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                                                      recommendations with Spark
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                                                    • Algorithms
                                                        -
                                                      • Matrix Decomposition
                                                        • +
                                                      • Distributed Matrix Decomposition
                                                      • Cholesky QR
                                                        • +
                                                      • SSVD
                                                      • Recommendations
                                                      • Recommender Overview
                                                      • Intro to cooccurrence-based
                                                        recommendations with Spark
                                                        • Modified: websites/staging/mahout/trunk/content/overview.html ============================================================================== --- websites/staging/mahout/trunk/content/overview.html (original) +++ websites/staging/mahout/trunk/content/overview.html Fri Mar 27 23:01:23 2015 @@ -165,8 +165,9 @@
                                                      • Algorithms
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                                                        • Matrix Decomposition
                                                          • +
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                                                        • Cholesky QR
                                                          • +
                                                        • SSVD
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                                                        • Recommender Overview
                                                        • Intro to cooccurrence-based
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                                                        • Algorithms
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                                                          • Matrix Decomposition
                                                            • +
                                                          • Distributed Matrix Decomposition
                                                          • Cholesky QR
                                                            • +
                                                          • SSVD
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                                                          • Recommender Overview
                                                          • Intro to cooccurrence-based
                                                            recommendations with Spark
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                                                            + + + + +
                                                            +
                                                            +

                                                            Distributed Stochastic Singular Value Decomposition

                                                            +

                                                            Intro

                                                            +

                                                            Mahout has a distributed implementation of Stochastic Singular Value Decomposition [1].

                                                            +

                                                            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 \(\mathbf{\Omega}\) + using Gaussian unit vectors per one of suggestions in [Halko, Martinsson, Tropp].

                                                              +
                                                              2. +
                                                            2. +

                                                              \(\mathbf{Y=A\boldsymbol{\Omega}},\,\mathbf{Y}\in\mathbb{R}^{m\times\left(k+p\right)}\)

                                                              +
                                                              3. +
                                                            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 \(\mathbf{Q}=\mbox{qr}\left(\mathbf{Y}\right).\mathbf{Q}\)

                                                              +
                                                              4. +
                                                            4. +

                                                              \(\mathbf{B}_{0}=\mathbf{Q}^{\top}\mathbf{A}:\,\,\mathbf{B}\in\mathbb{R}^{\left(k+p\right)\times n}\).

                                                              +
                                                              5. +
                                                            5. +

                                                              If \(q>0\) + repeat: for \(i=1..q\): + \(\mathbf{B}_{i}^{\top}=\mathbf{A}^{\top}\mbox{qr}\left(\mathbf{A}\mathbf{B}_{i-1}^{\top}\right).\mathbf{Q}\) + (power iterations step).

                                                              +
                                                              6. +
                                                            6. +

                                                              Compute Eigensolution of a small Hermitian \(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}=\mathbf{\hat{U}}\boldsymbol{\Lambda}\mathbf{\hat{U}}^{\top}\), + \(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\).

                                                              +
                                                              7. +
                                                            7. +

                                                              Singular values \(\mathbf{\boldsymbol{\Sigma}}=\boldsymbol{\Lambda}^{0.5}\), + or, in other words, \(s_{i}=\sqrt{\sigma_{i}}\).

                                                              +
                                                              8. +
                                                            8. +

                                                              If needed, compute \(\mathbf{U}=\mathbf{Q}\hat{\mathbf{U}}\).

                                                              +
                                                              9. +
                                                            9. +

                                                              If needed, compute \(\mathbf{V}=\mathbf{B}_{q}^{\top}\hat{\mathbf{U}}\boldsymbol{\Sigma}^{-1}\). +Another way is \(\mathbf{V}=\mathbf{A}^{\top}\mathbf{U}\boldsymbol{\Sigma}^{-1}\).

