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<!DOCTYPE html>
<!--
Created using JS Bin
http://jsbin.com
Copyright (c) 2016 by zhzhang (http://jsbin.com/patuyod/562/edit)
Released under the MIT license: http://jsbin.mit-license.org
-->
<meta name="robots" content="noindex">
<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width">
<title>JS Bin</title>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Lato:400,100,700,300" />
<link href='https://fonts.googleapis.com/css?family=Merriweather:400,400italic' rel='stylesheet' type='text/css'>
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.3.0/styles/zenburn.min.css">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/hint.css/2.2.1/hint.css" />
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<script type="text/javascript" async
src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-MML-AM_CHTML">
</script>
$$\newcommand{\bra}[1]{\langle #1 \lvert}$$
$$\newcommand{\ket}[1]{\rvert #1 \rangle}$$
$$\DeclareMathOperator{\tr}{Tr}$$
</head>
<body>
<div class="presentation">
<div class="full-slide title-slide">
<h1 class="disappear">
Density Matrices for
</h1>
<h1 class="entailment-title">
Lexical Entailment
</h1>
<div class="disappear">
21st June, 2016
<br>
Zhuoran (Jordan) Zhang, Laura Rimell, and Tamara Polajnar
<br>
MPhil in Advanced Computer Science
<br>
<br>
<img src="figures/uni_logo.png"/>
</div>
<div class="content entailment">
<div class="hypernym">Hypernymy</div>
<div class="disappear2">Meronymy</div>
<div class="disappear2">Attribute</div>
<div class="disappear2">Causality</div>
</div>
<div class="content entail-example">
<div>An <span class="hypo">apple</span> is a <span class="hyper">fruit</span>.</div>
<div class="example fruits">I ate the <span class="hypo apple">apple</span><span class="hyper fruit">fruit</span>.</div>
<div class="dih">
<p>Distributional Inclusion Hypothesis*</p>
<p class="dih-def">A word \(v\) entails another word \(w\) iff all of the <em>characteristic features</em> of \(v\) are expected to appear with \(w\).</p>
</div>
</div>
<div class="footnote">*Maayan Geffet and Ido Dagan. 2005. The distributional inclusion hypotheses and lexical entailment.</div>
</div>
<div class="full-slide vsms">
<h1 class="top">
Vector Space Models
</h1>
<div class="goldfish-ex">
<div class="content">
<div class="example vsm">My pet fish is a <span class="target goldfish hint--top" data-hint="TARGET WORD">goldfish</span>.</div>
<div class="example vsm">The <span class="target">goldfish</span> lives in the aquarium.</div>
</div>
<div class="vectors">
<div>\(\begin{bmatrix}pet&fish&aquarium\end{bmatrix}\)</div>
<div>\(\begin{bmatrix}1&1&0\end{bmatrix}\)</div>
<div>\(\begin{bmatrix}0&0&1\end{bmatrix}\)</div>
<div>\(\downarrow\)</div>
<div>\(\begin{bmatrix}1&1&1\end{bmatrix}\)</div>
</div>
</div>
<div class="cat-ex">
<div class="content">
<div class="example vsm">The <span class="target">cat</span> ate the fish.</div>
<div class="example vsm">I have a pet <span class="target">cat</span>.</div>
</div>
<div class="vectors">
<div>\(\begin{bmatrix}pet&fish&aquarium\end{bmatrix}\)</div>
<div>\(\begin{bmatrix}0&1&0\end{bmatrix}\)</div>
<div>\(\begin{bmatrix}1&0&0\end{bmatrix}\)</div>
<div>\(\downarrow\)</div>
<div>\(\begin{bmatrix}1&1&0\end{bmatrix}\)</div>
</div>
</div>
<div class="goldfish-vec">goldfish: \(\begin{bmatrix}1&1&1\end{bmatrix}\)</div>
</div>
<div class="full-slide balkir">
<h1 class="top">
Density Matrices
</h1>
<div class="content example outer-prod">My pet fish is a <span class="target">goldfish</span>. \(\rightarrow\begin{bmatrix}1\\1\\0\end{bmatrix}\cdot\begin{bmatrix}1&1&0\end{bmatrix}=\begin{bmatrix}1&1&0\\1&1&0\\0&0&0\end{bmatrix}\)</div>
<div class="example balkir-matrices centered">
<div>\(\rho_{goldfish} = \begin{bmatrix}1&1&0\\1&1&0\\0&0&1\end{bmatrix}\)</div>
<br>
<div>\(\rho_{cat} = \begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}\)</div>
</div>
<div class="subspace-img-container">
<img src="figures/overlap-tmp.png"/>
