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Solves one or more linear least-squares problems.

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.

matrix A Tensor. Must be one of the following types: float64, float32, half, complex64, complex128. Shape is [..., M, N].
rhs A Tensor. Must have the same type as matrix. Shape is [..., M, K].
l2_regularizer A Tensor of type float64. Scalar tensor.
fast An optional bool. Defaults to True.
name A name for the operation (optional).

A Tensor. Has the same type as matrix.

Numpy Compatibility

Equivalent to np.linalg.lstsq