# tfp.substrates.jax.math.pivoted_cholesky

Computes the (partial) pivoted cholesky decomposition of `matrix`.

The pivoted Cholesky is a low rank approximation of the Cholesky decomposition of `matrix`, i.e. as described in [(Harbrecht et al., 2012)]. The currently-worst-approximated diagonal element is selected as the pivot at each iteration. This yields from a `[B1...Bn, N, N]` shaped `matrix` a ```[B1...Bn, N, K]``` shaped rank-`K` approximation `lr` such that `lr @ lr.T ~= matrix`. Note that, unlike the Cholesky decomposition, `lr` is not triangular even in a rectangular-matrix sense. However, under a permutation it could be made triangular (it has one more zero in each column as you move to the right).

Such a matrix can be useful as a preconditioner for conjugate gradient optimization, i.e. as in [(Wang et al. 2019)], as matmuls and solves can be cheaply done via the Woodbury matrix identity, as implemented by `tf.linalg.LinearOperatorLowRankUpdate`.

`matrix` Floating point `Tensor` batch of symmetric, positive definite matrices.
`max_rank` Scalar `int` `Tensor`, the rank at which to truncate the approximation.
`diag_rtol` Scalar floating point `Tensor` (same dtype as `matrix`). If the errors of all diagonal elements of `lr @ lr.T` are each lower than `element * diag_rtol`, iteration is permitted to terminate early.
`name` Optional name for the op.

`lr` Low rank pivoted Cholesky approximation of `matrix`.

: H Harbrecht, M Peters, R Schneider. On the low-rank approximation by the pivoted Cholesky decomposition. Applied numerical mathematics, 62(4):428-440, 2012.

: K. A. Wang et al. Exact Gaussian Processes on a Million Data Points. arXiv preprint arXiv:1903.08114, 2019. https://arxiv.org/abs/1903.08114

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