The Symmetric Tensor Approximation problem (STA) consists of approximating a symmetric tensor or a homogeneous polynomial by a linear combination of symmetric rank-1 tensors or powers of linear forms of low symmetric rank. We present two new Riemannian Newton-type methods for low rank approximation of symmetric tensor with complex coefficients.The first method uses the parametrization of the set of tensors of rank at most $r$ by weights and unit vectors.Exploiting the properties of the apolar product on homogeneous polynomials combined with efficient tools from complex optimization, we provide an explicit and tractable formulation of the Riemannian gradient and Hessian, leading to Newton iterations with local quadratic convergence. We prove that under some regularity conditions on non-defective tensors in the neighborhood of the initial point, the Newton iteration (completed with a trust-region scheme) is converging to a local minimum.The second method is a Riemannian Gauss--Newton method on the Cartesian product of Veronese manifolds. An explicit orthonormal basis of the tangent space of this Riemannian manifold is described. We deduce the Riemannian gradient and the Gauss--Newton approximation of the Riemannian Hessian. We present a new retraction operator on the Veronese manifold.We analyze the numerical behavior of these methods, with an initial point provided by Simultaneous Matrix Diagonalisation (SMD).Numerical experiments show the good numerical behavior of the two methods in different cases and in comparison with existing state-of-the-art methods.
翻译:对称 Tensor 匹配问题( STA ), 近似于对称强力或同质多元度, 由对称一至强的线性组合或对称低等的线性形式力量组成。 我们展示了两种新型Riemannian 牛顿型方法, 用于对称强力和复杂系数的低级近似。 首种方法使用以重量和单位矢量表示的对称强力。 将同质多元度的极产物的特性与复杂优化的高效工具结合起来进行探讨, 我们提供了利曼度梯度和赫斯的直线性形式力量组合。 我们证明, 在初始点附近一些不偏差的强力强力强力条件下, 牛顿级变异性( 与信任的直径直径分析计划相结合) 正在将纯度多端多面多端多端多色度多色度的极产的极值特性进行演化。 第二种方法是Riemann- 梯度平面的直径直径直径直径直径直方的直径直径直方的直径直径直方的直径直方数据。 我们证明, 在的直方的直径直方的直方的直径直方的直方的直方的直方的直方计算方法。