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# torch.linalg.pinv

torch.linalg.pinv(A, *, atol=None, rtol=None, hermitian=False, out=None) → Tensor

  Computes the pseudoinverse (Moore-Penrose inverse) of a matrix.

  The pseudoinverse may be defined algebraically but it is more computationally convenient to understand it through the SVD

  Supports input of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if `A` is a batch of matrices then the output has the same batch dimensions.

  If `hermitian`= True, `A` is assumed to be Hermitian if complex or symmetric if real, but this is not checked internally. Instead, just the lower triangular part of the matrix is used in the computations.

  The singular values (or the norm of the eigenvalues when `hermitian`= True) that are below max⁡(atol,σ 1⋅rtol)\max(\text{atol}, \sigma_1 \cdot \text{rtol})max(atol,σ 1​⋅rtol) threshold are treated as zero and discarded in the computation, where σ 1\sigma_1 σ 1​ is the largest singular value (or eigenvalue).

  If `rtol` is not specified and `A` is a matrix of dimensions (m, n), the relative tolerance is set to be rtol=max⁡(m,n)ε\text{rtol} = \max(m, n) \varepsilon rtol=max(m,n)ε and ε\varepsilon ε is the epsilon value for the dtype of `A` (see `finfo`). If `rtol` is not specified and `atol` is specified to be larger than zero then `rtol` is set to zero.

  If `atol` or `rtol` is a `torch.Tensor`, its shape must be broadcastable to that of the singular values of `A` as returned by `torch.linalg.svd()`.

  Note

  This function uses `torch.linalg.svd()` if `hermitian`= False and `torch.linalg.eigh()` if `hermitian`= True. For CUDA inputs, this function synchronizes that device with the CPU.

  Note

  Consider using `torch.linalg.lstsq()` if possible for multiplying a matrix on the left by the pseudoinverse, as:

      torch.linalg.lstsq(A, B).solution == A.pinv() @ B

  It is always preferred to use `lstsq()` when possible, as it is faster and more numerically stable than computing the pseudoinverse explicitly.

  Note

  This function has NumPy compatible variant linalg.pinv(A, rcond, hermitian=False). However, use of the positional argument `rcond` is deprecated in favor of `rtol`.

  Warning

  This function uses internally `torch.linalg.svd()` (or `torch.linalg.eigh()` when `hermitian`= True), so its derivative has the same problems as those of these functions. See the warnings in `torch.linalg.svd()` and `torch.linalg.eigh()` for more details.

  See also

  `torch.linalg.inv()` computes the inverse of a square matrix.

  `torch.linalg.lstsq()` computes `A`.pinv() @ `B` with a numerically stable algorithm.

  Parameters:

    - A (Tensor) – tensor of shape (*, m, n) where * is zero or more batch dimensions.

    - rcond (float, Tensor, optional) – [NumPy Compat]. Alias for `rtol`. Default: None.

  Keyword Arguments:

    - atol (float, Tensor, optional) – the absolute tolerance value. When None it’s considered to be zero. Default: None.

    - rtol (float, Tensor, optional) – the relative tolerance value. See above for the value it takes when None. Default: None.

    - hermitian (bool, optional) – indicates whether `A` is Hermitian if complex or symmetric if real. Default: False.

    - out (Tensor, optional) – output tensor. Ignored if None. Default: None.

  Examples:

      >>> A = torch.randn(3, 5)
      >>> A
      tensor([[ 0.5495,  0.0979, -1.4092, -0.1128,  0.4132],
              [-1.1143, -0.3662,  0.3042,  1.6374, -0.9294],
              [-0.3269, -0.5745, -0.0382, -0.5922, -0.6759]])
      >>> torch.linalg.pinv(A)
      tensor([[ 0.0600, -0.1933, -0.2090],
              [-0.0903, -0.0817, -0.4752],
              [-0.7124, -0.1631, -0.2272],
              [ 0.1356,  0.3933, -0.5023],
              [-0.0308, -0.1725, -0.5216]])

      >>> A = torch.randn(2, 6, 3)
      >>> Apinv = torch.linalg.pinv(A)
      >>> torch.dist(Apinv @ A, torch.eye(3))
      tensor(8.5633e-07)

      >>> A = torch.randn(3, 3, dtype=torch.complex64)
      >>> A = A + A.T.conj()  # creates a Hermitian matrix
      >>> Apinv = torch.linalg.pinv(A, hermitian=True)
      >>> torch.dist(Apinv @ A, torch.eye(3))
      tensor(1.0830e-06)

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