<div class="distinguishing hint--bottom" data-hint="Distinguishing Context"></div>
</div>
<div class="footnote">Example and figures taken from : Esma Balkir, Dimitri Kartsaklis, and Mehrnoosh Sadrzadeh. 2015. Sentence entailment in compositional distributional semantics.</div>
<div class="eigendecomp">
Target Word: <span class="target example">apple</span><br>
<strong>Eigenvalue #1: 0.049</strong><br>
ii used use big introduced ibm cultivar microsoft discord apple<br>
<strong>Eigenvalue #6: 0.018</strong><br>
trees pear cherry tree peach plum orchard orchards apricot oak <br>
<strong>Eigenvalue #8: 0.017</strong><br>
macintosh products logo trees orange peach pear cherry apple microsoft <br>
</div>
<div class="repres">
Representativeness<br>
$$R(\rho_{x}||\rho_{y}) = \frac{1}{1+S(\rho_{x}||\rho_{y})}$$
<div class="entropy hint--bottom" data-hint="von Neumann Entropy">
$$S(\rho_{x}||\rho_{y}) = \tr{\rho_{x}\log\rho_{x}} - \tr{\rho_{x}\log\rho_{y}}$$
</div>
</div>
</div>
<div class="full-slide generation">
<h1 class="top">
Matrix Generation
</h1>
<div class="content">
<table>
<tr>
<th>0,0</th>
<th>0,1</th>
<th class="datacell hint--right" data-hint="ALLIGATOR: 8, BANANA: 2, ...">0,2</th>
</tr>
<tr>
<th></th>
<th>1,1</th>
<th>1,2</th>
</tr>
<tr>
<th></th>
<th></th>
<th>2,2</th>
</tr>
</table>
<br>
<table>
<tr>
<th class="datacell hint--bottom" data-hint="(3,3): 1, (4,5): 1, ...">ALLIGATOR</th>
<th>BANANA</th>
<th>...</th>
</tr>
</table>
</div>
<div class="pgfs">
$$\left[\begin{array}{ccc|c}25&15&10&5\\15&9&6&3\\10&6&4&2\\\hline5&3&2&1\end{array}\right]$$
</div>
</div>
<div class="full-slide main-results">
<h1 class="top">
Results
</h1>
<div class="left-panel">
Hypernymy Direction<br>
\(alligator\xrightarrow{?}reptile\)
\(dishwasher\xrightarrow{?}appliance\)
<br>
<br>
<strong>57.2% Accuracy</strong>
<br>
<br>
Data Set: Baroni and Lenci Evaluation of Semantic Spaces (BLESS)
</div>
<div class="right-panel">
Hypernymy Precision
<br>
Random Hypernyms<br>
\(alligator\xrightarrow{?}rectifier\)
<br>
<strong>61.9% Precision</strong>
<br>
<br>
Shuffled Hypernyms<br>
\(alligator\xrightarrow{?}appliance\)
\(dishwasher\xrightarrow{?}reptile\)
<br>
<strong>61.7% Precision</strong>
</div>
</div>
<div class="full-slide weeds">
<h1 class="top">
Other Measures
</h1>
<div class="left-panel">
<br>
\(WeedsPrec(x,y) = \frac{\sum\limits_{c\in F(x)\cap F(y)}w(c,x)}{\sum\limits_{c\in F(x)}w(c,x)}\)
<br>
<br>
<br>
<table>
<tr>
<th>x</th>
<th class="overlap">3</th>
<th>0</th>
<th>0</th>
<th class="overlap">2</th>
<th>5</th>
</tr>
<tr>
<th>y</th>
<th class="overlap">2</th>
<th>3</th>
<th>8</th>
<th class="overlap">4</th>
<th>0</th>
</tr>
</table>
</div>
<div class="right-panel">
<br>
\(WeedsPrec_{quantum}(x,y)\)<br>
\(= \sum_{i\in I_{y,+}}\bra{y_{i}}\rho_{x}\ket{y_{i}}\)
<br>
<br>
<img src="figures/intersection.png"/>
</div>
</div>
<div class="full-slide other-results">
<h1 class="top">
Results
</h1>
<div class="centered">
Average Precision, BLESS hypernyms vs. BLESS noun-pairs
<table>
<tr>
<th>Measure</th>
<th>Our Best</th>
<th>Lenci '12</th>
</tr>
<tr>
<th>WeedsPrec</th>
<th>19.6</th>
<th>40</th>
</tr>
<tr>
<th>ClarkeDE</th>
<th>22.0</th>
<th>39</th>
</tr>
<tr>
<th>invCL</th>
<th>23.0</th>
<th>40</th>
</tr>
</table>
</div>
</div>
<div class="concl">
<h1 class="top">
Conclusions
</h1>
<div class="content">
<ul>
<li>Can be used as a distributional semantic model.</li>
<li>Can encode disparate contexts.</li>
<li>Springboard for future research.</li>
</ul>
</div>
</div>
<div class="final">
<h1 class="top">
Acknowledgements
</h1>
<div class="content">
Laura Rimell and Tamara Polajnar for encouraging me throughout the project, and teaching me how to be a better researcher.
<br>
<br>
Esma Balkir for her email correspondence about her theoretical work.
</div>
</div>
</div>
<script src="https://code.jquery.com/jquery-2.1.4.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery.transit/0.9.12/jquery.transit.min.js"></script>
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<script src="presentation.js"></script>
</body>
</html>