lucid.linalg.qr

lucid.linalg.qr(a: Tensor, /) → tuple[Tensor, Tensor]

The qr function computes the QR decomposition of a matrix, decomposing it into an orthogonal matrix Q and an upper triangular matrix R.

Function Signature

def qr(a: Tensor) -> tuple[Tensor, Tensor]

Parameters

  • a (Tensor):

    The input tensor, which must be a two-dimensional matrix.

Returns

  • Tuple[Tensor, Tensor]: A tuple containing two tensors:

    • Q (Tensor): An orthogonal matrix where the columns are orthonormal vectors.

    • R (Tensor): An upper triangular matrix.

Forward Calculation

The forward calculation for qr decomposes a matrix \(\mathbf{A}\) into the product of an orthogonal matrix \(\mathbf{Q}\) and an upper triangular matrix \(\mathbf{R}\):

\[\mathbf{A} = \mathbf{Q} \mathbf{R}\]

where: - \(\mathbf{Q}\) is an orthogonal matrix (\(\mathbf{Q}^\top \mathbf{Q} = \mathbf{I}\)), - \(\mathbf{R}\) is an upper triangular matrix.

Backward Gradient Calculation

Given the QR decomposition \(\mathbf{A} = \mathbf{Q} \mathbf{R}\), the gradients of Q and R with respect to the input matrix \(\mathbf{A}\) can be computed as follows:

  1. Gradient of Q with respect to \(\mathbf{A}\):

    The gradient is derived based on the orthogonality of Q. It involves projecting the gradient onto the space orthogonal to Q.

  2. Gradient of R with respect to \(\mathbf{A}\):

    Since R is upper triangular, the gradient computation takes into account the structure of R to ensure that the gradients respect the upper triangular form.

These gradients are propagated through Q and R during backpropagation, allowing for the optimization of parameters in models that involve QR decomposition.

Raises

Attention

  • ValueError: If the input tensor is not a two-dimensional matrix.

  • LinAlgError: If the QR decomposition cannot be computed (e.g., if the input matrix contains NaNs or infinities).

Example

>>> import lucid
>>> a = lucid.Tensor([[1.0, 2.0], [3.0, 4.0]])
>>> q, r = lucid.linalg.qr(a)
>>> print(q)
Tensor([[-0.31622777, -0.9486833 ],
        [-0.9486833 ,  0.31622777]])
>>> print(r)
Tensor([[-3.16227766, -4.42718872],
        [ 0.        ,  0.63245553]])

Note

  • QR decomposition is useful for solving linear systems, least squares problems, and eigenvalue algorithms.

  • The input tensor must have two dimensions, i.e., \(a.ndim == 2\).

  • The matrix Q is orthogonal, meaning \(\mathbf{Q}^\top \mathbf{Q} = \mathbf{I}\).

  • The matrix R is upper triangular, which means all elements below the main diagonal are zero.

  • This function does not support batch processing; each input must be a single matrix.