Small Tensor Research Articles

3D Tensor Auto-encoder with Application to Video Compression

Auto-encoder has been widely used to compress high-dimensional data such as the images and videos. However, the traditional auto-encoder network needs to store a large number of parameters. Namely, when the input data is of dimension n , the number of parameters in an auto-encoder is in general O ( n ). In this article, we introduce a network structure called 3D Tensor Auto-Encoder (3DTAE). Unlike the traditional auto-encoder, in which a video is represented as a vector, our 3DTAE considers videos as 3D tensors to directly pass tensor objects through the network. The weights of each layer are represented by three small matrices, and thus the number of parameters in 3DTAE is just O ( n 1/3). The compact nature of 3DTAE fits well the needs of video compression. Given an ensemble of high-dimensional videos, we represent them as 3DTAE networks plus some small core tensors, and we further quantize the network parameters and the core tensors to get the final compressed data. Experimental results verify the efficiency of 3DTAE.

A Dimensionality Reduction Algorithm for Unstructured Campus Big Data Fusion

Data modeling and dimensionality reduction are important research points in the field of big data. At present, there is no effective model to realize the consistent representation and fusion of different types of data of students in unstructured campus big data. In addition, in the process of big data processing, the amount of data is too large and the intermediate results are too complex, which seriously affects the efficiency of big data dimension reduction. To solve the above problems, this paper proposes an incremental high order singular value decomposition dimensionality (icHOSVD) reduction algorithm for unstructured campus big data. In this algorithm, the characteristics of audio, video, image and text data in unstructured campus student data are tensioned to form a sub-tensor model, and the semi-tensor product is used to fuse the sub-tensor model into a unified model as the individual student tensor model. On the basis of individual model fusion, the campus big data fusion model was segmented, and each segmented small tensor model was dimensioned by icHOSVD reduction to obtain an approximate tensor as the symmetric tensor that could replace the original tensor, so as to solve the problem of large volume of tensor fusion model and repeated calculation of intermediate results in data processing. The experimental results show that the proposed algorithm can effectively reduce the computational complexity and improve the performance compared with traditional data dimension reduction algorithms. The research results can be applied to campus big data analysis and decision-making.

Almost complex manifolds with small Nijenhuis tensor

Abstract We give several explicit examples of compact manifolds with a 1-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the C 0-norm. The 4-dimensional examples possess no complex structure, whereas the 6-dimensional example does not possess a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.

The Strain/Stress Fields of a Subsurface Rectangular Dislocation Loop Parallel to the Surface of a Half Medium: Analytical Solution with Verification

The strain and stress fields of a rectangular dislocation loop in an isotropic solid that is a semi-infinite medium (half medium) are developed here for a Volterra-type dislocation. Specifically, the loop is parallel to the free surface of the solid. The elastic fields of the dislocation loop are developed by integrating the displacement equation of infinitesimals dislocation loops over a finite rectangular loop area below the free surface. The strains and stress then follow from the small strain tensor and Hooke’s law for isotropic materials, respectively. In this paper, analytical verification and numerical verification for the elastic fields are both demonstrated. Equilibrium equations and strain compatibility equations are applied in the verification. Also, a comparison with a newly-developed numerical method for dislocations near a free surface is performed as well. The developed solution is a function of the loop depth beneath the surface and can be used as a fundamental solution to solve elasticity, plasticity or dislocation problems.