                                                              +
                                                              10. +
                                                            +

                                                            Implementation

                                                            +

                                                            Mahout dssvd(...) is implemented in the mahout math-scala algebraic optimizer which translates Mahout's R-like linear algebra operators into a physical plan for both Spark and H2O distributed engines.

                                                            +
                                                            def dssvd[K: ClassTag](drmA: DrmLike[K], k: Int, p: Int = 15, q: Int = 0):
+    (DrmLike[K], DrmLike[Int], Vector) = {
+
+    val drmAcp = drmA.checkpoint()
+
+    val m = drmAcp.nrow
+    val n = drmAcp.ncol
+    assert(k <= (m min n), "k cannot be greater than smaller of m, n.")
+    val pfxed = safeToNonNegInt((m min n) - k min p)
+
+    // Actual decomposition rank
+    val r = k + pfxed
+
+    // We represent Omega by its seed.
+    val omegaSeed = RandomUtils.getRandom().nextInt()
+
+    // Compute Y = A*Omega.  
+    var drmY = drmAcp.mapBlock(ncol = r) {
+        case (keys, blockA) =>
+            val blockY = blockA %*% Matrices.symmetricUniformView(n, r, omegaSeed)
+        keys -> blockY
+    }
+
+    var drmQ = dqrThin(drmY.checkpoint())._1
+
+    // Checkpoint Q if last iteration
+    if (q == 0) drmQ = drmQ.checkpoint()
+
+    var drmBt = drmAcp.t %*% drmQ
+
+    // Checkpoint B' if last iteration
+    if (q == 0) drmBt = drmBt.checkpoint()
+
+    for (i <- 0  until q) {
+        drmY = drmAcp %*% drmBt
+        drmQ = dqrThin(drmY.checkpoint())._1
+
+        // Checkpoint Q if last iteration
+        if (i == q - 1) drmQ = drmQ.checkpoint()
+
+        drmBt = drmAcp.t %*% drmQ
+
+        // Checkpoint B' if last iteration
+        if (i == q - 1) drmBt = drmBt.checkpoint()
+    }
+
+    val (inCoreUHat, d) = eigen(drmBt.t %*% drmBt)
+    val s = d.sqrt
+
+    // Since neither drmU nor drmV are actually computed until actually used
+    // we don't need the flags instructing compute (or not compute) either of the U,V outputs 
+    val drmU = drmQ %*% inCoreUHat
+    val drmV = drmBt %*% (inCoreUHat %*%: diagv(1 /: s))
+
+    (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.

                                                            +

                                                            Usage

                                                            +

                                                            The scala dssvd(...) method can easily be called in any Spark or H2O application built with the math-scala library and the corresponding Spark or H2O engine module as follows:

                                                            +
                                                            import org.apache.mahout.math._
+import org.decompsitions._
+
+
+val(drmU, drmV, s) = dssvd(drma, k = 40, q = 1)
+
                                                            + + +

                                                            References

                                                            +

                                                            [1]: Mahout Scala and Mahout Spark Bindings for Linear Algebra Subroutines

                                                            +

                                                            [2]: Halko, Martinsson, Tropp

                                                            +

                                                            [3]: Mahout Spark and Scala Bindings

                                                            +

                                                            [4]: Randomized methods for computing low-rank +approximations of matrices

                                                            +
                                                            +
                                                            +
                                                            +
                                                            +
                                                            +

                                                            + Copyright © 2014 The Apache Software Foundation, Licensed under + the Apache License, Version 2.0. +
                                                            + Apache and the Apache feather logos are trademarks of The Apache Software Foundation. +

