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The following commit(s) were added to refs/heads/master by this push:
new f7fe2c2 [WEBSITE] Mathjax on dssvd page.
f7fe2c2 is described below
commit f7fe2c250f1e06c165692734e034a7c048cbb774
Author: Trevor a.k.a @rawkintrevo <[email protected]>
AuthorDate: Thu Feb 28 13:24:15 2019 -0600
[WEBSITE] Mathjax on dssvd page.
---
.../latest/algorithms/linear-algebra/d-ssvd.md | 50 +++++++++++-----------
1 file changed, 25 insertions(+), 25 deletions(-)
diff --git a/website/docs/latest/algorithms/linear-algebra/d-ssvd.md
b/website/docs/latest/algorithms/linear-algebra/d-ssvd.md
index 364cd31..35b05d8 100644
--- a/website/docs/latest/algorithms/linear-algebra/d-ssvd.md
+++ b/website/docs/latest/algorithms/linear-algebra/d-ssvd.md
@@ -11,40 +11,40 @@ Mahout has a distributed implementation of Stochastic
Singular Value Decompositi
## Modified SSVD Algorithm
-Given an `\(m\times n\)`
-matrix `\(\mathbf{A}\)`, a target rank `\(k\in\mathbb{N}_{1}\)`
-, an oversampling parameter `\(p\in\mathbb{N}_{1}\)`,
-and the number of additional power iterations `\(q\in\mathbb{N}_{0}\)`,
-this procedure computes an `\(m\times\left(k+p\right)\)`
-SVD `\(\mathbf{A\approx U}\boldsymbol{\Sigma}\mathbf{V}^{\top}\)`:
-
- 1. Create seed for random `\(n\times\left(k+p\right)\)`
- matrix `\(\boldsymbol{\Omega}\)`. The seed defines matrix
`\(\mathbf{\Omega}\)`
+Given an <foo>\(m\times n\)</foo>
+matrix <foo>\(\mathbf{A}\)</foo>, a target rank
<foo>\(k\in\mathbb{N}_{1}\)</foo>
+, an oversampling parameter <foo>\(p\in\mathbb{N}_{1}\)</foo>,
+and the number of additional power iterations
<foo>\(q\in\mathbb{N}_{0}\)</foo>,
+this procedure computes an <foo>\(m\times\left(k+p\right)\)</foo>
+SVD <foo>\(\mathbf{A\approx U}\boldsymbol{\Sigma}\mathbf{V}^{\top}\)</foo>:
+
+ 1. Create seed for random <foo>\(n\times\left(k+p\right)\)</foo>
+ matrix <foo>\(\boldsymbol{\Omega}\)</foo>. The seed defines matrix
<foo>\(\mathbf{\Omega}\)</foo>
using Gaussian unit vectors per one of suggestions in [Halko, Martinsson,
Tropp].
- 2.
`\(\mathbf{Y=A\boldsymbol{\Omega}},\,\mathbf{Y}\in\mathbb{R}^{m\times\left(k+p\right)}\)`
+ 2.
<foo>\(\mathbf{Y=A\boldsymbol{\Omega}},\,\mathbf{Y}\in\mathbb{R}^{m\times\left(k+p\right)}\)</foo>
- 3. Column-orthonormalize `\(\mathbf{Y}\rightarrow\mathbf{Q}\)`
- by computing thin decomposition `\(\mathbf{Y}=\mathbf{Q}\mathbf{R}\)`.
- Also,
`\(\mathbf{Q}\in\mathbb{R}^{m\times\left(k+p\right)},\,\mathbf{R}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\)`;
denoted as `\(\mathbf{Q}=\mbox{qr}\left(\mathbf{Y}\right).\mathbf{Q}\)`
+ 3. Column-orthonormalize <foo>\(\mathbf{Y}\rightarrow\mathbf{Q}\)</foo>
+ by computing thin decomposition
<foo>\(\mathbf{Y}=\mathbf{Q}\mathbf{R}\)</foo>.
