AI 中文总结
本文构造了具有大质心的矩阵乘法张量等，开发了利用质心的几何技术，证明相关张量的最小边界秩，为矩阵乘法指数上界及Strassen激光方法提供了新进展。
AI 中文摘要
本文研究张量的基本不变量——质心，主要结果包括：（i）构造了具有极大质心的显式张量，此前学界曾猜想不存在此类张量；（ii）给出质心维度的上界，且该上界可由本文示例基本达到；（iii）开发了一种几何技术，可利用质心与“扩展质心”写出张量的边界秩分解；（iv）将该技术应用于本文构造的张量，证明它们具有最小边界秩，该技术通用性强，能从几何角度推导并改进此前的特殊分解；（v）构造了具有大质心的对称张量，并证明其在Buczyńska-Buczyński意义下是“wild”的。本文结果为矩阵乘法指数的新上界研究铺平了道路，该几何技术还能从旧张量出发为Strassen的激光方法构造新的“更优”张量，本文将其应用于Strassen和Schönhage的张量，得到了比原张量能给出更好矩阵乘法指数上界的更优张量。
英文摘要
This paper addresses centroids, which are fundamental invariants of tensors. Our main results are as follows: (i) The construction of explicit tensors with very large centroids, whereas previously it had been conjectured that none such exist. (ii) An upper bound on the dimension of the centroid that is essentially attained by our examples. (iii) The development of a geometric technique to write down border rank decomposition of tensors using centroids and "extended centroids". (iv) The technique is applied to tensors of this paper to prove they are of minimal border rank. The technique is versatile and enables us to geometrically derive and improve upon previous ad hoc decompositions. (v) The construction of symmetric tensors with large centroids and proof that they are wild in the sense of Buczyńska-Buczyński. Our results also pave the way for new upper bounds on the exponent of matrix multiplication. The geometric technique also constructs new "better" tensors for Strassen's laser method from old, and we apply this to the tensors of Strassen and Schönhage to get better tensors in the sense that they give better upper bounds on the exponent than the original tensors.
Comments52 pages, 3 figures
