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.

中文翻译：

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近似偶阶对称正半定张量()。


各种阶的张量可用于对应变和扩散等物理量以及曲率和其他几何起源量进行建模。根据建模量的物理属性，估计的张量通常需要满足正性约束，而只有偶数阶的张量才能满足该约束。尽管已知 2m(th) 阶对称正半定张量的空间 [公式：参见文本] 是凸锥，但直接对 [公式：参见文本] 施加正性约束通常在计算上并不简单，因为没有m > 1 时 [公式：参见文本] 的已知分析描述。在本文中，我们提出了一种新方法，通过近似 0 < m 情况下的锥体 [公式：参见文本]，对偶数阶张量实施正约束< 3，并使用半定张量的子集[公式：参见文本]，对 m ≥ 1 时的近似 Σ(2) (m) ⊂ Ω(2) (m) 进行显式表征，可写为m 阶张量的平方和。此外，我们还表明，这种近似会导致非负线性最小二乘 (NNLS) 优化问题，其复杂度等于 Σ(2) (m) 中生成器的数量。最后，我们通过实验验证了所提出的方法，并提出了一种从扩散加权磁共振图像计算 2m 阶扩散张量的应用程序。