tfp.experimental.distributions.marginal_fns.retrying_cholesky
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Computes a modified Cholesky decomposition for a batch of square matrices.
tfp.experimental.distributions.marginal_fns.retrying_cholesky(
matrix,
jitter=None,
max_iters=5,
unroll=1,
name='retrying_cholesky'
)
Given a symmetric matrix A, this function attempts to give a factorization
A + E = LL^T where L is lower triangular, LL^T is positive definite, and
E is small in some suitable sense. This is useful for nearly positive
definite symmetric matrices that are otherwise numerically difficult to
Cholesky factor.
In particular, this function first attempts a Cholesky decomposition of
the input matrix. If that decomposition fails, exponentially-increasing
diagonal jitter is added to the matrix until either a Cholesky decomposition
succeeds or until the maximum specified number of iterations is reached.
This function is similar in spirit to a true modified Cholesky factorization
([1], [2]). However, it does not use pivoting or other strategies to ensure
stability, so may not work well for e.g. ill-conditioned matrices. Further,
this function may perform multiple Cholesky factorizations, while a true
modified Cholesky can be done with only slightly more work than a single
decomposition.
References
[1]: Nicholas Higham. What is a modified Cholesky factorization?
https://nhigham.com/2020/12/22/what-is-a-modified-cholesky-factorization/
[2]: Sheung Hun Cheng and Nicholas Higham, A Modified Cholesky Algorithm Based
on a Symmetric Indefinite Factorization, SIAM J. Matrix Anal. Appl. 19(4),
1097–1110, 1998.
Args |
|---|
matrix
|
A batch of symmetric square matrices, with shape [..., n, n].
|
jitter
|
Initial jitter to add to the diagonal. Default: 1e-6, unless
matrix.dtype is float64, in which case the default is 1e-10.
|
max_iters
|
Maximum number of times to retry the Cholesky decomposition
with larger diagonal jitter. Default: 5.
|
unroll
|
If specified, unroll >1 iterations per while_loop step. This can be
advantageous on GPU, to avoid host roundtripping.
|
name
|
Python str name prefixed to Ops created by this function.
Default value: 'retrying_cholesky'.
|
Returns |
|---|
triangular_factor
|
A Tensor with shape [..., n, n]. The lower triangular
Cholesky factor, modified as above. If the Cholesky decomposition failed
for a batch member, then all lower triangular entries returned for that
batch member will be NaN.
|
diagonal_shift
|
A tensor of shape [...]. diag_shift[i] is the value
added to the diagonal of matrix[i] in computing triangular_factor[i].
|
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Last updated 2023-11-21 UTC.
[[["Easy to understand","easyToUnderstand","thumb-up"],["Solved my problem","solvedMyProblem","thumb-up"],["Other","otherUp","thumb-up"]],[["Missing the information I need","missingTheInformationINeed","thumb-down"],["Too complicated / too many steps","tooComplicatedTooManySteps","thumb-down"],["Out of date","outOfDate","thumb-down"],["Samples / code issue","samplesCodeIssue","thumb-down"],["Other","otherDown","thumb-down"]],["Last updated 2023-11-21 UTC."],[],[],null,["# tfp.experimental.distributions.marginal_fns.retrying_cholesky\n\n\u003cbr /\u003e\n\n|--------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| [View source on GitHub](https://github.com/tensorflow/probability/blob/v0.23.0/tensorflow_probability/python/experimental/distributions/marginal_fns.py#L221-L276) |\n\nComputes a modified Cholesky decomposition for a batch of square matrices. \n\n tfp.experimental.distributions.marginal_fns.retrying_cholesky(\n matrix,\n jitter=None,\n max_iters=5,\n unroll=1,\n name='retrying_cholesky'\n )\n\nGiven a symmetric matrix `A`, this function attempts to give a factorization\n`A + E = LL^T` where `L` is lower triangular, `LL^T` is positive definite, and\n`E` is small in some suitable sense. This is useful for nearly positive\ndefinite symmetric matrices that are otherwise numerically difficult to\nCholesky factor.\n\nIn particular, this function first attempts a Cholesky decomposition of\nthe input matrix. If that decomposition fails, exponentially-increasing\ndiagonal jitter is added to the matrix until either a Cholesky decomposition\nsucceeds or until the maximum specified number of iterations is reached.\n\nThis function is similar in spirit to a true modified Cholesky factorization\n(\\[1\\], \\[2\\]). However, it does not use pivoting or other strategies to ensure\nstability, so may not work well for e.g. ill-conditioned matrices. Further,\nthis function may perform multiple Cholesky factorizations, while a true\nmodified Cholesky can be done with only slightly more work than a single\ndecomposition.\n\n#### References\n\n\\[1\\]: Nicholas Higham. What is a modified Cholesky factorization?\n\u003chttps://nhigham.com/2020/12/22/what-is-a-modified-cholesky-factorization/\u003e\n\n\\[2\\]: Sheung Hun Cheng and Nicholas Higham, A Modified Cholesky Algorithm Based\non a Symmetric Indefinite Factorization, SIAM J. Matrix Anal. Appl. 19(4),\n1097--1110, 1998.\n\n\u003cbr /\u003e\n\n\u003cbr /\u003e\n\n\u003cbr /\u003e\n\n| Args ---- ||\n|-------------|-----------------------------------------------------------------------------------------------------------------------------|\n| `matrix` | A batch of symmetric square matrices, with shape `[..., n, n]`. |\n| `jitter` | Initial jitter to add to the diagonal. Default: 1e-6, unless `matrix.dtype` is float64, in which case the default is 1e-10. |\n| `max_iters` | Maximum number of times to retry the Cholesky decomposition with larger diagonal jitter. Default: 5. |\n| `unroll` | If specified, unroll \\\u003e1 iterations per while_loop step. This can be advantageous on GPU, to avoid host roundtripping. |\n| `name` | Python `str` name prefixed to Ops created by this function. Default value: 'retrying_cholesky'. |\n\n\u003cbr /\u003e\n\n\u003cbr /\u003e\n\n\u003cbr /\u003e\n\n\u003cbr /\u003e\n\n| Returns ------- ||\n|---------------------|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| `triangular_factor` | A Tensor with shape `[..., n, n]`. The lower triangular Cholesky factor, modified as above. If the Cholesky decomposition failed for a batch member, then all lower triangular entries returned for that batch member will be NaN. |\n| `diagonal_shift` | A tensor of shape `[...]`. `diag_shift[i]` is the value added to the diagonal of `matrix[i]` in computing `triangular_factor[i]`. |\n\n\u003cbr /\u003e"]]
View source on GitHub