Solves one or more linear least-squares problems.

tf.raw_ops.MatrixSolveLs( matrix, rhs, l2_regularizer, fast=True, name=None )

`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.

Args | |
---|---|

`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). |

Returns | |
---|---|

A `Tensor` . Has the same type as `matrix` . |

Equivalent to np.linalg.lstsq

© 2020 The TensorFlow Authors. All rights reserved.

Licensed under the Creative Commons Attribution License 3.0.

Code samples licensed under the Apache 2.0 License.

https://www.tensorflow.org/versions/r2.4/api_docs/python/tf/raw_ops/MatrixSolveLs