensors for which the additivity holds. For instance, over the base field C, it is the case if both tensors are of rank less or equal 7. This proves that a pair of 2×2 matrix multiplication tensors has the rank additivity property. We show also that the Alexeev-Forbes-Tsimerman substitution method preserves the structure of a direct sum of tensors.

TENSOR AND VECTOR APPROACHES TO OBJECTS RECOGNITION BY INVERSE FEATURE FILTERS

The investigation of the extraction of image objects features by filters based on tensor and vector data presentation is considered. The tensor data is obtained as a sum of rank-one tensors, given by the tensor product of the vector of lexicographic representation of image fragments pixels with itself. The accumulated tensor is approximated by one rank tensor obtained using singular values decomposition. It has been shown that the main vector of the decomposition can be considered as the object feature vector. The vector data is obtained by accumulating analogous vectors of image fragments pixels. The accumulated vector is also considered as an object feature. The filter banks of a set of objects are obtained by regularized inversion of the matrices compiled by object features vectors. Optimized regularization of the inversion is used to expand the regions of object features capture with minimal error. The object fragments and corresponding feature vectors are selected through a training iterative process. The tensor and vector approaches create two channels for recognition. High efficiency of object recognition can be achieved by choosing the filter capture band and creating filter branches according to the given bands. The filters create a convolutional network to recognize a set of objects. It has been shown that the obtained filters have an advantage over known correlation filters when recognizing objects with small fragments.

Null Raychaudhuri: canonical structure and the dressing time

We initiate a study of gravity focusing on generic null hypersurfaces, non-perturbatively in the Newton coupling. We present an off-shell account of the extended phase space of the theory, which includes the expected spin-2 data as well as spin-0, spin-1 and arbitrary matter degrees of freedom. We construct the charges and the corresponding kinematic Poisson brackets, employing a Beltrami parameterization of the spin-2 modes. We explicitly show that the constraint algebra closes, the details of which depend on the non-perturbative mixing between spin-0 and spin-2 modes. Finally we show that the spin zero sector encodes a notion of a clock, called dressing time, which is dynamical and conjugate to the constraint.It is well-known that the null Raychaudhuri equation describes how the geometric data of a null hypersurface evolve in null time in response to gravitational radiation and external matter. Our analysis leads to three complementary viewpoints on this equation. First, it can be understood as a Carrollian stress tensor conservation equation. Second, we construct spin-0, spin-2 and matter stress tensors that act as generators of null time reparametrizations for each sector. This leads to the perspective that the null Raychaudhuri equation can be understood as imposing that the sum of CFT-like stress tensors vanishes. Third, we solve the Raychaudhuri constraint non-perturbatively. The solution relates the dressing time to the spin-2 and matter boost charge operators.Finally we establish that the corner charge corresponding to the boost operator in the dressing time frame is monotonic. These results show that the notion of an observer can be thought of as emerging from the gravitational degrees of freedom themselves. We briefly mention that the construction offers new insights into focusing conjectures.

Dimension-free bounds for sums of independent matrices and simple tensors via the variational principle

Dimension-free bounds for sums of independent matrices and simple tensors via the variational principle

Kernels of Algebraic Curvature Tensors of Symmetric and Skew-Symmetric Builds

The kernel of an algebraic curvature tensor is a fundamental subspace that can be used to distinguish between different algebraic curvature tensors. Kernels of algebraic curvature tensors built only of canonical algebraic curvature tensors of a single build have been studied in detail. We consider the kernel of an algebraic curvature tensor R that is a sum of canonical algebraic curvature tensors of symmetric and skew-symmetric build. An obvious way to ensure that the kernel of R is nontrivial is to choose the involved bilinear forms such that the intersection of their kernels is nontrivial. We present a construction wherein this intersection is trivial but the kernel of R is nontrivial. We also show how many bilinear forms satisfying certain conditions are needed in order for R to have a kernel of any allowable dimension.

The sum of width-one tensors

The sum of width-one tensors

Existence of the eigenvalues of a tensor sum of the Friedrichs models with rank 2 perturbation

Existence of the eigenvalues of a tensor sum of the Friedrichs models with rank 2 perturbation

Topological interference management via low-rank tensor completion for time-varying topology networks

In this paper, we introduce a tensor-based topological interference management framework, in which a novel low-rank tensor completion (LRTC) model is proposed to manage interferences in time-varying topology networks. We use particular channel decomposition to design the precoding tensor and interference suppression tensor for TIM with multi-antenna users. Furthermore, we formulate the low-rank tensor completion model as a tensor rank minimization problem, which is then solved by resorting to the tensor nuclear norm (TNN) and the tensor sum of nuclear norm (SNN). Extensive simulation results show that the proposed two low-rank tensor completion TIM algorithms have superior degree of freedom (DoF) and sum rate performance in general time-varying networks.

