Practical Criteria for H-Tensors and Their Application

Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method.

Biophysical compartment models for single-shell diffusion MRI in the human brain: a model fitting comparison

Clinically oriented studies commonly acquire diffusion MRI (dMRI) data with a single non-zero b-value (i.e. single-shell) and diffusion weighting of b=1000 s/mm2. To produce microstructural parameter maps, the tensor model is usually used, despite known limitations. Although compartment models have demonstrated improved fits in multi-shell dMRI data, they are rarely used for single-shell parameter maps, where their effectiveness is unclear from the literature. Here, various compartment models combining isotropic balls and symmetric tensors were fitted to single-shell dMRI data to investigate model fitting optimization and extract the most information possible. Full testing was performed in 5 subjects, and 3 subjects with multi-shell data were included for comparison. The results were tested and confirmed in a further 50 subjects. The Markov chain Monte Carlo (MCMC) model fitting technique outperformed non-linear least squares. Using MCMC, the 2-fibre-orientation mono-exponential ball & stick model (BSME 2) provided artifact-free, stable results, in little processing time. The analogous ball & zeppelin model (BZ2) also produced stable, low-noise parameter maps, though it required much greater computing resources (50 000 burn-in steps). In single-shell data, the gamma-distributed diffusivity ball & stick model (BSGD 2) underperformed relative to other models, despite being an often-used software default. It produced artifacts in the diffusivity maps even with extremely long processing times. Neither increased diffusion weighting nor a greater number of gradient orientations improved BSGD 2 fits. In white matter (WM), the tensor produced the best fit as measured by Bayesian information criterion. This result contrasts with studies using multi-shell data. However, in crossing fibre regions the tensor confounded geometric effects with fractional anisotropy (FA): the planar/linear WM FA ratio was 49%, while BZ2 and BSME 2 retained 76% and 83% of restricted fraction, respectively. As a result, the BZ2 and BSME 2 models are strong candidates to optimize information extraction from single-shell dMRI studies.

On decompositions and approximations of conjugate partial-symmetric tensors

Hermitian matrices have played an important role in matrix theory and complex quadratic optimization. The high-order generalization of Hermitian matrices, conjugate partial-symmetric (CPS) tensors, have shown growing interest recently in tensor theory and computation, particularly in application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, computing rank-one decompositions and approximations, as well as their applications. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an explicit method to compute such rank-one decompositions. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon’s conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximations of CPS tensors. Numerical experiments from data sets in radar wave form design, elasticity tensor, and quantum entanglement are performed to justify the capability of our methods.

A Block-Sparse Tensor Train Format for Sample-Efficient High-Dimensional Polynomial Regression

Low-rank tensors are an established framework for the parametrization of multivariate polynomials. We propose to extend this framework by including the concept of block-sparsity to efficiently parametrize homogeneous, multivariate polynomials with low-rank tensors. This provides a representation of general multivariate polynomials as a sum of homogeneous, multivariate polynomials, represented by block-sparse, low-rank tensors. We show that this sum can be concisely represented by a single block-sparse, low-rank tensor.We further prove cases, where low-rank tensors are particularly well suited by showing that for banded symmetric tensors of homogeneous polynomials the block sizes in the block-sparse multivariate polynomial space can be bounded independent of the number of variables.We showcase this format by applying it to high-dimensional least squares regression problems where it demonstrates improved computational resource utilization and sample efficiency.

Partition functions of p-forms from Harish-Chandra characters

We show that the determinant of the co-exact p-form on spheres and anti-de Sitter spaces can be written as an integral transform of bulk and edge Harish-Chandra characters. The edge character of a co-exact p-form contains characters of anti-symmetric tensors of rank lower to p all the way to the zero-form. Using this result we evaluate the partition function of p-forms and demonstrate that they obey known properties under Hodge duality. We show that the partition function of conformal forms in even d + 1 dimensions, on hyperbolic cylinders can be written as integral transforms involving only the bulk characters. This supports earlier observations that entanglement entropy evaluated using partition functions on hyperbolic cylinders do not contain contributions from the edge modes. For conformal coupled scalars we demonstrate that the character integral representation of the free energy on hyperbolic cylinders and branched spheres coincide. Finally we propose a character integral representation for the partition function of p-forms on branched spheres.

Counting Tensor Rank Decompositions

Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor Q with an error allowance Δ is to find vectors ϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all such possible ϕi is an interesting quantity which measures the amount of possible decompositions for a tensor Q within an allowance. While it would be difficult to evaluate this quantity for each Q, we find an explicit formula for a similar quantity by integrating over all Q of unit norm. The expression as a function of Δ is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.

Identifying the Positive Definiteness of Even-Order Weakly Symmetric Tensors via Z-Eigenvalue Inclusion Sets

The positive definiteness of even-order weakly symmetric tensors plays important roles in asymptotic stability of time-invariant polynomial systems. In this paper, we establish two Brauer-type Z-eigenvalue inclusion sets with parameters by Z-identity tensors, and show that these inclusion sets are sharper than existing results. Based on the new Z-eigenvalue inclusion sets, we propose some sufficient conditions for testing the positive definiteness of even-order weakly symmetric tensors, as well as the asymptotic stability of time-invariant polynomial systems. The given numerical experiments are reported to show the efficiency of our results.