                                                            +
                                                            +
                                                            + + + + + + Modified: websites/staging/mahout/trunk/content/users/algorithms/intro-cooccurrence-spark.html ============================================================================== --- websites/staging/mahout/trunk/content/users/algorithms/intro-cooccurrence-spark.html (original) +++ websites/staging/mahout/trunk/content/users/algorithms/intro-cooccurrence-spark.html Fri Mar 27 23:01:23 2015 @@ -165,8 +165,9 @@
                                                          • Algorithms
                                                              -
                                                            • Matrix Decomposition
                                                              • +
                                                            • Distributed Matrix Decomposition
                                                            • Cholesky QR
                                                              • +
                                                            • SSVD
                                                            • Recommendations
                                                            • Recommender Overview
                                                            • Intro to cooccurrence-based
                                                              recommendations with Spark
                                                              • Modified: websites/staging/mahout/trunk/content/users/algorithms/recommender-overview.html ============================================================================== --- websites/staging/mahout/trunk/content/users/algorithms/recommender-overview.html (original) +++ websites/staging/mahout/trunk/content/users/algorithms/recommender-overview.html Fri Mar 27 23:01:23 2015 @@ -165,8 +165,9 @@
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                                                              • Matrix Decomposition
                                                                • +
                                                              • Distributed Matrix Decomposition
                                                              • Cholesky QR
                                                                • +
                                                              • SSVD
                                                              • Recommendations
                                                              • Recommender Overview
                                                              • Intro to cooccurrence-based
                                                                recommendations with Spark
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                                                                • Distributed Matrix Decomposition
                                                                • Cholesky QR
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                                                                • SSVD
                                                                • Recommendations
                                                                • Recommender Overview
                                                                • Intro to cooccurrence-based
                                                                  recommendations with Spark
                                                                  • Modified: websites/staging/mahout/trunk/content/users/basics/algorithms.html ============================================================================== --- websites/staging/mahout/trunk/content/users/basics/algorithms.html (original) +++ websites/staging/mahout/trunk/content/users/basics/algorithms.html Fri Mar 27 23:01:23 2015 @@ -165,8 +165,9 @@
                                                                • Algorithms
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                                                                  • Matrix Decomposition
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                                                                  • Distributed Matrix Decomposition
                                                                  • Cholesky QR
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                                                                  • SSVD
                                                                  • Recommendations
                                                                  • Recommender Overview
                                                                  • Intro to cooccurrence-based
                                                                    recommendations with Spark
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                                                                    • Matrix Decomposition
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                                                                    • Distributed Matrix Decomposition
                                                                    • Cholesky QR
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                                                                    • Recommendations
                                                                    • Recommender Overview
                                                                    • Intro to cooccurrence-based
                                                                      recommendations with Spark
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                                                                      • Matrix Decomposition
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                                                                      • Distributed Matrix Decomposition
                                                                      • Cholesky QR
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                                                                      • SSVD
                                                                      • Recommendations
                                                                      • Recommender Overview
                                                                      • Intro to cooccurrence-based
                                                                        recommendations with Spark
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                                                                        • Matrix Decomposition
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                                                                        • Distributed Matrix Decomposition
                                                                        • Cholesky QR
                                                                          • +
                                                                        • SSVD
                                                                        • Recommendations
                                                                        • Recommender Overview
                                                                        • Intro to cooccurrence-based
                                                                          recommendations with Spark
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                                                                          • Recommender Overview
                                                                          • Intro to cooccurrence-based
                                                                            recommendations with Spark
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                                                                            • Cholesky QR
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                                                                            • SSVD
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                                                                            • Recommender Overview
                                                                            • Intro to cooccurrence-based
                                                                              recommendations with Spark
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                                                                              • Matrix Decomposition
                                                                                • +
                                                                              • Distributed Matrix Decomposition
                                                                              • Cholesky QR
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                                                                              • SSVD
                                                                              • Recommendations
                                                                              • Recommender Overview
                                                                              • Intro to cooccurrence-based
                                                                                recommendations with Spark
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                                                                                • Recommender Overview
                                                                                • Intro to cooccurrence-based
                                                                                  recommendations with Spark
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                                                                                  • Distributed Matrix Decomposition
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                                                                                  • Recommender Overview
                                                                                  • Intro to cooccurrence-based
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