Inexact Generalized Gauss–Newton for Scaling the Canonical Polyadic Decomposition With Non-Least-Squares Cost Functions

The canonical polyadic decomposition (CPD) allows one to extract compact and interpretable representations of tensors. Several optimization-based methods exist to fit the CPD of a tensor for the standard least-squares (LS) cost function. Extensions have been proposed for more general cost functions such as β-divergences as well. For these non-LS cost functions, a generalized Gauss-Newton (GGN) method has been developed. This is a second-order method that uses an approximation of the Hessian of the cost function to determine the next iterate and with this algorithm, fast convergence can be achieved close to the solution. While it is possible to construct the full Hessian approximation for small tensors, the exact GGN approach becomes too expensive for tensors with larger dimensions, as found in typical applications. In this paper, we therefore propose to use an inexact GGN method and provide several strategies to make this method scalable to large tensors. First, the approximation of the Hessian is only used implicitly and its multilinear structure is exploited during Hessian-vector products, which greatly improves the scalability of the method. Next, we show that by using a compressed instance of the GGN Hessian approximation, the computation time of the inexact GGN method can be lowered even more, with only limited influence on the convergence speed. We also propose dedicated preconditioners for the problem. Further, the maximum likelihood estimator for Rician distributed data is examined in detail as an example of an alternative cost function. This cost function is useful for the analysis of the moduli of complex data, as in functional magnetic resonance imaging, for instance. We compare the proposed method to the existing CPD methods and demonstrate the method's speed and effectiveness on synthetic and simulated real-life data. Finally, we show that the method can scale by using randomized block sampling.

Yet Another Tensor Toolbox for Discontinuous Galerkin Methods and Other Applications

The numerical solution of partial differential equations is at the heart of many grand challenges in supercomputing. Solvers based on high-order discontinuous Galerkin (DG) discretisation have been shown to scale on large supercomputers with excellent performance and efficiency if the implementation exploits all levels of parallelism and is tailored to the specific architecture. However, every year new supercomputers emerge and the list of hardware-specific considerations grows simultaneously with the list of desired features in a DG code. Thus, we believe that a sustainable DG code needs an abstraction layer to implement the numerical scheme in a suitable language. We explore the possibility to abstract the numerical scheme as small tensor operations, describe them in a domain-specific language (DSL) resembling the Einstein notation, and to map them to small General Matrix-Matrix Multiplication routines. The compiler for our DSL implements classic optimisations that are used for large tensor contractions, and we present novel optimisation techniques such as equivalent sparsity patterns and optimal index permutations for temporary tensors. Our application examples, which include the earthquake simulation software SeisSol, show that the generated kernels achieve over 50% peak performance of a recent 48-core Skylake system while the DSL considerably simplifies the implementation.

On the fragility of complex material systems

In this work we investigate, from both a macroscopic and a microscopic point of view, the fragility degree of complex material systems. In particular, we integrate a microscopic and a macroscopic approach. The former is formulated in the framework of non-equilibrium thermodynamics in order to obtain the phenomenological equations for the heat flux and viscous pressure tensor, as functions of the absolute temperature T and of the small strain tensor, in the isotropic case, where the shear viscosity is supposed to depend on T g /T . The latter is connected with a macroscopic definition of fragility, operatively introduced by Angell for glass-forming systems, and with the definition of thermal restraint, based on elastic incoherent neutron scattering data. On that score literature data of viscosity, as a function of concentration and temperature, and literature data of elastic intensity data are correlated with the system fragility and with the system thermal restraint. The obtained results are of interest for a wide range of innovative materials, such as, for example, carbon nanotubes and bioprotectant systems.

Large Scale Tensor Factorization via Parallel Sketches

Tensor factorization methods have recently gained increased popularity. A key feature that renders tensors attractive is the ability to directly model multi-relational data. In this work, we propose ParaSketch, a parallel tensor factorization algorithm that enables massive parallelism, to deal with large tensors. The idea is to compress the large tensor into multiple small tensors, decompose each small tensor in parallel, and combine the results to reconstruct the desired latent factors. Prior art in this direction entails potentially very high complexity in the (Gaussian) compression and final combining stages. Adopting sketching matrices for compression, the proposed method enjoys a dramatic reduction in compression complexity, and features a much lighter combining step. Moreover, theoretical analysis shows that the compressed tensors inherit latent identifiability under mild conditions, hence establishing correctness of the overall approach. Numerical experiments corroborate the theory and demonstrate the effectiveness of the proposed algorithm.

