In this paper we study the computation of both algebraic and non-algebraic tensor functions under the tensor-tensor multiplication with linear maps. In the case of algebraic tensor functions, we prove that the asymptotic exponent of both the tensor-tensor multiplication and the tensor polynomial evaluation problem under this multiplication is the same as that of the matrix multiplication, unless the linear map is injective. As for non-algebraic functions, we define the tensor geometric mean and the tensor Wasserstein mean for pseudo-positive-definite tensors under the tensor-tensor multiplication with invertible linear maps, and we show that the tensor geometric mean can be calculated by solving a specific Riccati tensor equation. Furthermore, we show that the tensor geometric mean does not satisfy the resultantal (determinantal) identity in general, which the matrix geometric mean always satisfies. Then we define a pseudo-SVD for the injective linear map case and we apply it on image data compression.

翻译：本文研究了在线性映射下的张量-张量乘法中，代数与非代数张量函数的计算问题。对于代数张量函数，我们证明除非线性映射是单射，否则该乘法下的张量-张量乘法与张量多项式求值问题的渐近指数均与矩阵乘法相同。针对非代数函数，我们在可逆线性映射下的张量-张量乘法中，为伪正定张量定义了张量几何平均与张量Wasserstein平均，并证明张量几何平均可通过求解特定Riccati张量方程计算。此外，我们证明张量几何平均通常不满足矩阵几何平均始终满足的结果式（行列式）恒等式。随后我们针对单射线性映射情形定义了伪奇异值分解，并将其应用于图像数据压缩。