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| <h1 class="title">SystemML Engine Developer Guide</h1> |
| |
| |
| <!-- |
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| --> |
| <ul id="markdown-toc"> |
| <li><a href="#building-systemml" id="markdown-toc-building-systemml">Building SystemML</a></li> |
| <li><a href="#testing-systemml" id="markdown-toc-testing-systemml">Testing SystemML</a></li> |
| <li><a href="#development-environment" id="markdown-toc-development-environment">Development Environment</a></li> |
| <li><a href="#python-mlcontext-api" id="markdown-toc-python-mlcontext-api">Python MLContext API</a></li> |
| <li><a href="#matrix-multiplication-operators" id="markdown-toc-matrix-multiplication-operators">Matrix Multiplication Operators</a> <ul> |
| <li><a href="#basic-matrix-multiplication-operators" id="markdown-toc-basic-matrix-multiplication-operators">Basic Matrix Multiplication Operators</a></li> |
| <li><a href="#complex-matrix-multiplication-operators" id="markdown-toc-complex-matrix-multiplication-operators">Complex Matrix Multiplication Operators</a></li> |
| <li><a href="#core-matrix-multiplication-primitives" id="markdown-toc-core-matrix-multiplication-primitives">Core Matrix Multiplication Primitives</a></li> |
| </ul> |
| </li> |
| </ul> |
| |
| <h2 id="building-systemml">Building SystemML</h2> |
| |
| <p>SystemML is built using <a href="http://maven.apache.org/">Apache Maven</a>. |
| SystemML will build on Linux, MacOS, or Windows, and requires Maven 3 and Java 7 (or higher). |
| To build SystemML, run:</p> |
| |
| <pre><code>mvn clean package |
| </code></pre> |
| |
| <p>To build the SystemML distributions, run:</p> |
| |
| <pre><code>mvn clean package -P distribution |
| </code></pre> |
| |
| <hr /> |
| |
| <h2 id="testing-systemml">Testing SystemML</h2> |
| |
| <p>SystemML features a comprehensive set of integration tests. To perform these tests, run:</p> |
| |
| <pre><code>mvn verify |
| </code></pre> |
| |
| <p>Note: these tests require <a href="https://www.r-project.org/">R</a> to be installed and available as part of the PATH variable on |
| the machine on which you are running these tests.</p> |
| |
| <p>If required, please install the following packages in R:</p> |
| |
| <pre><code>install.packages(c("batch", "bitops", "boot", "caTools", "data.table", "doMC", "doSNOW", "ggplot2", "glmnet", "lda", "Matrix", "matrixStats", "moments", "plotrix", "psych", "reshape", "topicmodels", "wordcloud"), dependencies=TRUE) |
| </code></pre> |
| |
| <hr /> |
| |
| <h2 id="development-environment">Development Environment</h2> |
| |
| <p>SystemML itself is written in Java and is managed using Maven. As a result, SystemML can readily be |
| imported into a standard development environment such as Eclipse and IntelliJ IDEA. |
| The <code>DMLScript</code> class serves as the main entrypoint to SystemML. Executing |
| <code>DMLScript</code> with no arguments displays usage information. A script file can be specified using the <code>-f</code> argument.</p> |
| |
| <p>In Eclipse, a Debug Configuration can be created with <code>DMLScript</code> as the Main class and any arguments specified as |
| Program arguments.</p> |
| |
| <p>Suppose that we have a <code>hello.dml</code> script containing the following:</p> |
| |
| <pre><code>print('hello ' + $1) |
| </code></pre> |
| |
| <p>This SystemML script can be debugged in Eclipse using a Debug Configuration such as the following:</p> |
| |
