torch.linalg.householder_product

torch.linalg.householder_product(A, tau, *, out=None) → Tensor

计算 Householder 矩阵乘积的前 n 列。

令 $\mathbb{K}$ 为 $\mathbb{R}$ 或 $\mathbb{C}$，令 $A \in \mathbb{K}^{m \times n}$ be a matrix with columns $a_i \in \mathbb{K}^m$ for $i=1,\ldots,m$ with $m \geq n$. Denote by $b_i$ the vector resulting from zeroing out the first $i-1$ components of $a_i$ and setting to 1 the $i$-th. For a vector $\tau \in \mathbb{K}^k$ with $k \leq n$, this function computes the first $n$ columns of the matrix

$$H_1H_2 ... H_k \qquad\text{with}\qquad H_i = \mathrm{I}_m - \tau_i b_i b_i^{\text{H}}$$

where $\mathrm{I}_m$ is the m-dimensional identity matrix and $b^{\text{H}}$ is the conjugate transpose when $b$ is complex, and the transpose when $b$ is real-valued. The output matrix is the same size as the input matrix A.

See Representation of Orthogonal or Unitary Matrices for further details.

支持float、double、cfloat和cdouble数据类型的输入。也支持矩阵的批次，如果输入是矩阵的批次，则输出具有相同的批次维度。

另请参阅

torch.geqrf() can be used together with this function to form the Q from the qr() decomposition.

torch.ormqr() is a related function that computes the matrix multiplication of a product of Householder matrices with another matrix. However, that function is not supported by autograd.

警告

Gradient computations are only well-defined if $\tau_i \neq \frac{1}{||a_i||^2}$. If this condition is not met, no error will be thrown, but the gradient produced may contain NaN.

参数

• A (Tensor) – 形状为 (*, m, n) 的张量，其中 * 是零个或多个批处理维度。

• tau (Tensor) – shape (*, k) 的张量，其中 * 是零个或多个批处理维度。

关键字参数

out (Tensor, optional) – 输出张量。如果为 None 则忽略。默认为 None。

引发

RuntimeError – If A doesn’t satisfy the requirement m >= n, or tau doesn’t satisfy the requirement n >= k.

示例

>>> A = torch.randn(2, 2)
>>> h, tau = torch.geqrf(A)
>>> Q = torch.linalg.householder_product(h, tau)
>>> torch.dist(Q, torch.linalg.qr(A).Q)
tensor(0.)

>>> h = torch.randn(3, 2, 2, dtype=torch.complex128)
>>> tau = torch.randn(3, 1, dtype=torch.complex128)
>>> Q = torch.linalg.householder_product(h, tau)
>>> Q
tensor([[[ 1.8034+0.4184j,  0.2588-1.0174j],
        [-0.6853+0.7953j,  2.0790+0.5620j]],

        [[ 1.4581+1.6989j, -1.5360+0.1193j],
        [ 1.3877-0.6691j,  1.3512+1.3024j]],

        [[ 1.4766+0.5783j,  0.0361+0.6587j],
        [ 0.6396+0.1612j,  1.3693+0.4481j]]], dtype=torch.complex128)