svn commit: r1781623 - /mahout/site/mahout_cms/trunk/content/users/algorithms/d-spca.mdtext

[ak...@apache.org]

Author: akm
Date: Fri Feb  3 23:35:13 2017
New Revision: 1781623

URL: http://svn.apache.org/viewvc?rev=1781623&view=rev
Log:
Formatting

Modified:
    mahout/site/mahout_cms/trunk/content/users/algorithms/d-spca.mdtext

Modified: mahout/site/mahout_cms/trunk/content/users/algorithms/d-spca.mdtext
URL: http://svn.apache.org/viewvc/mahout/site/mahout_cms/trunk/content/users/algorithms/d-spca.mdtext?rev=1781623&r1=1781622&r2=1781623&view=diff
==============================================================================
--- mahout/site/mahout_cms/trunk/content/users/algorithms/d-spca.mdtext (original)
+++ mahout/site/mahout_cms/trunk/content/users/algorithms/d-spca.mdtext Fri Feb  3 23:35:13 2017
@@ -12,22 +12,22 @@ Given an *m* `\(\times\)` *n* matrix `\(
 1. Create seed for random *n* `\(\times\)` *(k+p)* matrix `\(\Omega\)`.
 2. `\(s_\Omega \leftarrow \Omega^\top \mu\)`.
 3. `\(\mathbf{Y_0 \leftarrow A\Omega − 1 {s_\Omega}^\top, Y \in \mathbb{R}^{m\times(k+p)}}\)`.
-4. Column-orthonormalize `\(\mathbf{Y_0} \rightarrow \mathbf{Q}\)` by computing thin decomposition `\(\mathbf{Y_0} = \mathbf{QR}\)`. Also, `\(\mathbf{Q}\in\mathbb{R}^(m\times(k+p)), \mathbf{R}\in\mathbb{R}^((k+p)\times(k+p))\)`.
-5. `\(s_Q \leftarrow Q^\top 1\)`.
-6. `\(\mathbf{B_0} \leftarrow Q^\top A: B \in \mathbb{R}^((k+p)\times n)\)`.
-7. `\(s_B \leftarrow (B_0)^\top \mu\)`.
+4. Column-orthonormalize `\(\mathbf{Y_0} \rightarrow \mathbf{Q}\)` by computing thin decomposition `\(\mathbf{Y_0} = \mathbf{QR}\)`. Also, `\(\mathbf{Q}\in\mathbb{R}^{m\times(k+p)}, \mathbf{R}\in\mathbb{R}^{(k+p)\times(k+p)}\)`.
+5. `\(\mathbf{s_Q \leftarrow Q^\top 1}\)`.
+6. `\(\mathbf{B_0 \leftarrow Q^\top A: B \in \mathbb{R}^{(k+p)\times n}}\)`.
+7. `\(\mathbf{s_B \leftarrow {B_0}^\top \mu}\)`.
 8. For *i* in 1..*q* repeat (power iterations):
-    - For *j* in 1..*n* `\(apply(B_(i−1))_(∗j) \leftarrow (B_(i−1))_(∗j)−\mu_j s_Q\)`.
-    - `\(\mathbf{Y_i) \leftarrow \mathbf{(AB_(i−1)^\top)−1(s_B−\mu^\top \mu s_Q^\top)}\)`.
+    - For *j* in 1..*n* apply `\(\mathbf{(B_{i−1})_{∗j} \leftarrow (B_{i−1})_{∗j}−\mu_j s_Q}\)`.
+    - `\(\mathbf{Y_i \leftarrow (A{B_{i−1}}^\top)−1(s_B−\mu^\top \mu s_Q)^\top)}\)`.
     - Column-orthonormalize `\(\mathbf{Y_i} \rightarrow \mathbf{Q}\)` by computing thin decomposition `\(\mathbf{Y_i = QR}\)`.
     - `\(\mathbf{s_Q \leftarrow Q^\top 1}\)`.
     - `\(\mathbf{B_i \leftarrow Q^\top A}\)`.
-    - `\(\mathbf{s_B \leftarrow (B_i)^\top \mu}\)`.
-9. Let `\(\mathbf{C \triangleq s_Q (s_B)^\top}\)`. `\(\mathbf{M \leftarrow B_q (B_q)^\top − C − C^\top + \mu^\top \mu s_Q (s_Q)^\top}\)`.
-10. Compute an eigensolution of the small symmetric `\(\mathbf{M = \hat{U} \Lambda \hat{U}^\top: M \in \mathbb{R}^((k+p)\times(k+p))}\)`.
-11. The singular values `\(\Sigma = \Lambda^(\circ 0.5)\)`, or, in other words, `\(\mathbf{\sigma_i= \sqrt{\lambda_i}}\)`.
+    - `\(\mathbf{s_B \leftarrow {B_i}^\top \mu}\)`.
+9. Let `\(\mathbf{C \triangleq s_Q {s_B}^\top}\)`. `\(\mathbf{M \leftarrow B_q {B_q}^\top − C − C^\top + \mu^\top \mu s_Q {s_Q}^\top}\)`.
+10. Compute an eigensolution of the small symmetric `\(\mathbf{M = \hat{U} \Lambda \hat{U}^\top: M \in \mathbb{R}^{(k+p)\times(k+p)}}\)`.
+11. The singular values `\(\Sigma = \Lambda^{\circ 0.5}\)`, or, in other words, `\(\mathbf{\sigma_i= \sqrt{\lambda_i}}\)`.
 12. If needed, compute `\(\mathbf{U = Q\hat{U}}\)`.
-13. If needed, compute `\(\mathbf{V = B^\top \hat{U} \Sigma^(−1)}\)`. Another way is `\(\mathbf{V = A^\top U\Sigma^(−1)}\)1.
+13. If needed, compute `\(\mathbf{V = B^\top \hat{U} \Sigma^{−1}}\)`. Another way is `\(\mathbf{V = A^\top U\Sigma^{−1}}\).
 14. If needed, items converted to the PCA space can be computed as `\(\mathbf{U\Sigma}\)`.
 
 ## Implementation