| <div class="codetabs2"> |
| |
| <div data-lang="Eclipse Debug Configuration - Main"> |
| <p><img src="img/engine-dev-guide/dmlscript-debug-configuration-hello-world-main-class.png" alt="Eclipse Debug Configuration - Main" title="DMLScript Debug Configuration, Main class" /></p> |
| </div> |
| |
| <div data-lang="Eclipse Debug Configuration - Arguments"> |
| <p><img src="img/engine-dev-guide/dmlscript-debug-configuration-hello-world-program-arguments.png" alt="Eclipse Debug Configuration - Arguments" title="DMLScript Debug Configuration, Program arguments" /></p> |
| </div> |
| |
| </div> |
| |
| <hr /> |
| |
| <h2 id="python-mlcontext-api">Python MLContext API</h2> |
| |
| <p>When working with the Python MLContext API (see <code>src/main/python/systemml/mlcontext.py</code>) during development, |
| it can be useful to install the Python MLContext API in editable mode (<code>-e</code>). This allows Python updates |
| to take effect without requiring the SystemML python artifact to be built and installed.</p> |
| |
| <div class="highlight"><pre><code class="language-bash" data-lang="bash">mvn clean |
| pip3 install <span class="nt">-e</span> src/main/python |
| mvn clean package |
| <span class="nv">PYSPARK_PYTHON</span><span class="o">=</span>python3 pyspark <span class="nt">--driver-class-path</span> target/SystemML.jar</code></pre></div> |
| |
| <div class="codetabs"> |
| |
| <div data-lang="Python 3"> |
| |
| <div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">systemml</span> <span class="kn">import</span> <span class="n">MLContext</span><span class="p">,</span> <span class="n">dml</span> |
| <span class="n">ml</span> <span class="o">=</span> <span class="n">MLContext</span><span class="p">(</span><span class="n">sc</span><span class="p">)</span> |
| <span class="n">script</span> <span class="o">=</span> <span class="n">dml</span><span class="p">(</span><span class="s">"print('hello world')"</span><span class="p">)</span> |
| <span class="n">ml</span><span class="o">.</span><span class="n">execute</span><span class="p">(</span><span class="n">script</span><span class="p">)</span></code></pre></div> |
| |
| </div> |
| |
| <div data-lang="PySpark"> |
| |
| <div class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">Python</span> <span class="mf">3.5</span><span class="o">.</span><span class="mi">2</span> <span class="p">(</span><span class="n">default</span><span class="p">,</span> <span class="n">Jul</span> <span class="mi">28</span> <span class="mi">2016</span><span class="p">,</span> <span class="mi">21</span><span class="p">:</span><span class="mi">28</span><span class="p">:</span><span class="mo">07</span><span class="p">)</span> |
| <span class="p">[</span><span class="n">GCC</span> <span class="mf">4.2</span><span class="o">.</span><span class="mi">1</span> <span class="n">Compatible</span> <span class="n">Apple</span> <span class="n">LLVM</span> <span class="mf">7.0</span><span class="o">.</span><span class="mi">2</span> <span class="p">(</span><span class="n">clang</span><span class="o">-</span><span class="mf">700.1</span><span class="o">.</span><span class="mi">81</span><span class="p">)]</span> <span class="n">on</span> <span class="n">darwin</span> |
| <span class="n">Type</span> <span class="s">"help"</span><span class="p">,</span> <span class="s">"copyright"</span><span class="p">,</span> <span class="s">"credits"</span> <span class="ow">or</span> <span class="s">"license"</span> <span class="k">for</span> <span class="n">more</span> <span class="n">information</span><span class="o">.</span> |