Active Mass Ejection for Asteroid Manipulation and Deflection

The dynamics of asteroid deflection and manipulation using mass ejection are examined. First, a linearized small angular velocity and inertia tensor deviation model for close-to-principal-axis rotators is developed. The model is then applied to launching boulders off of asteroid surfaces, proving that the principal-axis-rotator characteristic can be kept while removing material off the surface. Mass ejections for asteroid deflection are presented and examined. This method, dubbed mass driver deflection, uses small discrete launches of masses off an asteroid to prevent its collision with Earth. By using material from the asteroid itself, such as boulders or regolith deposits, mass driver deflection substantially reduces the required mass of the deflection system. The analysis in the paper seeks to optimize the deflection efforts while minimizing unwanted effects on the deflected asteroid’s state: both rotational and orbital. The results show that deflection is possible in timeframes of several years in a variety of scenarios and that the deflection effects on the asteroid behavior do not pose a risk of disrupting the asteroid in a catastrophic way.

Robust Tensor Factorization for Color Image and Grayscale Video Recovery

Low-rank tensor completion (LRTC) plays an important role in many fields, such as machine learning, computer vision, image processing, and mathematical theory. Since rank minimization is an NP-hard problem, one strategy is that it is converted into a convex relaxation tensor nuclear norm (TNN) that requires the repeated calculation of time-consuming SVD, and the other is to convert it into some product of two smaller tensors that are easy to fall into the local minimum. In order to overcome the above shortcomings, we propose a robust tensor factorization (RTF) model for solving LRTC. In RTF, the noisy tensor data with missing entries is decomposed into low-rank tensor and noisy tensor, and then the low-rank tensor is equivalently decomposed into t-products (essentially vectors convolution) of two smaller tensors: orthogonal dictionary tensor and low-rank representation tensor. Meanwhile, the TNN of low-rank representation tensor is adopted to characterize the low-rank structure of the tensor data for preserving global information. Then, an effective iterative update algorithm based on the alternating direction method of multipliers (ADMM) is proposed to solve RTF. Finally, numerical experiments on image recovering and video completion tasks show the effectiveness of the proposed RTF model compared with several state-of-the-art tensor completion models.

On Strassen's Rank Additivity for Small Three-way Tensors

We address the problem of the additivity of the tensor rank. That is, for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov. The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces. In this article we prove that for some small three-way tensors the additivity holds. For instance, if the rank of one of the tensors is at most 6, then the additivity holds. Or, if one of the tensors lives in ${\mathbb C}^k\otimes {\mathbb C}^3\otimes {\mathbb C}^3$ for any $k$, then the additivity also holds. More generally, if one of the tensors is concise and its rank is at most 2 more than the dimension of one of the linear spaces, then additivity holds. In addition we also treat some cases of the additivity of the border rank of such tensors. In particular, we show that the additivity of the border rank holds if the direct sum tensor is contained in ${\mathbb C}^4\otimes {\mathbb C}^4\otimes {\mathbb C}^4$. Some of our results are valid over an arbitrary base field.

