We discover a geometric property of the space of tensors of fixed multilinear (Tucker) rank. Namely, it is shown that real tensors of the fixed multilinear rank form a minimal submanifold of the Euclidean space of tensors endowed with the Frobenius inner product. We also establish the absence of local extrema for linear functionals restricted to the submanifold of rank-one tensors, finding application in statistics.

中文翻译：

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固定多线性秩张量的极小性

我们发现了固定多线性 (Tucker) 秩张量空间的几何性质。即，证明了固定多线性秩的实张量构成欧几里德张量空间的最小子流形，该张量具有 Frobenius 内积。我们还确定了局限于一级张量子流形的线性泛函不存在局部极值，从而在统计学中得到应用。