Tensor Diagonalization Research Articles

Jacobi-type algorithms for homogeneous polynomial optimization on Stiefel manifolds with applications to tensor approximations

This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous polynomials subject to a constraint on a Stiefel manifold, we reformulate it as an optimization problem on a unitary group, which makes it possible to apply the gradient-based Jacobi-type (Jacobi-G) algorithm. Then, if the subproblem can always be represented as a quadratic form, we establish the global convergence of Jacobi-G under any one of three conditions. The convergence result for the first condition is an easy extension of the result by Usevich, Li, and Comon [SIAM J. Optim. 30 (2020), pp. 2998–3028], while other two conditions are new ones. This algorithm and the convergence properties apply to the well-known joint approximate symmetric tensor diagonalization. For the second class of homogeneous polynomials subject to constraints on the product of Stiefel manifolds, we reformulate it as an optimization problem on the product of unitary groups, and then develop a new gradient-based multiblock Jacobi-type (Jacobi-MG) algorithm to solve it. We establish the global convergence of Jacobi-MG under any one of the above three conditions, if the subproblem can always be represented as a quadratic form. This algorithm and the convergence properties are suitable to the well-known joint approximate tensor diagonalization. As the proximal variants of Jacobi-G and Jacobi-MG, we also propose the Jacobi-GP and Jacobi-MGP algorithms, and establish their global convergence without any further condition. Some numerical results are provided indicating the efficiency of the proposed algorithms.

Trace maximization algorithm for the approximate tensor diagonalization

In this paper, we develop a Jacobi-type algorithm for the approximate diagonalization of tensors of order via tensor trace maximization. For a general tensor, this is an alternating least squares algorithm and the rotation matrices are chosen in each mode one-by-one to maximize the tensor trace. On the other hand, for symmetric tensors, we discuss a structure-preserving variant of this algorithm where in each iteration the same rotation is applied in all modes. We show that both versions of the algorithm converge to the stationary points of the corresponding objective functions.

Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization

Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization

A generalized strain energy function using fractional powers: Application to isotropy, transverse isotropy, orthotropy, and residual stress symmetry

In this paper, we propose a generalized strain energy density function based on invariants of stretch tensor with arbitrary exponents. We employ polynomial, logarithmic and exponential functions of these invariants to develop the strain energy functions. We also study characteristics and applications of the proposed model for isotropy, transverse isotropy, orthotropy with a special focus on initial/residual stress symmetry. The proposed invariants generalize the existing strain energy potentials constructed with invariants of Cauchy stretch. We construct generalized strain energy functions for initial stress problems using initial stress symmetries. We obtain objectivity of strain energy for initially stressed transversely isotropic solids to derive the invariants and study resulting symmetry. By extending Dunford–Taylor integration based approach and tensor diagonalization approach, we obtain stress through differentiation of anisotropic scalar invariants with respect to a tensor. These approaches are usually applicable for derivative of isotropic tensor functions with respect to tensors. Using initial stress compatibility, we derive constraint equations for material parameters by evaluating the limits of Cauchy stress in the reference configuration. In order to apply the proposed model for initial stress problems, we further investigate bending and unbending of hyperelastic structures. We study bending of a rectangle to a cylinder and unbending of a cylinder to a rectangle in presence of initial/residual stress and observe both magnifying and moderating effects of initial stress for stress distribution and flexural characteristics. Using experimental data we substantiate the proposed model for seat foam, bovine pericardium and rabbit skin which represent compressible isotropic and incompressible orthotropic materials. For demonstration, we use both polynomial and exponential functions of these invariants. Furthermore, we develop a nonlinear finite element computational model for thermo-hyperelastic structure where thermal stress represents the initial stress. We corroborate this stress–strain data with the proposed model for initial stress symmetry and observe nice agreements.

Approximate Matrix and Tensor Diagonalization by Unitary Transformations: Convergence of Jacobi-Type Algorithms

We propose a gradient-based Jacobi algorithm for a class of maximization problems on the unitary group, with a focus on approximate diagonalization of complex matrices and tensors by unitary transformations. We provide weak convergence results, and prove local linear convergence of this algorithm. The convergence results also apply to the case of real-valued tensors.

