pytorch中张量操作对应的数学表示

张量就是多重线性函数,以下用2阶张量为例来说明pytorch的函数变换

$l: V^{*}\times V^{*} \rightarrow R$

$l=\sum k_{ij}e_{i}\otimes e_{j}$

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t = torch.Tensor([[1, 2],[3, 4]])

$l=1e_{1}\otimes e_{1}+2e_{1}\otimes e_{2}+3e_{2}\otimes e_{1}+4e_{2}\otimes e_{2}$

$l=\sum_{i,j\in\{1,2\}} l(e_{i}^{*}, e_{j}^{*})e_{i}\otimes e_{j}$

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torch.transpose(t, 0, 1)
[[1, 3], [2, 4]]

$l_{out}=1e_{1}\otimes e_{1}+2e_{2}\otimes e_{1}+3e_{1}\otimes e_{2}+4e_{2}\otimes e_{2}$

$l_{out}=1e_{1}\otimes e_{1}+3e_{1}\otimes e_{2}+2e_{2}\otimes e_{1}+4e_{2}\otimes e_{2}$

---

torch.cat((t, torch.Tensor([[5, 6]])), 0)
[[1, 2], [3, 4], [5, 6]]

$l_{in2}=5e_{3}\otimes e_{1}+6e_{3}\otimes e_{2}$

$l_{out}=1e_{1}\otimes e_{1}+2e_{1}\otimes e_{2}+3e_{2}\otimes e_{1}+4e_{2}\otimes e_{2} + 5e_{3}\otimes e_{1}+6e_{3}\otimes e_{2}$

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torch.cat((t, torch.Tensor([[5], [6]])), 1)
[[1, 2, 5], [3, 4, 6]]

$l_{in2}=5e_{1}\otimes e_{3}+6e_{2}\otimes e_{3}$

$l=1e_{1}\otimes e_{1}+2e_{1}\otimes e_{2}+3e_{2}\otimes e_{1}+4e_{2}\otimes e_{2} + 5e_{1}\otimes e_{3}+6e_{2}\otimes e_{3}$

$l=1e_{1}\otimes e_{1}+2e_{1}\otimes e_{2}+5e_{1}\otimes e_{3}+3e_{2}\otimes e_{1}+4e_{2}\otimes e_{2}+6e_{2}\otimes e_{3}$

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torch.split(t, 1, 0)
[[1, 2]], [[3, 4]]

$l=1e_{1}\otimes e_{1}+2e_{1}\otimes e_{2}+3e_{2}\otimes e_{1}+4e_{2}\otimes e_{2}$

$l_{out1}=1e_{1}\otimes e_{1}+2e_{1}\otimes e_{2}$

$l_{out2}=3e_{1}\otimes e_{1}+4e_{1}\otimes e_{2}$

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torch.split(t, 1, 1)
[[1], [3]], [[2], [4]]

$l=1e_{1}\otimes e_{1}+2e_{1}\otimes e_{2}+3e_{2}\otimes e_{1}+4e_{2}\otimes e_{2}$

$l_{out1}=1e_{1}\otimes e_{1}+3e_{2}\otimes e_{1}$

$l_{out2}=2e_{1}\otimes e_{1}+4e_{2}\otimes e_{1}$

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torch.gather(t, 0, torch.tensor([[1, 0], [1, 1]]))
[[3, 2], [3, 4]]

$l=1e_{1}\otimes e_{1}+2e_{1}\otimes e_{2}+3e_{2}\otimes e_{1}+4e_{2}\otimes e_{2}$

$l_{in2}=2e_{1}\otimes e_{1}+1e_{1}\otimes e_{2}+2e_{2}\otimes e_{1}+2e_{2}\otimes e_{2}$

$l_{out}=\sum_{i,j\in\{1,2\}} l(e_{l_{in2}(e_{i}^{*}, e_{j}^{*})}^{*}, e^{*}_{j}) e_{i}\otimes e_{j}$

$l_{out}=3e_{1}\otimes e_{1}+2e_{1}\otimes e_{2}+3e_{2}\otimes e_{1}+4e_{2}\otimes e_{2}$

---

torch.gather(t, 1, torch.tensor([[1, 0], [1, 1]]))
[[2, 1], [4, 4]]

$l=1e_{1}\otimes e_{1}+2e_{1}\otimes e_{2}+3e_{2}\otimes e_{1}+4e_{2}\otimes e_{2}$

$l_{in2}=2e_{1}\otimes e_{1}+1e_{1}\otimes e_{2}+2e_{2}\otimes e_{1}+2e_{2}\otimes e_{2}$

$l_{out}=\sum_{i,j\in\{1,2\}} l(e^{*}_{i}, e_{l_{in2}(e_{i}^{*}, e_{j}^{*})}^{*}) e_{i}\otimes e_{j}$

$l_{out}=2e_{1}\otimes e_{1}+1e_{1}\otimes e_{2}+4e_{2}\otimes e_{1}+4e_{2}\otimes e_{2}$