| <span class="n">Using</span> <span class="n">Spark</span><span class="s">'s default log4j profile: org/apache/spark/log4j-defaults.properties</span><span class="err"> |
| </span><span class="s">Setting default log level to "WARN".</span><span class="err"> |
| </span><span class="s">To adjust logging level use sc.setLogLevel(newLevel). For SparkR, use setLogLevel(newLevel).</span><span class="err"> |
| </span><span class="s">17/02/03 12:33:42 WARN NativeCodeLoader: Unable to load native-hadoop library for your platform... using builtin-java classes where applicable</span><span class="err"> |
| </span><span class="s">17/02/03 12:33:56 WARN ObjectStore: Failed to get database global_temp, returning NoSuchObjectException</span><span class="err"> |
| </span><span class="s">Welcome to</span><span class="err"> |
| </span><span class="s"> ____ __</span><span class="err"> |
| </span><span class="s"> / __/__ ___ _____/ /__</span><span class="err"> |
| </span><span class="s"> _</span><span class="err">\</span><span class="s"> </span><span class="err">\</span><span class="s">/ _ </span><span class="err">\</span><span class="s">/ _ `/ __/ '</span><span class="n">_</span><span class="o">/</span> |
| <span class="o">/</span><span class="n">__</span> <span class="o">/</span> <span class="o">.</span><span class="n">__</span><span class="o">/</span>\<span class="n">_</span><span class="p">,</span><span class="n">_</span><span class="o">/</span><span class="n">_</span><span class="o">/</span> <span class="o">/</span><span class="n">_</span><span class="o">/</span>\<span class="n">_</span>\ <span class="n">version</span> <span class="mf">2.1</span><span class="o">.</span><span class="mi">0</span> |
| <span class="o">/</span><span class="n">_</span><span class="o">/</span> |
| |
| <span class="n">Using</span> <span class="n">Python</span> <span class="n">version</span> <span class="mf">3.5</span><span class="o">.</span><span class="mi">2</span> <span class="p">(</span><span class="n">default</span><span class="p">,</span> <span class="n">Jul</span> <span class="mi">28</span> <span class="mi">2016</span> <span class="mi">21</span><span class="p">:</span><span class="mi">28</span><span class="p">:</span><span class="mo">07</span><span class="p">)</span> |
| <span class="n">SparkSession</span> <span class="n">available</span> <span class="k">as</span> <span class="s">'spark'</span><span class="o">.</span> |
| <span class="o">>>></span> <span class="kn">from</span> <span class="nn">systemml</span> <span class="kn">import</span> <span class="n">MLContext</span><span class="p">,</span> <span class="n">dml</span> |
| <span class="o">>>></span> <span class="n">ml</span> <span class="o">=</span> <span class="n">MLContext</span><span class="p">(</span><span class="n">sc</span><span class="p">)</span> |
| |
| <span class="n">Welcome</span> <span class="n">to</span> <span class="n">Apache</span> <span class="n">SystemML</span><span class="err">!</span> |
| |
| <span class="o">>>></span> <span class="n">script</span> <span class="o">=</span> <span class="n">dml</span><span class="p">(</span><span class="s">"print('hello world')"</span><span class="p">)</span> |
| <span class="o">>>></span> <span class="n">ml</span><span class="o">.</span><span class="n">execute</span><span class="p">(</span><span class="n">script</span><span class="p">)</span> |