Non-orthogonal tensor diagonalization

Tensor diagonalization means transforming a given tensor to an exactly or nearly diagonal form through multiplying the tensor by non-orthogonal invertible matrices along selected dimensions of the tensor. It has a link to an approximate joint diagonalization (AJD) of a set of matrices. In this paper, we derive (1) a new algorithm for a symmetric AJD, which is called two-sided symmetric diagonalization of an order-three tensor, (2) a similar algorithm for a non-symmetric AJD, also called a two-sided diagonalization of an order-three tensor, and (3) an algorithm for three-sided diagonalization of order-three or order-four tensors. The latter two algorithms may serve for canonical polyadic (CP) tensor decomposition, and in certain scenarios they can outperform traditional CP decomposition methods. Finally, we propose (4) similar algorithms for tensor block diagonalization, which is related to tensor block-term decomposition. The proposed algorithm can either outperform the existing block-term decomposition algorithms, or produce good initial points for their application.

Orientation-Independent Tensor Voting Analysis

Tensor voting framework is an approach for perceptual organization. A tensor can provide more information than scalar or vector in image processing. However, the structure of tensor also makes it not unique but orientation dependent. In this paper, to quantify properly the intrinsic orientation-independent voting process, we proposed a new description of the tensor fields, which consists of three rotationally invariant quantities. Instead of coordinate transformation, this approach does not require tensor diagonalization or eigenvalue calculation. Therefore, our approach is not susceptible to potential artifacts induced during these number manipulations, meanwhile simplified the voting process at the same time.

Analysis and correction of gradient nonlinearity bias in apparent diffusion coefficient measurements

Gradient nonlinearity of MRI systems leads to spatially dependent b-values and consequently high non-uniformity errors (10-20%) in apparent diffusion coefficient (ADC) measurements over clinically relevant field-of-views. This work seeks practical correction procedure that effectively reduces observed ADC bias for media of arbitrary anisotropy in the fewest measurements. All-inclusive bias analysis considers spatial and time-domain cross-terms for diffusion and imaging gradients. The proposed correction is based on rotation of the gradient nonlinearity tensor into the diffusion gradient frame where spatial bias of b-matrix can be approximated by its Euclidean norm. Correction efficiency of the proposed procedure is numerically evaluated for a range of model diffusion tensor anisotropies and orientations. Spatial dependence of nonlinearity correction terms accounts for the bulk (75-95%) of ADC bias for FA = 0.3-0.9. Residual ADC non-uniformity errors are amplified for anisotropic diffusion. This approximation obviates need for full diffusion tensor measurement and diagonalization to derive a corrected ADC. Practical scenarios are outlined for implementation of the correction on clinical MRI systems. The proposed simplified correction algorithm appears sufficient to control ADC non-uniformity errors in clinical studies using three orthogonal diffusion measurements. The most efficient reduction of ADC bias for anisotropic medium is achieved with non-lab-based diffusion gradients.

On the Stiffness of the Tangent Modulus Tensor in Elastoplasticity

In finite element analysis of pressure vessels undergoing elastoplastic deformation, low stiffness of the tangent modulus tensor will engender low stiffness in the tangent stiffness matrix, posing a risk of computational difficulties such as poor convergence. The current investigation presents the explicit tangent modulus tensor in an elastoplastic model based on a Von Mises yield surface with isotropic work hardening, and the associated flow rule. The stiffness of the tangent modulus tensor is assessed by deriving explicit expressions for its minimum eigenvalue using both tensor diagonalization and Rayleigh quotient minimization. The derived expressions are validated computationally. Using the minimum eigenvalue, the stiffness is found to depend on the current path in stress space. The results of the current investigation suggest a way of following a stress path, which bypasses low stiffness, while attaining the prescribed load.

