lucid.exp¶
The exp function computes the exponential of each element in the input tensor, i.e., (e^x), where (e) is the base of natural logarithms.
Function Signature¶
def exp(a: Tensor) -> Tensor
Parameters¶
a (Tensor): The input tensor whose elements will be exponentiated.
Returns¶
- Tensor:
A new Tensor with the exponential of each element of a. If a requires gradients, the resulting tensor will also require gradients.
Forward Calculation¶
The forward calculation for the exp operation is:
\[\mathbf{out}_i = e^{\mathbf{a}_i}\]
where \(\mathbf{a}_i\) are the elements of the input tensor a.
Backward Gradient Calculation¶
For a tensor a involved in the exp operation, the gradient with respect to the output (out) is:
\[\frac{\partial \mathbf{out}_i}{\partial \mathbf{a}_i} = e^{\mathbf{a}_i}\]
This means the gradient of a is the same as the value of the forward output.
Examples¶
Using exp on a tensor:
>>> import lucid
>>> a = Tensor([0.0, 1.0, 2.0], requires_grad=True)
>>> out = lucid.exp(a)
>>> print(out)
Tensor([1.0, 2.7182818, 7.389056], grad=None)
Backpropagation computes gradients for a:
>>> out.backward()
>>> print(a.grad)
[1.0, 2.7182818, 7.389056] # Corresponding to the exponential values of a
Using exp on a higher-dimensional tensor:
>>> import lucid
>>> a = Tensor([[0.0, 1.0], [2.0, 3.0]], requires_grad=True)
>>> out = lucid.exp(a)
>>> print(out)
Tensor([[1.0, 2.7182818], [7.389056, 20.085537]], grad=None)
Performing backpropagation:
>>> out.backward()
>>> print(a.grad)
[[1.0, 2.7182818], [7.389056, 20.085537]] # Matches the forward exponential values