| <span class="n">hello</span> <span class="n">world</span> |
| <span class="n">MLResults</span></code></pre></div> |
| |
| </div> |
| |
| </div> |
| |
| <hr /> |
| |
| <h2 id="matrix-multiplication-operators">Matrix Multiplication Operators</h2> |
| |
| <p>In the following, we give an overview of backend-specific physical matrix multiplication operators in SystemML as well as their internally used matrix multiplication block operations.</p> |
| |
| <h3 id="basic-matrix-multiplication-operators">Basic Matrix Multiplication Operators</h3> |
| |
| <p>An <code>AggBinaryOp</code> hop can be compiled into the following physical operators.</p> |
| |
| <p><strong>1. Physical Operators in CP (single node, control program)</strong></p> |
| |
| <table> |
| <thead> |
| <tr> |
| <th>Name</th> |
| <th>Description</th> |
| <th>Operation</th> |
| </tr> |
| </thead> |
| <tbody> |
| <tr> |
| <td><code>MM</code></td> |
| <td>basic matrix multiplication</td> |
| <td><code>mm</code></td> |
| </tr> |
| <tr> |
| <td><code>MMChain</code></td> |
| <td>matrix multiplication chain</td> |
| <td><code>mmchain</code></td> |
| </tr> |
| <tr> |
| <td><code>TSMM</code></td> |
| <td>transpose-self matrix multiplication</td> |
| <td><code>tsmm</code></td> |
| </tr> |
| <tr> |
| <td><code>PMM</code></td> |
| <td>permutation matrix multiplication</td> |
| <td><code>pmm</code></td> |
| </tr> |
| </tbody> |
| </table> |
| |
| <p><strong>2. Physical Operator in MR (distributed, mapreduce)</strong></p> |
| |
| <table> |
| <thead> |
| <tr> |
| <th>Name</th> |
| <th>Description</th> |
| <th>Operation</th> |
| </tr> |
| </thead> |
| <tbody> |
| <tr> |
| <td><code>MapMM</code></td> |
| <td>map-side matrix multiplication, w/ or w/o agg</td> |
| <td><code>mm</code></td> |
| </tr> |
| <tr> |
| <td><code>MapMMChain</code></td> |
| <td>map-side matrix chain multiplication</td> |
| <td><code>mmchain</code></td> |
| </tr> |
| <tr> |
| <td><code>TSMM</code></td> |
| <td>map-side transpose-self matrix multiplication</td> |
| <td><code>tsmm</code></td> |
| </tr> |
| <tr> |
| <td><code>PMM</code></td> |
| <td>map-side permutation matrix multiplication</td> |
| <td><code>pmm</code></td> |
| </tr> |
| <tr> |
| <td><code>CPMM</code></td> |
| <td>cross-product matrix multiplication, 2 jobs</td> |
| <td><code>mm</code></td> |
| </tr> |
| <tr> |
| <td><code>RMM</code></td> |
| <td>replication-based matrix multiplication, 1 job</td> |
| <td><code>mm</code></td> |
| </tr> |
| </tbody> |
| </table> |
| |
| <p><strong>3. Physical Operators in SPARK (distributed, spark)</strong></p> |
| |
| <table> |
| <thead> |
| <tr> |
| <th>Name</th> |
| <th>Description</th> |
| <th>Operation</th> |
| </tr> |
| </thead> |
| <tbody> |
| <tr> |
| <td><code>MapMM</code></td> |
| <td>see MR, flatmap/mappartitions/maptopair + reduce/reducebykey/no_aggregation</td> |
| <td><code>mm</code></td> |
| </tr> |
| <tr> |
| <td><code>MapMMChain</code></td> |
| <td>see MR, mapvalues/maptopair + reduce</td> |
| <td><code>mmchain</code></td> |
| </tr> |
| <tr> |
| <td><code>TSMM</code></td> |
| <td>see MR, mapvalues + reduce</td> |
| <td><code>tsmm</code></td> |
| </tr> |
| <tr> |
| <td><code>PMM</code></td> |
| <td>see MR, flatmaptopair + reducebykey</td> |
| <td><code>pmm</code></td> |
| </tr> |
| <tr> |
| <td><code>CPMM</code></td> |
| <td>see MR, 2 x maptopair + join + maptopair + reduce/reducebykey</td> |
| <td><code>mm</code></td> |
| </tr> |
| <tr> |
| <td><code>RMM</code></td> |