Dehomogenization for completely positive tensors

A real symmetric tensor is completely positive (CP) if it is a sum of symmetric tensor powers of nonnegative vectors. We propose a dehomogenization approach for studying CP tensors. This gives new Moment-SOS relaxations for CP tensors. Detection for CP tensors and the linear conic optimization with CP tensor cones can be solved more efficiently by the dehomogenization approach.

A generalization of Kruskal’s theorem on tensor decomposition

Abstract Kruskal’s theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. We prove a ‘splitting theorem’ for sets of product tensors, in which the k-rank condition of Kruskal’s theorem is weakened to the standard notion of rank, and the conclusion of uniqueness is relaxed to the statement that the set of product tensors splits (i.e., is disconnected as a matroid). Our splitting theorem implies a generalization of Kruskal’s theorem. While several extensions of Kruskal’s theorem are already present in the literature, all of these use Kruskal’s original permutation lemma and hence still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization uses a completely new proof technique, contains many of these extensions and can certify uniqueness below this threshold. We obtain several other useful results on tensor decompositions as consequences of our splitting theorem. We prove sharp lower bounds on tensor rank and Waring rank, which extend Sylvester’s matrix rank inequality to tensors. We also prove novel uniqueness results for nonrank tensor decompositions.

High-Dimensional Seismic Data Reconstruction Based on Linear Radon Transform–Constrained Tensor CANDECOM/PARAFAC Decomposition

Random noise and missing seismic traces are common in field seismic data, which seriously affects the subsequent seismic processing flow. The complete noise-free high-dimensional seismic dataset in the frequency–space (f-x) domain under the local linear assumption are regarded as a low-rank tensor, and each high dimensional seismic dataset containing only one linear event is a rank-1 tensor. The tensor CANDECOM/PARAFAC decomposition (CPD) method estimates complete noise-free seismic signals by characterizing high-dimensional seismic signals as the sum of several rank-1 tensors. In order to improve the stability and effect of the tensor CPD algorithm, this paper proposes a linear Radon transform–constrained tensor CPD method (RCPD) by using the sparsity of factor matrix in the Radon domain after high-dimensional seismic signal tensor CPD and uses alternating direction multiplier method (ADMM) to solve the established optimization problem. This proposed method is an essential realization of the high-dimensional linear Radon transform, and the results of synthetic and field data reconstruction prove the effectiveness of the proposed method.

Salt‐and‐pepper denoising method for colour images based on tensor low‐rank prior and implicit regularization

Most of the information obtained by humans comes from colour images. However, salt-and-pepper noise (SPN) during signal acquisition, encoding, transmission, and decoding easily interferes with the quality of colour images. Most existing SPN denoising methods decompose a colour image into three independent matrices according to the colour channel and then recover each channel signal independently, ignoring the strong data correlation between channels. In addition, most existing SPN denoising methods apply only a single model-driven or data-driven approach and fail to take the advantages of their combination fully. Therefore, we first regard a colour image contaminated by SPN as the sum of an SPN tensor and a tensor with missing data. In this manner, we transform the denoising problem into a low-rank tensor reconstruction problem. We then introduce a model-driven-based parallel matrix factorization low-rank tensor reconstruction algorithm and a data-driven-based FFDNet denoising network to restore the colour image better. The proposed method not only enhances the similarity of the colour image channels but also explores the deep prior of the colour image to capture the image details. Finally, the proposed method is compared with some advanced denoising methods. The results show that the proposed method achieves a competitive denoising performance.

Multilayer Sparsity-Based Tensor Decomposition for Low-Rank Tensor Completion.

Existing methods for tensor completion (TC) have limited ability for characterizing low-rank (LR) structures. To depict the complex hierarchical knowledge with implicit sparsity attributes hidden in a tensor, we propose a new multilayer sparsity-based tensor decomposition (MLSTD) for the low-rank tensor completion (LRTC). The method encodes the structured sparsity of a tensor by the multiple-layer representation. Specifically, we use the CANDECOMP/PARAFAC (CP) model to decompose a tensor into an ensemble of the sum of rank-1 tensors, and the number of rank-1 components is easily interpreted as the first-layer sparsity measure. Presumably, the factor matrices are smooth since local piecewise property exists in within-mode correlation. In subspace, the local smoothness can be regarded as the second-layer sparsity. To describe the refined structures of factor/subspace sparsity, we introduce a new sparsity insight of subspace smoothness: a self-adaptive low-rank matrix factorization (LRMF) scheme, called the third-layer sparsity. By the progressive description of the sparsity structure, we formulate an MLSTD model and embed it into the LRTC problem. Then, an effective alternating direction method of multipliers (ADMM) algorithm is designed for the MLSTD minimization problem. Various experiments in RGB images, hyperspectral images (HSIs), and videos substantiate that the proposed LRTC methods are superior to state-of-the-art methods.

