tf.matrix_solve_ls(matrix, rhs, l2_regularizer=0.0, fast=True, name=None)
Solves one or more linear least-squares problems.
matrix
is a tensor of shape [..., M, N]
whose inner-most 2 dimensions form M
-by-N
matrices. Rhs is a tensor of shape [..., M, K]
whose inner-most 2 dimensions form M
-by-K
matrices. The computed output is a Tensor
of shape [..., N, K]
whose inner-most 2 dimensions form M
-by-K
matrices that solve the equations matrix[..., :, :] * output[..., :, :] = rhs[..., :, :]
in the least squares sense.
Below we will use the following notation for each pair of matrix and right-hand sides in the batch:
matrix
=\(A \in \Re^{m \times n}\), rhs
=\(B \in \Re^{m \times k}\), output
=\(X \in \Re^{n \times k}\), l2_regularizer
=\(\lambda\).
If fast
is True
, then the solution is computed by solving the normal equations using Cholesky decomposition. Specifically, if \(m \ge n\) then \(X = (A^T A + \lambda I)^{-1} A^T B\), 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 \(X = A^T (A A^T + \lambda I)^{-1} B\), which (for \(\lambda = 0\)) is the minimum-norm solution to the under-determined linear system, i.e. \(X = \mathrm{argmin}_{Z \in \Re^{n \times k}} ||Z||_F^2 \), 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.
Args:
-
matrix
:Tensor
of shape[..., M, N]
. -
rhs
:Tensor
of shape[..., M, K]
. -
l2_regularizer
: 0-Ddouble
Tensor
. Ignored iffast=False
. -
fast
: bool. Defaults toTrue
. -
name
: string, optional name of the operation.
Returns:
-
output
:Tensor
of shape[..., N, K]
whose inner-most 2 dimensions formM
-by-K
matrices that solve the equationsmatrix[..., :, :] * output[..., :, :] = rhs[..., :, :]
in the least squares sense.
Please login to continue.