| <td>see MR, 2 x flatmap + join + maptopair + reducebykey</td> |
| <td><code>mm</code></td> |
| </tr> |
| <tr> |
| <td><code>ZIPMM</code></td> |
| <td>partitioning-preserving 1-1 zipping mm, join + mapvalues + reduce</td> |
| <td><code>mm</code></td> |
| </tr> |
| </tbody> |
| </table> |
| |
| <h3 id="complex-matrix-multiplication-operators">Complex Matrix Multiplication Operators</h3> |
| |
| <p>A <code>QuaternaryOp</code> hop can be compiled into the following physical operators. Note that <code>wsloss</code>, <code>wsigmoid</code>, <code>wdivmm</code> have different semantics though. The main goal of these operators is to prevent the creation of dense “outer” products via selective computation over a sparse driver (sparse matrix and sparse-safe operation).</p> |
| |
| <p><strong>1. Physical Operators in CP (single node, control program)</strong></p> |
| |
| <table> |
| <thead> |
| <tr> |
| <th>Name</th> |
| <th>Description</th> |
| <th>Operation</th> |
| </tr> |
| </thead> |
| <tbody> |
| <tr> |
| <td><code>WSLoss</code></td> |
| <td>weighted squared loss</td> |
| <td><code>wsloss</code></td> |
| </tr> |
| <tr> |
| <td><code>WSigmoid</code></td> |
| <td>weighted sigmoid</td> |
| <td><code>wsigmoid</code></td> |
| </tr> |
| <tr> |
| <td><code>WDivMM</code></td> |
| <td>weighted divide matrix multiplication</td> |
| <td><code>wdivmm</code></td> |
| </tr> |
| <tr> |
| <td><code>WCeMM</code></td> |
| <td>weighted cross entropy matrix multiplication</td> |
| <td><code>wcemm</code></td> |
| </tr> |
| <tr> |
| <td><code>WuMM</code></td> |
| <td>weighted unary op matrix multiplication</td> |
| <td><code>wumm</code></td> |
| </tr> |
| </tbody> |
| </table> |
| |
| <p><strong>2. Physical Operator in MR (distributed, mapreduce)</strong></p> |
| |
| <table> |
| <thead> |
| <tr> |
| <th>Name</th> |
| <th>Description</th> |
| <th>Operation</th> |
| </tr> |
| </thead> |
| <tbody> |
| <tr> |
| <td><code>MapWSLoss</code></td> |
| <td>map-side weighted squared loss</td> |
| <td><code>wsloss</code></td> |
| </tr> |
| <tr> |
| <td><code>RedWSLoss</code></td> |
| <td>reduce-side weighted squared loss</td> |
| <td><code>wsloss</code></td> |
| </tr> |
| <tr> |
| <td><code>MapWSigmoid</code></td> |
| <td>map-side weighted sigmoid</td> |
| <td><code>wsigmoid</code></td> |
| </tr> |
| <tr> |
| <td><code>RedWSigmoid</code></td> |
| <td>reduce-side weighted sigmoid</td> |
| <td><code>wsigmoid</code></td> |
| </tr> |
| <tr> |
| <td><code>MapWDivMM</code></td> |
| <td>map-side weighted divide matrix mult</td> |
| <td><code>wdivmm</code></td> |
| </tr> |
| <tr> |
| <td><code>RedWDivMM</code></td> |
| <td>reduce-side weighted divide matrix mult</td> |
| <td><code>wdivmm</code></td> |
| </tr> |
| <tr> |
| <td><code>MapWCeMM</code></td> |
| <td>map-side weighted cross entr. matrix mult</td> |
| <td><code>wcemm</code></td> |
| </tr> |
| <tr> |
| <td><code>RedWCeMM</code></td> |
| <td>reduce-side w. cross entr. matrix mult</td> |
| <td><code>wcemm</code></td> |
| </tr> |
| <tr> |
| <td><code>MapWuMM</code></td> |
| <td>map-side weighted unary op matrix mult</td> |
| <td><code>wumm</code></td> |
| </tr> |
| <tr> |
| <td><code>RedWuMM</code></td> |
| <td>reduce-side weighted unary op matrix mult</td> |
| <td><code>wumm</code></td> |
| </tr> |
| </tbody> |
| </table> |
| |
| <p><strong>3. Physical Operators in SPARK (distributed, spark)</strong></p> |
| |
| <table> |
| <thead> |
| <tr> |
| <th>Name</th> |
| <th>Description</th> |
| <th>Operation</th> |
| </tr> |
| </thead> |