Separable Symmetric Tensors and Separable Anti-symmetric Tensors

In this paper, we first initialize the S-product of tensors to unify the outer product, contractive product, and the inner product of tensors. Then, we introduce the separable symmetry tensors and separable anti-symmetry tensors, which are defined, respectively, as the sum and the algebraic sum of rank-one tensors generated by the tensor product of some vectors. We offer a class of tensors to achieve the upper bound for \(\texttt {rank}({\mathcal {A}}) \leqslant 6\) for all tensors of size \(3\times 3\times 3\). We also show that each \(3\times 3\times 3\) anti-symmetric tensor is separable.

A sub-sampled tensor method for nonconvex optimization

Abstract A significant theoretical advantage of high-order optimization methods is their superior convergence guarantees. For instance, third-order regularized methods reach an $(\epsilon _1,\epsilon _2,\epsilon _3)$third-order critical point in at most ${\mathcal {O}} (\max (\epsilon _1^{-4/3}, \epsilon _2^{-2}, \epsilon _3^{-4} ) )$ iterations. However, the cost of computing high-order derivatives is prohibitively expensive in real applications, including for instance many real-world machine learning tasks. In order to address this problem, we present a sub-sampled optimization method that uses a third-order regularized model to find local minima of smooth and potentially nonconvex objective functions with a finite-sum structure. This algorithm uses sub-sampled derivatives instead of exact quantities and is guaranteed to converge to a third-order critical point. Our analysis relies on a novel tensor concentration inequality for sums of tensors of any order that makes explicit use of the finite-sum structure of the objective function.

Black bounces, wormholes, and partly phantom scalar fields

Simpson and Visser recently proposed a phenomenological way to avoid some kinds of space-time singularities by replacing a parameter whose zero value corresponds to a singularity (say, $r$) with the manifestly nonzero expression $r(u) = \sqrt{u^2 + b^2}$, where $u$ is a new coordinate, and $b =$ \const $>0$. This trick, generically leading to a regular minimum of $r$ beyond a black hole horizon (called a "black bounce"), may hopefully mimic some expected results of quantum gravity, and was previously applied to regularize the Schwarzschild, Reissner-Nordstr\"om, Kerr and some other metrics. In this paper it is applied to regularize the Fisher solution with a massless canonical scalar field in general relativity (resulting in a traversable wormhole) and a family of static, spherically symmetric dilatonic black holes (resulting in regular black holes and wormholes). These new regular metrics represent exact solutions of general relativity with a sum of stress-energy tensors of a scalar field with a nonzero self-interaction potential and a magnetic field in the framework of nonlinear electrodynamics with a Lagrangian function $L(F)$, $F = F_{\mu\nu} F^{\mu\nu}$. A novel feature in the present study is that the scalar fields involved have "trapped ghost" properties, that is, are phantom in a strong-field region and canonical outside it, with a smooth transition between the regions. It is also shown that any static, spherically symmetric metric can be obtained as an exact solution to the Einstein equations with the stress-energy tensor of the above field combination.

Convenient Tail Bounds for Sums of Random Tensors

This work prepares new probability bounds for sums of random, independent, Hermitian tensors. These probability bounds characterize large-deviation behavior of the extreme eigenvalue of the sums of random tensors. We extend Laplace transform method and Lieb's concavity theorem from matrices to tensors, and apply these tools to generalize the classical bounds associated with the names Chernoff, Bennett, and Bernstein from the scalar to the tensor setting. Tail bounds for the norm of a sum of random rectangular tensors are also derived from corollaries of random Hermitian tensors cases. The proof mechanism can also be applied to tensor-valued martingales and tensor-based Azuma, Hoeffding and McDiarmid inequalities are established.

F(T,Θ) gravity and generalized gravitational baryogenesis

Where is the anti-matter in our universe? This is the most significant open problem of the modern era. This paper is devoted to the study of cosmological impact of gravitational baryogenesis and its generalized case driven by [Davoudiasl et al., Phys. Rev. Lett. 93, 201301 (2004)]. In this class of baryogenesis, the Ricci scalar [Formula: see text] is derivatively coupled to the baryon current [Formula: see text] as [Formula: see text]. We examine the imbalance of baryon and anti-baryon of the universe in the context of [Formula: see text] gravity, where [Formula: see text] and [Formula: see text] are the torsion scalar and the trace of the energy–momentum tensor, respectively. We assume two [Formula: see text] models in which the first one contains the sum of second-order torsion tensors and the trace of energy–momentum tensor while the second one is the product of these terms. For all cases, we find the ratio of baryon number to entropy by considering the power-law scale factor. Finally, we compare our results with the observational value of baryon-number-density-to-entropy ratio. The baryon-to-entropy ratio against the first model lies in the range [Formula: see text] while for the second model it contains [Formula: see text]. For generalized case of gravitational baryogenesis, we also obtain results that are compatible with the observation limit.