Low-Tubal-Rank Tensor Completion Using Alternating Minimization

The low-tubal-rank tensor model has been recently proposed for real-world multidimensional data. In this paper, we study the low-tubal-rank tensor completion problem, i.e., to recover a third-order tensor by observing a subset of its elements selected uniformly at random. We propose a fast iterative algorithm, called Tubal-AltMin , that is inspired by a similar approach for low-rank matrix completion. The unknown low-tubal-rank tensor is represented as the product of two much smaller tensors with the low-tubal-rank property being automatically incorporated, and Tubal-AltMin alternates between estimating those two tensors using tensor least squares minimization. First, we note that tensor least squares minimization is different from its matrix counterpart and nontrivial as the circular convolution operator of the low-tubal-rank tensor model is intertwined with the sub-sampling operator. Secondly, the theoretical performance guarantee is challenging since Tubal-AltMin is iterative and nonconvex. We prove that 1) Tubal-AltMin generates a best rank- $r$ approximate up to any predefined accuracy $\epsilon $ at an exponential rate, and 2) for an $n \times n \times k$ tensor $\mathcal {M}$ with tubal-rank $r \ll n$ , the required sampling complexity is $O((nr^{2}k ||\mathcal {M}||_{F}^{2} \log ^{3}~n) / \overline {\sigma }_{rk}^{2})$ , where $\overline {\sigma }_{rk}$ is the $rk$ -th singular value of the block diagonal matrix representation of $\mathcal {M}$ in the frequency domain, and the computational complexity is $O(n^{2}r^{2}k^{3} \log n \log (n/\epsilon))$ . Finally, on both synthetic data and real-world video data, evaluation results show that compared with tensor-nuclear norm minimization using alternating direction method of multipliers (TNN-ADMM), Tubal-AltMin-Simple (a simplified implementation of Tubal-AltMin) improves the recovery error by several orders of magnitude. In experiments, Tubal-AltMin-Simple is faster than TNN-ADMM by a factor of 5 for a $200 \times 200 \times 20$ tensor.

Even-order Toeplitz tensor: framework for multidimensional structured linear systems

In this paper, we introduce tensors with Toeplitz structure. These structured tensors occur in different kinds of applications such as discretization of multidimensional PDE’s or Fredholm integral equations with an invariant kernel. We investigate the main properties of the new structured tensor and show the tensor contractive product with such tenors can be carried out with the fast Fourier transform. Also, we show that approximation of Toeplitz tensors with a specially structured tensor (that will be named $$\circledast $$-product tensors) can be reduced to the rank-1 approximation of a smaller tensor. Tensor equations with such $$\circledast $$-product coefficient tenors can be solved by a direct method. So, this approximation of a Toeplitz tensor can be used to find an approximate solution of the original tensor equation or can be used as a preconditioner. Our main goal is to show the ability of the tensor framework to handle structured multidimensional problems in their original format.

On isospectral compactness in conformal class for 4-manifolds

Let [Formula: see text] be a closed 4-manifold with positive Yamabe invariant and with [Formula: see text]-small Weyl curvature tensor. Let [Formula: see text] be any metric in the conformal class of [Formula: see text] whose scalar curvature is [Formula: see text]-close to a constant. We prove that the set of Riemannian metrics in the conformal class [Formula: see text] that are isospectral to [Formula: see text] is compact in the [Formula: see text] topology.

How to produce an arbitrarily small tensor to scalar ratio

In this paper, construct a toy a model which demonstrates that large field single scalar inflation can produce an arbitrarily small scalar to tensor ratio in the window of e-foldings recoverable from CMB experiments. This is done by generalizing the [Formula: see text]-attractor models to allow the potential to approach a constant as rapidly as we desire for super-Planckian field values. This implies that a non-detection of [Formula: see text] alone can never rule out entirely the theory of large field inflation.

A sphere theorem for Bach-flat manifolds with positive constant scalar curvature

A classic gap theorem for complete non-compact Ricci flat manifolds by M. Anderson suggests that one can get the rigidity of certain spaces simply by passing local curvature estimates on geodesic balls to global. Using similar ideas, Kim showed that a 4-dimensional complete non-compact Bach-flat manifold with vanishing scalar curvature and small L2-curvature tensor has to be flat. Unfortunately, this method does not generalize to compact manifolds without boundary. Applying a different approach, we show that a closed Bach-flat Riemannian manifold with positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either L∞ or Ln2-norm. These results generalize a rigidity theorem of positive Einstein manifolds due to M.-A. Singer. As an application, we can partially recover the well-known Chang–Gursky–Yang's 4-dimensional conformal sphere theorem.

