tfp.substrates.numpy.math.gram_schmidt
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Implementation of the modified Gram-Schmidt orthonormalization algorithm.
tfp.substrates.numpy.math.gram_schmidt(
vectors, num_vectors=None
)
We assume here that the vectors are linearly independent. Zero vectors will be
left unchanged, but will also consume an iteration against num_vectors.
From [1]: "MGS is numerically equivalent to Householder QR factorization
applied to the matrix A augmented with a square matrix of zero elements on
top."
Historical note, see [1]: "modified" Gram-Schmidt was derived by Laplace [2],
for elimination and not as an orthogonalization algorithm. "Classical"
Gram-Schmidt actually came later [2]. Classical Gram-Schmidt has a sometimes
catastrophic loss of orthogonality for badly conditioned matrices, which is
discussed further in [1].
References
[1] Bjorck, A. (1994). Numerics of gram-schmidt orthogonalization. Linear
Algebra and Its Applications, 197, 297-316.
[2] P. S. Laplace, Thiorie Analytique des Probabilites. Premier Supple'ment,
Mme. Courtier, Paris, 1816.
[3] E. Schmidt, über die Auflosung linearer Gleichungen mit unendlich vielen
Unbekannten, Rend. Circ. Mat. Pulermo (1) 25:53-77 (1908).
Args |
|---|
vectors
|
A Tensor of shape [..., d, n] of d-dim column vectors to
orthonormalize.
|
num_vectors
|
Optional, number of leading vectors of the result to make
orthogonal. If unspecified, then num_vectors = n, implying that each
vector except for the last will be used in sequence.
|
Returns |
|---|
A Tensor of shape [..., d, n] corresponding to the orthonormalization.
|
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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complicated / too many steps","tooComplicatedTooManySteps","thumb-down"],["Out of date","outOfDate","thumb-down"],["Samples / code issue","samplesCodeIssue","thumb-down"],["Other","otherDown","thumb-down"]],["Last updated 2023-11-21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.substrates.numpy.math.gram_schmidt\n\n\u003cbr /\u003e\n\n|------------------------------------------------------------------------------------------------------------------------------------------------------|\n| [View source on GitHub](https://github.com/tensorflow/probability/blob/v0.23.0/tensorflow_probability/substrates/numpy/math/gram_schmidt.py#L28-L85) |\n\nImplementation of the modified Gram-Schmidt orthonormalization algorithm.\n\n#### View aliases\n\n\n**Main aliases**\n\n[`tfp.experimental.substrates.numpy.math.gram_schmidt`](https://www.tensorflow.org/probability/api_docs/python/tfp/substrates/numpy/math/gram_schmidt)\n\n\u003cbr /\u003e\n\n tfp.substrates.numpy.math.gram_schmidt(\n vectors, num_vectors=None\n )\n\nWe assume here that the vectors are linearly independent. Zero vectors will be\nleft unchanged, but will also consume an iteration against `num_vectors`.\n\nFrom \\[1\\]: \"MGS is numerically equivalent to Householder QR factorization\napplied to the matrix A augmented with a square matrix of zero elements on\ntop.\"\n\nHistorical note, see \\[1\\]: \"modified\" Gram-Schmidt was derived by Laplace \\[2\\],\nfor elimination and not as an orthogonalization algorithm. \"Classical\"\nGram-Schmidt actually came later \\[2\\]. Classical Gram-Schmidt has a sometimes\ncatastrophic loss of orthogonality for badly conditioned matrices, which is\ndiscussed further in \\[1\\].\n\n#### References\n\n\\[1\\] Bjorck, A. (1994). Numerics of gram-schmidt orthogonalization. Linear\nAlgebra and Its Applications, 197, 297-316.\n\n\\[2\\] P. S. Laplace, Thiorie Analytique des Probabilites. Premier Supple'ment,\nMme. Courtier, Paris, 1816.\n\n\\[3\\] E. Schmidt, über die Auflosung linearer Gleichungen mit unendlich vielen\nUnbekannten, Rend. Circ. Mat. Pulermo (1) 25:53-77 (1908).\n\n\u003cbr /\u003e\n\n\u003cbr /\u003e\n\n\u003cbr /\u003e\n\n| Args ---- ||\n|---------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| `vectors` | A Tensor of shape `[..., d, n]` of `d`-dim column vectors to orthonormalize. |\n| `num_vectors` | Optional, number of leading vectors of the result to make orthogonal. If unspecified, then `num_vectors = n`, implying that each vector except for the last will be used in sequence. |\n\n\u003cbr /\u003e\n\n\u003cbr /\u003e\n\n\u003cbr /\u003e\n\n\u003cbr /\u003e\n\n| Returns ------- ||\n|---|---|\n| A Tensor of shape `[..., d, n]` corresponding to the orthonormalization. ||\n\n\u003cbr /\u003e"]]
View source on GitHub