linalg_lstsq {torch}

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

Description

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

Usage

linalg_lstsq(A, B, rcond = NULL, ..., driver = NULL)

Arguments

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 = NULL, rcond is set to the machine precision of the dtype of A times max(m, n). Default: NULL.

...

currently unused.

driver

(str, optional): name of the LAPACK/MAGMA method to be used. If NULL, 'gelsy' is used for CPU inputs and 'gels' for CUDA inputs. Default: NULL.

Details

Math could not be displayed. Please visit the package website.

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).

See also the full description of these drivers

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 rcond \cdot \sigma_1. If rcond = NULL (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 list 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.

Value

A list ⁠(solution, residuals, rank, singular_values)⁠.

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.

Note

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

See Also

Other linalg: linalg_cholesky_ex(), linalg_cholesky(), linalg_det(), linalg_eigh(), linalg_eigvalsh(), linalg_eigvals(), linalg_eig(), linalg_householder_product(), linalg_inv_ex(), linalg_inv(), linalg_matrix_norm(), linalg_matrix_power(), linalg_matrix_rank(), linalg_multi_dot(), linalg_norm(), linalg_pinv(), linalg_qr(), linalg_slogdet(), linalg_solve_triangular(), linalg_solve(), linalg_svdvals(), linalg_svd(), linalg_tensorinv(), linalg_tensorsolve(), linalg_vector_norm()

Examples

if (torch_is_installed()) {
A <- torch_tensor(rbind(c(10, 2, 3), c(3, 10, 5), c(5, 6, 12)))$unsqueeze(1) # shape (1, 3, 3)
B <- torch_stack(list(
  rbind(c(2, 5, 1), c(3, 2, 1), c(5, 1, 9)),
  rbind(c(4, 2, 9), c(2, 0, 3), c(2, 5, 3))
), dim = 1) # shape (2, 3, 3)
X <- linalg_lstsq(A, B)$solution # A is broadcasted to shape (2, 3, 3)
}