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op {
graph_op_name: "MatrixSolveLs"
in_arg {
name: "matrix"
description: <<END
Shape is `[..., M, N]`.
END
}
in_arg {
name: "rhs"
description: <<END
Shape is `[..., M, K]`.
END
}
in_arg {
name: "l2_regularizer"
description: <<END
Scalar tensor.
@compatibility(numpy)
Equivalent to np.linalg.lstsq
@end_compatibility
END
}
out_arg {
name: "output"
description: <<END
Shape is `[..., N, K]`.
END
}
summary: "Solves one or more linear least-squares problems."
description: <<END
`matrix` is a tensor of shape `[..., M, N]` whose inner-most 2 dimensions
form real or complex matrices of size `[M, N]`. `Rhs` is a tensor of the same
type as `matrix` and shape `[..., M, K]`.
The output is a tensor shape `[..., N, K]` where each output matrix solves
each of the equations
`matrix[..., :, :]` * `output[..., :, :]` = `rhs[..., :, :]`
in the least squares sense.
We use the following notation for (complex) matrix and right-hand sides
in the batch:
`matrix`=\\(A \in \mathbb{C}^{m \times n}\\),
`rhs`=\\(B \in \mathbb{C}^{m \times k}\\),
`output`=\\(X \in \mathbb{C}^{n \times k}\\),
`l2_regularizer`=\\(\lambda \in \mathbb{R}\\).
If `fast` is `True`, then the solution is computed by solving the normal
equations using Cholesky decomposition. Specifically, if \\(m \ge n\\) then
\\(X = (A^H A + \lambda I)^{-1} A^H B\\), which solves the least-squares
problem \\(X = \mathrm{argmin}_{Z \in \Re^{n \times k} } ||A Z - B||_F^2 + \lambda ||Z||_F^2\\).
If \\(m \lt n\\) then `output` is computed as
\\(X = A^H (A A^H + \lambda I)^{-1} B\\), which (for \\(\lambda = 0\\)) is the
minimum-norm solution to the under-determined linear system, i.e.
\\(X = \mathrm{argmin}_{Z \in \mathbb{C}^{n \times k} } ||Z||_F^2 \\),
subject to \\(A Z = B\\). Notice that the fast path is only numerically stable
when \\(A\\) is numerically full rank and has a condition number
\\(\mathrm{cond}(A) \lt \frac{1}{\sqrt{\epsilon_{mach} } }\\) or \\(\lambda\\) is
sufficiently large.
If `fast` is `False` an algorithm based on the numerically robust complete
orthogonal decomposition is used. This computes the minimum-norm
least-squares solution, even when \\(A\\) is rank deficient. This path is
typically 6-7 times slower than the fast path. If `fast` is `False` then
`l2_regularizer` is ignored.
END
}
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