tf.linalg.lstsq

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 stabl