| <tbody> |
| <tr> |
| <td><code>MapWSLoss</code></td> |
| <td>see MR, mappartitions + reduce</td> |
| <td><code>wsloss</code></td> |
| </tr> |
| <tr> |
| <td><code>RedWSLoss</code></td> |
| <td>see MR, 1/2x flatmaptopair + 1-3x join + maptopair + reduce</td> |
| <td><code>wsloss</code></td> |
| </tr> |
| <tr> |
| <td><code>MapWSigmoid</code></td> |
| <td>see MR, mappartitions</td> |
| <td><code>wsigmoid</code></td> |
| </tr> |
| <tr> |
| <td><code>RedWSigmoid</code></td> |
| <td>see MR, 1/2x flatmaptopair + 1/2x join + maptopair</td> |
| <td><code>wsigmoid</code></td> |
| </tr> |
| <tr> |
| <td><code>MapWDivMM</code></td> |
| <td>see MR, mappartitions + reducebykey</td> |
| <td><code>wdivmm</code></td> |
| </tr> |
| <tr> |
| <td><code>RedWDivMM</code></td> |
| <td>see MR, 1/2x flatmaptopair + 1/2x join + maptopair + reducebykey</td> |
| <td><code>wdivmm</code></td> |
| </tr> |
| <tr> |
| <td><code>MapWCeMM</code></td> |
| <td>see MR, mappartitions + reduce</td> |
| <td><code>wcemm</code></td> |
| </tr> |
| <tr> |
| <td><code>RedWCeMM</code></td> |
| <td>see MR, 1/2x flatmaptopair + 1/2x join + maptopair + reduce</td> |
| <td><code>wcemm</code></td> |
| </tr> |
| <tr> |
| <td><code>MapWuMM</code></td> |
| <td>see MR, mappartitions</td> |
| <td><code>wumm</code></td> |
| </tr> |
| <tr> |
| <td><code>RedWuMM</code></td> |
| <td>see MR, 1/2x flatmaptopair + 1/2x join + maptopair</td> |
| <td><code>wumm</code></td> |
| </tr> |
| </tbody> |
| </table> |
| |
| <h3 id="core-matrix-multiplication-primitives">Core Matrix Multiplication Primitives</h3> |
| |
| <table> |
| <thead> |
| <tr> |
| <th>#</th> |
| <th>Operation</th> |
| <th>Equations</th> |
| <th>Description</th> |
| </tr> |
| </thead> |
| <tbody> |
| <tr style="vertical-align: top"> |
| <td>1</td> |
| <td><code class="boldcell">mm</code></td> |
| <td><code>(a) <span class="boldcell">A%*%B</span></code></td> |
| <td> |
| - sequential / multi-threaded (same block ops, par over rows in A)<br /> |
| - dense-dense, dense-sparse, sparse-dense, sparse-sparse, ultra-sparse<br /> |
| - ~20 special cases for matrix-vector, vector-vector, etc |
| </td> |
| </tr> |
| <tr style="vertical-align: top"> |
| <td>2</td> |
| <td><code class="boldcell">mmchain</code></td> |
| <td> |
| <code>(a) <span class="boldcell">t(X)%*%(X%*%v)</span></code><br /> |
| <code>(b) <span class="boldcell">t(X)%*%(w*(X%*%v))</span></code> |
| </td> |
| <td> |
| - sequential / multi-threaded (same block ops, par over rows in X)<br /> |
| - dense / sparse x 2 patterns |
| </td> |
| </tr> |
| <tr style="vertical-align: top"> |
| <td>3</td> |
| <td><code class="boldcell">tsmm</code></td> |
| <td> |
| <code>(a) <span class="boldcell">t(X)%*%X</span></code><br /> |
| <code>(b) <span class="boldcell">X%*%t(X)</span></code> |
| </td> |
| <td> |
| - sequential / multi-threaded (same block ops, par over rows in R, 2x tasks)<br /> |
| - dense / sparse x 2 patterns; special cases for dot products |
| </td> |
| </tr> |
| <tr style="vertical-align: top"> |
| <td>4</td> |
| <td><code class="boldcell">pmm</code></td> |
| <td><code>(a) <span class="boldcell">removeEmpty(diag(v),"rows")%*%X</span></code></td> |
| <td> |
| - sequential / multi-threaded (same block ops, par over rows in X)<br /> |
| - sparse-sparse, dense-dense, sparse-dense |
| </td> |
| </tr> |
| <tr style="vertical-align: top"> |
| <td>5</td> |
| <td><code class="boldcell">wsloss</code></td> |
| <td> |
| <code>(a) <span class="boldcell">sum(W*(X-U%*%t(V))^2)</span></code><br /> |
