Positive Semi-definite Tensors Research Articles

An even order symmetric B tensor is positive definite

It is easily checkable if a given tensor is a B tensor, or a B0 tensor or not. In this paper, we show that a symmetric B tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors, and a symmetric B0 tensor can always be decomposed to the sum of a diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors. When the order is even, this implies that the corresponding B tensor is positive definite, and the corresponding B0 tensor is positive semi-definite. This gives a checkable sufficient condition for positive definite and semi-definite tensors. This approach is different from the approach in the literature for proving a symmetric B matrix is positive definite, as that matrix approach cannot be extended to the tensor case.

Symmetric nonnegative tensors and copositive tensors

We first prove two new spectral properties for symmetric nonnegative tensors. We prove a maximum property for the largest H-eigenvalue of a symmetric nonnegative tensor, and establish some bounds for this eigenvalue via row sums of that tensor. We show that if an eigenvalue of a symmetric nonnegative tensor has a positive H-eigenvector, then this eigenvalue is the largest H-eigenvalue of that tensor. We also give a necessary and sufficient condition for this. We then introduce copositive tensors. This concept extends the concept of copositive matrices. Symmetric nonnegative tensors and positive semi-definite tensors are examples of copositive tensors. The diagonal elements of a copositive tensor must be nonnegative. We show that if each sum of a diagonal element and all the negative off-diagonal elements in the same row of a real symmetric tensor is nonnegative, then that tensor is a copositive tensor. Some further properties of copositive tensors are discussed.

Symmetric positive semi-definite Cartesian Tensor fiber orientation distributions (CT-FOD)

A novel method for estimating a field of fiber orientation distribution (FOD) based on signal de-convolution from a given set of diffusion weighted magnetic resonance (DW-MR) images is presented. We model the FOD by higher order Cartesian tensor basis using a parametrization that explicitly enforces the positive semi-definite property to the computed FOD. The computed Cartesian tensors, dubbed Cartesian Tensor-FOD (CT-FOD), are symmetric positive semi-definite tensors whose coefficients can be efficiently estimated by solving a linear system with non-negative constraints. Next, we show how to use our method for converting higher-order diffusion tensors to CT-FODs, which is an essential task since the maxima of higher-order tensors do not correspond to the underlying fiber orientations. Finally, we propose a diffusion anisotropy index computed directly from CT-FODs using higher order tensor distance measures thus consolidating the whole analysis pipeline of diffusion imaging solely using CT-FODs. We evaluate our method qualitatively and quantitatively using simulated DW-MR images, phantom images, and human brain real dataset. The results conclusively demonstrate the superiority of the proposed technique over several existing multi-fiber reconstruction methods.

APPROXIMATING SYMMETRIC POSITIVE SEMIDEFINITE TENSORS OF EVEN ORDER().

Tensors of various orders can be used for modeling physical quantities such as strain and diffusion as well as curvature and other quantities of geometric origin. Depending on the physical properties of the modeled quantity, the estimated tensors are often required to satisfy the positivity constraint, which can be satisfied only with tensors of even order. Although the space [Formula: see text] of 2m(th)-order symmetric positive semi-definite tensors is known to be a convex cone, enforcing positivity constraint directly on [Formula: see text] is usually not straightforward computationally because there is no known analytic description of [Formula: see text] for m > 1. In this paper, we propose a novel approach for enforcing the positivity constraint on even-order tensors by approximating the cone [Formula: see text] for the cases 0 < m < 3, and presenting an explicit characterization of the approximation Σ(2) (m) ⊂ Ω(2) (m) for m ≥ 1, using the subset [Formula: see text] of semi-definite tensors that can be written as a sum of squares of tensors of order m. Furthermore, we show that this approximation leads to a non-negative linear least-squares (NNLS) optimization problem with the complexity that equals the number of generators in Σ(2) (m). Finally, we experimentally validate the proposed approach and we present an application for computing 2m(th)-order diffusion tensors from Diffusion Weighted Magnetic Resonance Images.

Perception-Based Visualization of Manifold-Valued Medical Images Using Distance-Preserving Dimensionality Reduction

A method for visualizing manifold-valued medical image data is proposed. The method operates on images in which each pixel is assumed to be sampled from an underlying manifold. For example, each pixel may contain a high dimensional vector, such as the time activity curve (TAC) in a dynamic positron emission tomography (dPET) or a dynamic single photon emission computed tomography (dSPECT) image, or the positive semi-definite tensor in a diffusion tensor magnetic resonance image (DTMRI). A nonlinear mapping reduces the dimensionality of the pixel data to achieve two goals: distance preservation and embedding into a perceptual color space. We use multidimensional scaling distance-preserving mapping to render similar pixels (e.g., DT or TAC pixels) with perceptually similar colors. The 3D CIELAB perceptual color space is adopted as the range of the distance preserving mapping, with a final similarity transform mapping colors to a maximum gamut size. Similarity between pixels is either determined analytically as geodesics on the manifold of pixels or is approximated using manifold learning techniques. In particular, dissimilarity between DTMRI pixels is evaluated via a Log-Euclidean Riemannian metric respecting the manifold of the rank 3, second-order positive semi-definite DTs, whereas the dissimilarity between TACs is approximated via ISOMAP. We demonstrate our approach via artificial high-dimensional, manifold-valued data, as well as case studies of normal and pathological clinical brain and heart DTMRI, dPET, and dSPECT images. Our results demonstrate the effectiveness of our approach in capturing, in a perceptually meaningful way, important features in the data.

A direct proof of uniqueness of square-root of a positive semi-definite tensor

Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues, eigenvectors and the spectral decomposition of the second-order symmetric tensor.

Symmetric positive 4th order tensors & their estimation from diffusion weighted MRI.

In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. It is now well known that this 2nd-order approximation fails to approximate complex local tissue structures, such as fibers crossings. In this paper we employ a 4th order symmetric positive semi-definite (PSD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the PSD property. There have been several published articles in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positive semi-definite constraint, which is a fundamental constraint since negative values of the diffusivity coefficients are not meaningful. In our methods, we parameterize the 4th order tensors as a sum of squares of quadratic forms by using the so called Gram matrix method from linear algebra and its relation to the Hilbert's theorem on ternary quartics. This parametric representation is then used in a nonlinear-least squares formulation to estimate the PSD tensors of order 4 from the data. We define a metric for the higher-order tensors and employ it for regularization across the lattice. Finally, performance of this model is depicted on synthetic data as well as real DW-MRI from an isolated rat hippocampus.

Positive semi-definiteness of generalized anti-circulant tensors

Anti-circulant tensors have applications in exponential data fitting. They are special Hankel tensors. In this paper, we extend the definition of anti-circulant tensors to generalized anti-circulant tensors by introducing a circulant index $r$ such that the entries of the generating vector of a Hankel tensor are circulant with module $r$. In the special case when $r =n$, where $n$ is the dimension of the Hankel tensor, the generalized anticirculant tensor reduces to the anti-circulant tensor. Hence, generalized anti-circulant tensors are still special Hankel tensors. For the cases that $GCD(m, r) =1$, $GCD(m, r) = 2$ and some other cases, including the matrix case that $m=2$, we give necessary and sufficient conditions for positive semi-definiteness of even order generalized anti-circulant tensors, and show that in these cases, they are SOS tensors. This shows that, in these cases, there are no PNS (positive semidefinite tensors which are not sum of squares) Hankel tensors.