lucid.tril¶

lucid.tril(a: Tensor, /, diagonal: int = 0) → Tensor¶

The tril function extracts the lower triangular part of a tensor, setting all elements above a specified diagonal to zero.

Function Signature¶

def tril(input_: Tensor, diagonal: int = 0) -> Tensor

Parameters¶

input_ (Tensor): The input tensor, which must be at least 2-dimensional.
diagonal (int, optional): Specifies which diagonal to consider (default: 0). - 0: Keeps the main diagonal. - >0: Includes elements above the main diagonal. - <0: Excludes more lower rows.

Returns¶

Tensor: A new tensor with the same shape as the input, but with all elements above the specified diagonal set to zero.

Forward Calculation¶

The tril operation applies a lower-triangular mask to the input tensor:

\[\begin{split}\mathbf{out}_{ij} = \begin{cases} \mathbf{X}_{ij}, & \text{if } i - j \geq -\text{diagonal} \\ 0, & \text{otherwise} \end{cases}\end{split}\]

Backward Gradient Calculation¶

The gradient for the tril operation ensures that only the lower triangular portion receives gradients, setting the upper triangular part to zero:

\[\begin{split}\frac{\partial \mathbf{out}}{\partial \mathbf{X}} = \mathbf{M}, \quad \text{where} \quad M_{ij} = \begin{cases} 1, & \text{if } i - j \geq -\text{diagonal} \\ 0, & \text{otherwise} \end{cases}\end{split}\]

Example¶

>>> import lucid
>>> a = Tensor([[1.0, 2.0, 3.0],
...             [4.0, 5.0, 6.0],
...             [7.0, 8.0, 9.0]], requires_grad=True)
>>> lower_triangular = lucid.tril(a)
>>> print(lower_triangular)
Tensor([[1. 0. 0.]
        [4. 5. 0.]
        [7. 8. 9.]], grad=None)

>>> lower_triangular = lucid.tril(a, diagonal=1)
>>> print(lower_triangular)
Tensor([[1. 2. 0.]
        [4. 5. 6.]
        [7. 8. 9.]], grad=None)

Note

Only elements in the lower triangular part (including the specified diagonal) are retained; the rest are set to zero.
The gradient is passed only through the lower triangular portion, ensuring correct backpropagation.
This function is useful for constructing masked matrices in deep learning applications.