# torch.linalg.lstsq¶

torch.linalg.lstsq(A, B, rcond=None, *, driver=None)

Computes a solution to the least squares problem of a system of linear equations.

Letting $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, the least squares problem for a linear system $AX = B$ with $A \in \mathbb{K}^{m \times n}, B \in \mathbb{K}^{m \times k}$ is defined as

$\min_{X \in \mathbb{K}^{n \times k}} \|AX - B\|_F$

where $\|-\|_F$ denotes the Frobenius norm.

Supports inputs of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if the inputs are batches of matrices then the output has the same batch dimensions.

driver chooses the LAPACK/MAGMA function that will be used. For CPU inputs the valid values are ‘gels’, ‘gelsy’, ‘gelsd, ‘gelss’. For CUDA input, the only valid driver is ‘gels’, which assumes that A is full-rank. To choose the best driver on CPU consider:

• If A is well-conditioned (its condition number is not too large), or you do not mind some precision loss.

• For a general matrix: ‘gelsy’ (QR with pivoting) (default)

• If A is full-rank: ‘gels’ (QR)

• If A is not well-conditioned.

• ‘gelsd’ (tridiagonal reduction and SVD)

• But if you run into memory issues: ‘gelss’ (full SVD).

rcond is used to determine the effective rank of the matrices in A when driver is one of (‘gelsy’, ‘gelsd’, ‘gelss’). In this case, if $\sigma_i$ are the singular values of A in decreasing order, $\sigma_i$ will be rounded down to zero if $\sigma_i \leq \text{rcond} \cdot \sigma_1$. If rcond= None (default), rcond is set to the machine precision of the dtype of A.

This function returns the solution to the problem and some extra information in a named tuple of four tensors (solution, residuals, rank, singular_values). For inputs A, B of shape (*, m, n), (*, m, k) respectively, it cointains

• solution: the least squares solution. It has shape (*, n, k).

• residuals: the squared residuals of the solutions, that is, $\|AX - B\|_F^2$. It has shape equal to the batch dimensions of A. It is computed when m > n and every matrix in A is full-rank, otherwise, it is an empty tensor. If A is a batch of matrices and any matrix in the batch is not full rank, then an empty tensor is returned. This behavior may change in a future PyTorch release.

• rank: tensor of ranks of the matrices in A. It has shape equal to the batch dimensions of A. It is computed when driver is one of (‘gelsy’, ‘gelsd’, ‘gelss’), otherwise it is an empty tensor.

• singular_values: tensor of singular values of the matrices in A. It has shape (*, min(m, n)). It is computed when driver is one of (‘gelsd’, ‘gelss’), otherwise it is an empty tensor.

Note

This function computes X = A.pinverse() @ B in a faster and more numerically stable way than performing the computations separately.

Warning

The default value of rcond may change in a future PyTorch release. It is therefore recommended to use a fixed value to avoid potential breaking changes.

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

• B (Tensor) – rhs tensor of shape (*, m, k) where * is zero or more batch dimensions.

• rcond (float, optional) – used to determine the effective rank of A. If rcond= None, rcond is set to the machine precision of the dtype of A times max(m, n). Default: None.

Keyword Arguments

driver (str, optional) – name of the LAPACK/MAGMA method to be used. If None, ‘gelsy’ is used for CPU inputs and ‘gels’ for CUDA inputs. Default: None.

Returns

A named tuple (solution, residuals, rank, singular_values).

Examples:

>>> A = torch.tensor([[[10, 2, 3], [3, 10, 5], [5, 6, 12]]], dtype=torch.float) # shape (1, 3, 3)
>>> B = torch.tensor([[[2, 5, 1], [3, 2, 1], [5, 1, 9]],
[[4, 2, 9], [2, 0, 3], [2, 5, 3]]], dtype=torch.float) # shape (2, 3, 3)
>>> X = torch.linalg.lstsq(A, B).solution # A is broadcasted to shape (2, 3, 3)
>>> torch.dist(X, torch.linalg.pinv(A) @ B)
tensor(2.0862e-07)

>>> S = torch.linalg.lstsq(A, B, driver='gelsd').singular_values
>>> torch.dist(S, torch.linalg.svdvals(A))
tensor(5.7220e-06)

>>> A[:, 0].zero_()  # Decrease the rank of A
>>> rank = torch.linalg.lstsq(A, B).rank
>>> rank
tensor([2])