SU‐GG‐J‐150: Analyses of Multiple Q‐Sampled Diffusion Tensor Amplitudes for Pseudoprogressive GBM

Motivation: The current temozolomide (TMZ) concomitant with radiotherapy, can lead to earlier disruption of the blood‐brain barrier (BBB) than the conventional radiotherapy alone. 1∼2 months after concomitant radio‐chemotherapy, the BBB disruption from the therapeutic response may be radiologically evidenced as an apparently progressive (i.e. pseudoprogressive) lesion. There are few convincing functional imaging studies, e.g. perfusion MRI, F‐18 PET, that can clearly differentiate true tumor from pseudoprogressive one. The multiple‐q‐sampled diffusion tensor imaging (DTT) analyses, i.e. multiple “diffusion weighted echo‐amplitudes” analyzed in q‐space along the resolved diffusion characterized directions, can associate a certain diffusion population with diffusion spectral function of “q”. The hypotheses of this study: (1) large diffusive displacements with dispersive characteristic directions correspond to the BBB or edema area; (2) the range of anisotropic displacement becoming larger reflects deteriorating white matter (WM) glioma infiltration. Method: Each of 10 q‐sampled‐amplitudes (q: 0∼380cm−1) with 6 DTI‐sensitized‐directions is analyzed by 3D‐ellipsoidal diffusion model. Then Eigen‐values and Eigenvectors for each of amplitudes sampled with various q‐values are resolved by tensor diagonalization processes. Results: The DTI data sampled by q‐value lower than 180cm−1 illustrates the averaged population over a wider diffusive range without anisotropic enhancement and the one higher q‐value (e.g. q>250cm−1) excluding larger displacement spectrum shows anisotropic specification. As scanned by conventional T2‐weighted imaging or apparent‐diffusion‐coefficient mapping, GBM‐brain images are overwhelmingly dominated by larger diffusive components and incapable of differentiating BBB from infiltrated WM. Our preliminary results show the multiple q‐sampled DTI analyses can distinguish the regions of increasing diffusive displacements with dispersive diffusive directions (assuming BBB or edema) from the increasing secretion along WM fibers (assuming infiltrating WM). Conclusion: The spectrum of diffusive displacement are possibly differentiated by multiple q‐sampled DTI, and this technique can be utilized for a better diagnosis and prognosis for GBM treatment studies or WM diseases.

Very long period seismic signals observed before the caldera formation with the 2000 Miyake‐jima volcanic activity, Japan

Very long period seismic signals, whose waveform consists of an initial impulsive signal and a later oscillatory wave with 0.2 or 0.4 Hz in dominant frequency, were observed before the caldera formation in the 2000 activity of Miyake‐jima volcano, Japan. The results of waveform inversion show that the initial and later parts can be explained by a northward single force of 1.5 × 108 N working at a depth of 2 km beneath the summit and a moment tensor solution at a depth of 5 km below and 2 km southwest of the summit with ∼1012 Nm, respectively. A clear positive correlation of the amplitudes between the two sources strongly suggests that the shallow single force triggers the deeper moment source in spite of the several km distance between the two sources. To analyze the source time functions of the moment tensor that do not always oscillate in phase, we introduce a new method of moment tensor diagonalization which is performed in the frequency domain. According to the analysis, the two principal components have similar amplitudes and are greater than the third principal component, suggesting an axially symmetric oscillation. One of the possible systems is a combination of two cracks intersecting perpendicularly. Our interpretation is that the single force was generated when magma containing rock blocks suddenly began to move in a choked subsurface magma path, and the resultant pressure waves propagated and excited a resonance oscillation of the two cracks.

Computation of the fractional anisotropy and mean diffusivity maps without tensor decoding and diagonalization: Theoretical analysis and validation.