| <code>(b) <span class="boldcell">sum((X-W*(U%*%t(V)))^2)</span></code><br /> |
| <code>(c) <span class="boldcell">sum((X-(U%*%t(V)))^2))</span></code><br /> |
| <code>(d) <span class="boldcell">sum(W*(U%*%t(V)-X)^2)</span></code><br /> |
| <code>(e) <span class="boldcell">sum((W*(U%*%t(V))-X)^2)</span></code><br /> |
| <code>(f) <span class="boldcell">sum(((U%*%t(V))-X)^2)</span></code> |
| </td> |
| <td> |
| - sequential / multi-threaded (same block ops, par over rows in W/X)<br /> |
| - all dense, sparse-dense factors, sparse/dense-* x 3 patterns<br /> |
| - special patterns for (a) and (d) if W is X!=0 |
| </td> |
| </tr> |
| <tr style="vertical-align: top"> |
| <td>6</td> |
| <td><code class="boldcell">wsigmoid</code></td> |
| <td> |
| <code>(a) <span class="boldcell">W*sigmoid(Y%*%t(X))</span></code><br /> |
| <code>(b) <span class="boldcell">W*sigmoid(-(Y%*%t(X)))</span></code><br /> |
| <code>(c) <span class="boldcell">W*log(sigmoid(Y%*%t(X)))</span></code><br /> |
| <code>(d) <span class="boldcell">W*log(sigmoid(-(Y%*%t(X))))</span></code> |
| </td> |
| <td> |
| - sequential / multi-threaded (same block ops, par over rows in W)<br /> |
| - all dense, sparse-dense factors, sparse/dense-* x 4 patterns |
| </td> |
| </tr> |
| <tr style="vertical-align: top"> |
| <td>7</td> |
| <td><code class="boldcell">wdivmm</code></td> |
| <td> |
| <code>(a) <span class="boldcell">t(t(U)%*%(W/(U%*%t(V))))</span></code><br /> |
| <code>(b) <span class="boldcell">(W/(U%*%t(V)))%*%V</span></code><br /> |
| <code>(c) <span class="boldcell">t(t(U)%*%(W*(U%*%t(V))))</span></code><br /> |
| <code>(d) <span class="boldcell">(W*(U%*%t(V)))%*%V</span></code><br /> |
| <code>(e) <span class="boldcell">W*(U%*%t(V))</span></code><br /> |
| <code>(f) <span class="boldcell">t(t(U)%*%((X!=0)*(U%*%t(V)-X)))</span></code><br /> |
| <code>(g) <span class="boldcell">(X!=0)*(U%*%t(V)-X)%*%V</span></code><br /> |
| <code>(h) <span class="boldcell">t(t(U)%*%(W*(U%*%t(V)-X)))</span></code><br /> |
| <code>(i) <span class="boldcell">(W*(U%*%t(V)-X))%*%V</span></code><br /> |
| <code>(j) <span class="boldcell">t(t(U)%*%(W/(U%*%t(V)+x)))</span></code><br /> |
| <code>(k) <span class="boldcell">(W/(U%*%t(V)+x))%*%V</span></code> |
| </td> |
| <td> |
| - sequential / multi-threaded (same block ops, par over rows in X)<br /> |
| - all dense, sparse-dense factors, sparse/dense-* x 9 patterns |
| </td> |
| </tr> |
| <tr style="vertical-align: top"> |
| <td>8</td> |
| <td><code class="boldcell">wcemm</code></td> |
| <td> |
| <code>(a) <span class="boldcell">sum(X*log(U%*%t(V)))</span></code><br /> |
| <code>(b) <span class="boldcell">sum(X*log(U%*%t(V)+epsilon))</span></code> |
| </td> |
| <td> |
| - sequential / multi-threaded (same block ops, par over rows in X)<br /> |
| - all dense, sparse-dense factors, sparse/dense-*, 1 pattern |
| </td> |
| </tr> |
| <tr style="vertical-align: top"> |
| <td>9</td> |
| <td><code class="boldcell">wumm</code></td> |
| <td> |
| <code>(a) <span class="boldcell">X*uop(U%*%t(V))</span></code><br /> |
| <code>(b) <span class="boldcell">X/uop(U%*%t(V))</span></code> |
| </td> |
| <td> |
| - any unary operator, e.g., X*exp(U%*%t(V)) or X*(U%*%t(V))^2<br /> |
| - sequential / multi-threaded (same block ops, par over rows in X)<br /> |
| - all dense, sparse-dense factors, sparse/dense-*, 2 pattern |
| </td> |
| </tr> |
| </tbody> |
| </table> |
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