+ Also,
<foo>\(\mathbf{Q}\in\mathbb{R}^{m\times\left(k+p\right)},\,\mathbf{R}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\)</foo>;
denoted as
<foo>\(\mathbf{Q}=\mbox{qr}\left(\mathbf{Y}\right).\mathbf{Q}\)</foo>
- 4.
`\(\mathbf{B}_{0}=\mathbf{Q}^{\top}\mathbf{A}:\,\,\mathbf{B}\in\mathbb{R}^{\left(k+p\right)\times
n}\)`.
+ 4.
<foo>\(\mathbf{B}_{0}=\mathbf{Q}^{\top}\mathbf{A}:\,\,\mathbf{B}\in\mathbb{R}^{\left(k+p\right)\times
n}\)</foo>.
- 5. If `\(q>0\)`
- repeat: for `\(i=1..q\)`:
-
`\(\mathbf{B}_{i}^{\top}=\mathbf{A}^{\top}\mbox{qr}\left(\mathbf{A}\mathbf{B}_{i-1}^{\top}\right).\mathbf{Q}\)`
+ 5. If <foo>\(q>0\)</foo>
+ repeat: for <foo>\(i=1..q\)</foo>:
+
<foo>\(\mathbf{B}_{i}^{\top}=\mathbf{A}^{\top}\mbox{qr}\left(\mathbf{A}\mathbf{B}_{i-1}^{\top}\right).\mathbf{Q}\)</foo>
(power iterations step).
- 6. Compute Eigensolution of a small Hermitian
`\(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}=\mathbf{\hat{U}}\boldsymbol{\Lambda}\mathbf{\hat{U}}^{\top}\)`,
-
`\(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\)`.
+ 6. Compute Eigensolution of a small Hermitian
<foo>\(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}=\mathbf{\hat{U}}\boldsymbol{\Lambda}\mathbf{\hat{U}}^{\top}\)</foo>,
+
<foo>\(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\)</foo>.
- 7. Singular values
`\(\mathbf{\boldsymbol{\Sigma}}=\boldsymbol{\Lambda}^{0.5}\)`,
- or, in other words, `\(s_{i}=\sqrt{\sigma_{i}}\)`.
+ 7. Singular values
<foo>\(\mathbf{\boldsymbol{\Sigma}}=\boldsymbol{\Lambda}^{0.5}\)</foo>,
+ or, in other words, <foo>\(s_{i}=\sqrt{\sigma_{i}}\)</foo>.
- 8. If needed, compute `\(\mathbf{U}=\mathbf{Q}\hat{\mathbf{U}}\)`.
+ 8. If needed, compute <foo>\(\mathbf{U}=\mathbf{Q}\hat{\mathbf{U}}\)</foo>.
- 9. If needed, compute
`\(\mathbf{V}=\mathbf{B}_{q}^{\top}\hat{\mathbf{U}}\boldsymbol{\Sigma}^{-1}\)`.
-Another way is
`\(\mathbf{V}=\mathbf{A}^{\top}\mathbf{U}\boldsymbol{\Sigma}^{-1}\)`.
+ 9. If needed, compute
<foo>\(\mathbf{V}=\mathbf{B}_{q}^{\top}\hat{\mathbf{U}}\boldsymbol{\Sigma}^{-1}\)</foo>.
+Another way is
<foo>\(\mathbf{V}=\mathbf{A}^{\top}\mathbf{U}\boldsymbol{\Sigma}^{-1}\)</foo>.
@@ -110,7 +110,7 @@ Mahout `dssvd(...)` is implemented in the mahout
`math-scala` algebraic optimize
(drmU(::, 0 until k), drmV(::, 0 until k), s(0 until k))
}
-Note: As a side effect of checkpointing, U and V values are returned as
logical operators (i.e. they are neither checkpointed nor computed). Therefore
there is no physical work actually done to compute `\(\mathbf{U}\)` or
`\(\mathbf{V}\)` until they are used in a subsequent expression.
+Note: As a side effect of checkpointing, U and V values are returned as
logical operators (i.e. they are neither checkpointed nor computed). Therefore
there is no physical work actually done to compute <foo>\(\mathbf{U}\)</foo> or
<foo>\(\mathbf{V}\)</foo> until they are used in a subsequent expression.
## Usage