Diffusion tensor MRI (DT-MRI) is a promising modality for in vivo mapping of the organization of deep tissues. The most commonly used DT-MRI invariant maps are the mean diffusivity, mu(D), relative anisotropy (RA), and fractional anisotropy (FA). Because of the computational burden, anisotropy maps are generally computed offline. The availability of a simple procedure to compute RA, FA, and mu(D) online would make DT-MRI more useful in clinical applications that require immediate feedback. In this study, analytical expressions that relate the commonly used tensor anisotropy measures obtained from the decoded and diagonalized DT with those obtained from the first and second moments of the measured diffusion-weighted (DW) data are derived. Specifically, it is shown that for the principal icosahedron encoding scheme, RA is related to the mean and standard deviation (SD) of the DW measurements that can be computed online. Since FA is commonly used as an anisotropy measure, an analytical expression relating RA and FA was derived from the tensor invariants. These results were validated using both Monte Carlo simulations and high-resolution, normal whole-brain DT-MRI measurements acquired with different b-factors, encoding schemes, and signal-to-noise ratio (SNR) levels. The bias introduced by the rotationally variant encoding schemes into the diffusion measures is also investigated.

A generalization of joint-diagonalization criteria for source separation

In the field of blind source separation, joint-diagonalization-based approaches constitute an important framework, leading to useful algorithms such as the popular joint approximate diagonalization of eigenmatrices (JADE) and simultaneous third-order tensor diagonalization (STOTD) algorithms. However, they are often restricted to the case of cumulants of order four. In this paper, we extend the results leading to JADE and STOTD to cumulants of any order greater than or equal to three by exhibiting a new family of contrast functions that constitutes then a unified framework for the above known results. This also leads us to generalize some links between contrast functions and joint-diagonalization criteria on which these algorithms are based. In turn, one contrast of the new family allows us to show that a function previously proposed as a separation criterion is also a contrast. Moreover, for the two generalized JADE and STOTD contrasts, the analytical optimal solution in the case of two sources is derived and shown to keep the same simple expression, whatever the cumulant order. Finally, some computer simulations illustrate the potential advantage one can take by considering statistics of different orders for the joint-diagonalization of cumulant matrices.

Independent component analysis and (simultaneous) third-order tensor diagonalization

Comon's (1994) well-known scheme for independent component analysis (ICA) is based on the maximal diagonalization, in a least-squares sense, of a higher-order cumulant tensor. In a previous paper, we proved that for fourth-order cumulants, the computation of an elementary Jacobi rotation is equivalent to the computation of the best rank-1 approximation of a fourth-order tensor. In this paper, we show that for third-order tensors, the computation of an elementary Jacobi rotation is again equivalent to a best rank-1 approximation; however, here, it is a matrix that has to be approximated. This favorable computational load makes it attractive to do "something third-order-like" for fourth-order cumulant tensors as well. We show that simultaneous optimal diagonalization of "third-order tensor slices" of the fourth-order cumulant is a suitable strategy. This "simultaneous third-order tensor diagonalization" approach (STOTD) is similar in spirit to the efficient JADE-algorithm.

Orientation-independent diffusion imaging without tensor diagonalization: anisotropy definitions based on physical attributes of the diffusion ellipsoid.

Diffusion tensor imaging can provide a complete description of the diffusion process in tissue. However, this description is not unique but is orientation dependent, and, to quantify properly the intrinsic orientation-independent diffusion properties of the tissue, a set of three rotationally invariant quantities is needed. Instead of using the tensor eigenvalues for this, we define a new set consisting of scaled invariants that have the proper magnitude of actual diffusion constants and that are directly related to the physical attributes of the diffusion ellipsoid, namely, its average radius, surface, and volume. Using these three physical invariants, a new family of anisotropy measures is defined that are normalized between zero (isotropic) and one (completely anisotropic). Because rotational invariants are used, this approach does not require tensor diagonalization and eigenvalue determination and is therefore not susceptible to potential artifacts induced during these number manipulations. The relationship between the new anisotropy definitions and existing orientation-independent anisotropy indices obtained from eigenvalues is discussed, after which the new approach is evaluated for a group of healthy volunteers.

Tensor Diagonalization, A Useful Tool in Signal Processing

Tensors appear more and more often in signal processing problems, and especially spatial processing, which typically involves multichannel modeling. Even if it is not always obvious that tensor algebra is the best framework to address a problem, there are cases where no choice is left. Blind identification of multichannel non monic MA models is given as an illustrating example of this claim.