Normal projected entangled pair states generating the same state

Tensor networks (TNs) are generated by a set of small rank tensors and define many-body quantum states in a succinct form. The corresponding map is not one-to-one: different sets of tensors may generate the very same state. A fundamental question in the study of TNs naturally arises: what is then the relation between those sets? The answer to this question in one-dimensional setups has found several applications, like the characterization of local and global symmetries, the classification of phases of matter and unitary evolutions, or the determination of the fixed points of renormalization procedures. Here we answer this question for projected entangled pair states in any dimension and lattice geometry (including, for example, the Kagome lattice, hyperbolic lattices, or tree tensor networks), as long as the tensors generating the states are normal, which constitute an important and generic class.

An Order of Smallness of the Poynting Effect from the Standpoint of the Tensor Nonlinear Functions Apparatus

A class of constitutive relations is considered that connect symmetric stress and small strain tensors in three-dimensional space using an isotropic potential tensor nonlinear function of a rather general form. Various definitions of tensor nonlinearity are given and their equivalence is shown. From the standpoint of the mathematical apparatus of the theory of tensor nonlinear functions, the interpretation of the Poynting effect known in experimental mechanics and similar phenomena has been carried out. It is proved that these effects are not necessarily the result of the tensor nonlinearity of the defining relations, but may be due to the dependence on one of the material functions on the quadratic invariant, which is absent, for example, in the physically linear case. From here conclusions are drawn about the order of smallness of these effects. The possibility of modeling the Poynting effect by tensor-linear defining relations is discussed.

Detection of deformation anisotropy of tuff by a single triaxial test on a single specimen

Rock anisotropy has conventionally been determined by numerous tests using many specimens sampled at several orientations, however, it is both costly and time consuming. The anisotropic deformation properties (transversely isotropic elasticity and its dominant orientation) of a tuffaceous rock sample obtained in Utsunomiya city, Japan, and characterized by a clear bedding plane, were determined by a newly proposed method, namely, a single triaxial test using a single specimen. Using this method, the dominant orientations of anisotropy are determined by the principal orientations of the strain during isotropic consolidation, and the elastic parameters can be determined by stresses and strains during triaxial compression. This method required the implementation of a triaxial testing apparatus with a newly designed cap including a slider mechanism and low-friction sheets. The six components of the symmetrical, small strain tensors of the tuff specimen were measured using nine strain gauges, while the 3D principal strains were evaluated during uniaxial and consolidated drained triaxial tests. To verify the proposed method, the anisotropic properties of four specimens sampled from a cubic block at different orientations, as characterized by the proposed method, were compared with those determined by multiple uniaxial tests using the specimens sampled for different orientations. The strain responses of the uniaxial tests demonstrated that the Young's modulus of the tuff in the bedding orientation is 1.5 times greater than that for the perpendicular orientation. The orientation of the principal strains deviate from the loading orientation due to the orientation of the bedding plane, and the orientations during isotropic consolidation were inclined in accordance with the dip orientation of the bedding plane. Similar values were evaluated for each anisotropic property determined by both a single triaxial test on each specimen and multiple uniaxial tests. The obtained values demonstrated that the Young's moduli for the bedding orientation are 1.5–2.7 times larger than that in the perpendicular orientation.

On infinitesimal and finite deformations in shape memory alloys

Two material models for shape memory alloys are developed and are evaluated by comparing and contrasting their predictions in three standard problems involving martensitic transformations. Both models are based on generalized plasticity and comprise a von Mises type of expression for the loading surfaces, a linear evolution law for the material martensite fraction, and a hyperelastic constitutive equation. The first model is an infinitesimal one, based on the usual additive decomposition of the small strain tensor into elastic and inelastic (transformation-induced) parts, while the second is a finite one based on the consistent use of the “physical” (intrinsic material) metric concept. This study reveals that in the first and second problem—uniaxial tension and simple shear—and under small and moderate levels of strain, both models predict almost identical response, while for higher levels of strain the models still predict comparable response, even though their basic kinematical assumptions differ vastly. The third problem, comprising infinitesimal shear with finite rotation, is considered next. In this case, it is shown that while the finite model yields the physically correct response, the infinitesimal model yields completely erroneous results.