AI 中文总结
本文研究基对齐双正交（bato）张量，证明其一般分解可识别、截断为临界低秩近似，确定其集合维数与闭包不可约分量的对应关系，并指出行列式等重要张量属于bato张量。
AI 中文摘要
我们研究基对齐双正交（bato）张量，这类张量可表示为临界秩1近似的和，其因子是各扁平化操作的奇异向量。因此，bato张量既存在Tucker分解，也存在规范多元分解，且二者相互关联。我们证明一般bato分解是可识别的，其截断形式为临界低bato秩近似；还计算了bato张量集合的维数，并将其扎里斯基闭包的不可约分量与最大部分拉丁超矩形的同痕类对应起来。许多重要张量均为bato张量，例如行列式、矩阵乘法张量及其他代数的结构张量。
英文摘要
We study basis-aligned two-orthogonal (bato) tensors, which can be written as a sum of critical rank-one approximations whose factors are singular vectors of their flattenings. As such, bato tensors admit a Tucker decomposition and a canonical polyadic decomposition that are related to each other. We prove that generic bato decompositions are identifiable and that their truncations are critical low-bato-rank approximations. We also compute the dimension of the set of bato tensors, and identify the irreducible components of its Zariski closure with isotopy classes of maximal partial Latin hyperrectangles. Many important tensors are bato, such as determinants, matrix multiplication tensors, and other structure tensors of algebras.
Comments25 pages, 1 figure, 1 table
