tf.matrix_solve_ls

Aliases:

  • tf.linalg.lstsq
  • tf.matrix_solve_ls
tf.matrix_solve_ls(
    matrix,
    rhs,
    l2_regularizer=0.0,
    fast=True,
    name=None
)

Defined in tensorflow/python/ops/linalg_ops.py.

See the guide: Math > Matrix Math Functions

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-D double Tensor. Ignored if fast=False.
  • fast: bool. Defaults to True.
  • name: string, optional name of the operation.

Returns:

  • output: 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.

Raises:

  • NotImplementedError: matrix_solve_ls is currently disabled for complex128 and l2_regularizer != 0 due to poor accuracy.