论文标题
部分可观测时空混沌系统的无模型预测
Rank Two Approximations of $2 \times 2 \times 2$ Tensors over $\mathbb{R}$
论文作者
论文摘要
我们提供了无坐标的证据，证明实际$ 2 \ times 2 \ times 2 $等级三个张量对于Frobenius规范没有最佳排名。首先通过考虑$ {\ text {gl}（v^1）\ times \ text {gl}（v^2）\ times \ times \ text {gl}（v^3）} $ orbit类的$ {我们的无坐标证明通过开发一种可以更容易地将其推广到更高维度$ n_1 \ times n_2 \ times n_3 $ tensor空间的证明方法来扩展这种已知结果。
We provide a coordinate-free proof that real $2 \times 2 \times 2$ rank three tensors do not have optimal rank two approximations with respect to the Frobenius norm. This result was first proved in by considering the ${\text{GL}(V^1) \times \text{GL}(V^2) \times \text{GL}(V^3)}$ orbit classes of ${V^1 \otimes V^2 \otimes V^3}$ and the $2 \times 2 \times 2$ hyperdeterminant. Our coordinate-free proof expands on this known result by developing a proof method that can be generalized more readily to higher dimensional $n_1 \times n_2 \times n_3$ tensor spaces.
