实操指南：用Python的Torch库理解并运用gradient函数

最编程 2024-02-08 10:53:02

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使用 second-order 精确中心差法估计一维或多维函数 $g : R^{n} \to R$ 的梯度。

$g$ 的梯度是使用样本估计的。默认情况下，当不指定 spacing 时，样本完全由 input 说明，输入坐标到输出的映射与张量索引到值的映射相同。例如，对于三维 input，所说明的函数是 $g : R^{3} \to R$ 和 $g (1, 2, 3) == in p u t [1, 2, 3]$ 。

当指定spacing 时，它会修改input 与输入坐标之间的关系。这在下面的“Keyword Arguments” 部分中有详细说明。

通过独立估计 $g$ 的每个偏导数来估计梯度。如果 $g$ 在 $C^{3}$ 中(它至少有3个连续导数)，这个估计是准确的，并且可以通过提供更接近的样本来改进估计。在数学上，偏导数的每个内点的值是使用带余数的泰勒定理估计的。让 $x$ 为内部点， $x + h_{r}$ 为与其相邻的点， $f (x + h_{r})$ 处的部分梯度估计使用：

f (x + h_{r}) = f (x) + h_{r} f^{'} (x) + h_{r}^{2} \frac{f ^{''} ( x )}{2} + h_{r}^{3} \frac{f ^{'''} ( x _{r} )}{6}

其中 $x_{r}$ 是区间 $[x, x + h_{r}]$ 中的一个数字，并使用 $f \in C^{3}$ 我们得出以下事实：

f^{'} (x) \approx \frac{h _{l} ^{2} f ( x + h _{r} ) - h _{r} ^{2} f ( x - h _{l} ) + ( h _{r} ^{2} - h _{l} ^{2} ) f ( x )}{h _{r} h _{l} ^{2} + h _{r} ^{2} h _{l}}

注意

我们以相同的方式估计复域 $g : C^{n} \to C$ 中函数的梯度。

边界点处的每个偏导数的计算方式不同。请参阅下面的edge_order。

例子：

>>> # Estimates the gradient of f(x)=x^2 at points [-2, -1, 2, 4]
>>> coordinates = (torch.tensor([-2., -1., 1., 4.]),)
>>> values = torch.tensor([4., 1., 1., 16.], )
>>> torch.gradient(values, spacing = coordinates)
(tensor([-3., -2., 2., 5.]),)

>>> # Estimates the gradient of the R^2 -> R function whose samples are
>>> # described by the tensor t. Implicit coordinates are [0, 1] for the outermost
>>> # dimension and [0, 1, 2, 3] for the innermost dimension, and function estimates
>>> # partial derivative for both dimensions.
>>> t = torch.tensor([[1, 2, 4, 8], [10, 20, 40, 80]])
>>> torch.gradient(t)
(tensor([[ 9., 18., 36., 72.],
         [ 9., 18., 36., 72.]]),
 tensor([[ 1.0000, 1.5000, 3.0000, 4.0000],
         [10.0000, 15.0000, 30.0000, 40.0000]]))

>>> # A scalar value for spacing modifies the relationship between tensor indices
>>> # and input coordinates by multiplying the indices to find the
>>> # coordinates. For example, below the indices of the innermost
>>> # 0, 1, 2, 3 translate to coordinates of [0, 2, 4, 6], and the indices of
>>> # the outermost dimension 0, 1 translate to coordinates of [0, 2].
>>> torch.gradient(t, spacing = 2.0) # dim = None (implicitly [0, 1])
(tensor([[ 4.5000, 9.0000, 18.0000, 36.0000],
          [ 4.5000, 9.0000, 18.0000, 36.0000]]),
 tensor([[ 0.5000, 0.7500, 1.5000, 2.0000],
          [ 5.0000, 7.5000, 15.0000, 20.0000]]))
>>> # doubling the spacing between samples halves the estimated partial gradients.

>>>
>>> # Estimates only the partial derivative for dimension 1
>>> torch.gradient(t, dim = 1) # spacing = None (implicitly 1.)
(tensor([[ 1.0000, 1.5000, 3.0000, 4.0000],
         [10.0000, 15.0000, 30.0000, 40.0000]]),)

>>> # When spacing is a list of scalars, the relationship between the tensor
>>> # indices and input coordinates changes based on dimension.
>>> # For example, below, the indices of the innermost dimension 0, 1, 2, 3 translate
>>> # to coordinates of [0, 3, 6, 9], and the indices of the outermost dimension
>>> # 0, 1 translate to coordinates of [0, 2].
>>> torch.gradient(t, spacing = [3., 2.])
(tensor([[ 4.5000, 9.0000, 18.0000, 36.0000],
         [ 4.5000, 9.0000, 18.0000, 36.0000]]),
 tensor([[ 0.3333, 0.5000, 1.0000, 1.3333],
         [ 3.3333, 5.0000, 10.0000, 13.3333]]))

>>> # The following example is a replication of the previous one with explicit
>>> # coordinates.
>>> coords = (torch.tensor([0, 2]), torch.tensor([0, 3, 6, 9]))
>>> torch.gradient(t, spacing = coords)
(tensor([[ 4.5000, 9.0000, 18.0000, 36.0000],
         [ 4.5000, 9.0000, 18.0000, 36.0000]]),
 tensor([[ 0.3333, 0.5000, 1.0000, 1.3333],
         [ 3.3333, 5.0000, 10.0000, 13.3333]]))

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