negative sampling.tex (3600B)
1 \tikzset{ 2 nsnode/.style={inner sep=0.5mm}, 3 nshighlight/.style={draw=#1, fill=#1!30!white, inner sep=0}, 4 } 5 \begin{frame}{Negative Sampling Approximation} 6 \begin{tikzpicture}[remember picture, overlay, on background layer] 7 \only<2->{ 8 \coordinate (negsampl-dataset-left) at ($(pic cs:negsampl-1) + (-15mm, 0)$); 9 \coordinate (negsampl-dataset-i) at ($(pic cs:negsampl-1) + (-6mm, 0.3em)$); 10 \node[nshighlight=Dark2-A, fit=(negsampl-dataset)] (negsampl-dataset-n) {}; 11 \draw[arrow, Dark2-A, rounded corners=1mm] (negsampl-dataset-n.west) -- (negsampl-dataset-n.west-|negsampl-dataset-left) -- (negsampl-dataset-i-|negsampl-dataset-left) -- (negsampl-dataset-i); 12 } 13 \only<3->{ 14 \coordinate (negsampl-classifier-left) at ($(pic cs:negsampl-2) + (-14mm, 0)$); 15 \coordinate (negsampl-classifier-i) at ($(pic cs:negsampl-2) + (-6mm, 0.3em)$); 16 \node[nshighlight=Dark2-B, fit=(negsampl-classifier)] (negsampl-classifier-n) {}; 17 \draw[arrow, Dark2-B, rounded corners=1mm] (negsampl-classifier-n.west) -- (negsampl-classifier-n.west-|negsampl-classifier-left) -- (negsampl-classifier-i-|negsampl-classifier-left) -- (negsampl-classifier-i); 18 } 19 \only<4->{ 20 \coordinate (negsampl-positive-right) at ($(negsampl-negative2.east) + (11mm, 0)$); 21 \coordinate (negsampl-positive-i) at ($(pic cs:negsampl-3) + (1mm, 0.3em)$); 22 \node[nshighlight=Dark2-C, fit=(negsampl-positive)] (negsampl-positive-n) {}; 23 \draw[arrow, Dark2-C, rounded corners=1mm] (negsampl-positive-n.east) -- (negsampl-positive-n.east-|negsampl-positive-right) -- (negsampl-positive-i-|negsampl-positive-right) -- (negsampl-positive-i); 24 } 25 \only<5->{ 26 \coordinate (negsampl-negative-right) at ($(negsampl-negative2.east) + (10mm, 0)$); 27 \coordinate (negsampl-negative-i) at ($(pic cs:negsampl-4) + (1mm, 0.3em)$); 28 \node[nshighlight=Dark2-D, fit=(negsampl-negative1) (negsampl-negative2)] (negsampl-negative-n) {}; 29 \draw[arrow, Dark2-D, rounded corners=1mm] (negsampl-negative-n.east) -- (negsampl-negative-n.east-|negsampl-negative-right) -- (negsampl-negative-i-|negsampl-negative-right) -- (negsampl-negative-i); 30 } 31 \end{tikzpicture} 32 \begin{equation*} 33 \begin{split} 34 \loss{ep}(\vctr{\theta}, \vctr{\phi}) = 35 \expectation_{\raisebox{-3mm}[0mm][0mm]{\(\substack{\tikzmarknode[nsnode]{negsampl-dataset}{(\rndm{s}, \rndm{e}_1, \rndm{e}_2)\sim \uniformDistribution(\dataSet)}\\\tikzmarknode[nsnode]{negsampl-classifier}{\rndm{r}\sim \operatorname{\text{PCNN}}(\rndm{s}; \vctr{\phi})}}\)}} 36 \bigg[ & \tikzmarknode{negsampl-positive}{- \log \sigmoid \left( \psi(\rndm{e}_1, \rndm{r}, \rndm{e}_2; \vctr{\theta})\right)} \\ 37 & \tikzmarknode{negsampl-negative1}{- \sum_{j=1}^k \expectation_{\rndm{e}'\sim\uniformDistribution_\dataSet(\entitySet)} \left[ \log \sigmoid \left( - \psi(\rndm{e}_1, \rndm{r}, \rndm{e}'; \vctr{\theta})\right) \right]} \\ 38 & \tikzmarknode{negsampl-negative2}{- \sum_{j=1}^k \expectation_{\rndm{e}'\sim\uniformDistribution_\dataSet(\entitySet)} \left[ \log \sigmoid \left( - \psi(\rndm{e}', \rndm{r}, \rndm{e}_2; \vctr{\theta})\right) \right]} \bigg] 39 \end{split} 40 \end{equation*} 41 42 \bigskip 43 44 \pause 45 46 \begin{center} 47 \begin{minipage}{11cm} 48 \begin{enumerate}[<+->] 49 \item \tikzmark{negsampl-1}Take a sample uniformly from the dataset. 50 \item \tikzmark{negsampl-2}Sample a relation \(\rndm{r}\) from the output of the PCNN classifier. 51 \item Increase the energy of this fact.\tikzmark{negsampl-3} 52 \item Decrease the energy of negative facts. (\hypothesis{\(1\to1\)})\tikzmark{negsampl-4} 53 \end{enumerate} 54 \end{minipage} 55 \end{center} 56 \end{frame}