Update 05-inductive.php

chapters/05-inductive.php

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 				<dd>Given two tensors \(\mathcal{X} \in \mathbb{R}^{a_1,\ldots,a_n}\) and \(\mathcal{Y} \in \mathbb{R}^{a_1,\ldots,a_n}\), the <em>Hadamard product</em> \(\mathcal{X} \odot \mathcal{Y}\) is defined as a tensor in \(\mathbb{R}^{a_1,\ldots,a_n}\), with each element computed as \((\mathcal{X} \odot \mathcal{Y})_{i_1\ldots i_{n}} \coloneqq (\mathcal{X})_{i_1\ldots i_{n}} (\mathcal{Y})_{i_1\ldots i_{n}}\).</dd>
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-			<p>Other embedding techniques – namely RotatE&nbsp;<?php echo $references->cite("SunDNT19"); ?> and ComplEx&nbsp;<?php echo $references->cite("TrouillonWRGB16"); ?> – uses <em>complex space</em> based on complex numbers. With a slight abuse of notation, the definitions of vectors, matrices and tensors can be modified by replacing the set of real numbers \(\mathbb{R}\) by the set of complex numbers \(\mathbb{C}\), giving rise to complex vectors, complex matrices, and complex tensors. In this case, we denote by \(\mathrm{Re}(\cdot)\) the real part of a complex number. Given a complex vector \(\mathbf{x} \in \mathbb{C}^I\), we denote by \(\overline{\mathbf{x}}\) its complex conjugate (swapping the sign of the imaginary part of each element). Complex analogues of the aforementioned operators can then be defined by replacing the multiplication and addition of real numbers with the analogous operators for complex numbers, where RotateE&nbsp;<?php echo $references->cite("SunDNT19"); ?> uses the complex Hadamard product, and ComplEx&nbsp;<?php echo $references->cite("TrouillonWRGB16"); ?> uses complex matrix multiplication.</p>
+			<p>Other embedding techniques – namely RotatE&nbsp;<?php echo $references->cite("SunDNT19"); ?> and ComplEx&nbsp;<?php echo $references->cite("TrouillonWRGB16"); ?> – uses <em>complex space</em> based on complex numbers. With a slight abuse of notation, the definitions of vectors, matrices and tensors can be modified by replacing the set of real numbers \(\mathbb{R}\) by the set of complex numbers \(\mathbb{C}\), giving rise to complex vectors, complex matrices, and complex tensors. In this case, we denote by \(\mathrm{Re}(\cdot)\) the real part of a complex number. Given a complex vector \(\mathbf{x} \in \mathbb{C}^I\), we denote by \(\overline{\mathbf{x}}\) its complex conjugate (swapping the sign of the imaginary part of each element). Complex analogues of the aforementioned operators can then be defined by replacing the multiplication and addition of real numbers with the analogous operators for complex numbers, where RotatE&nbsp;<?php echo $references->cite("SunDNT19"); ?> uses the complex Hadamard product, and ComplEx&nbsp;<?php echo $references->cite("TrouillonWRGB16"); ?> uses complex matrix multiplication.</p>
 			<p>One embedding technique – MuRP&nbsp;<?php echo $references->cite("BalazevicAH19"); ?> – uses hyperbolic space, specifically based on the Poincaré ball. As this is the only embedding we cover that uses this space, and the formalisms are lengthy (covering the Poincaré ball, Möbius addition, Möbius matrix–vector multiplication, logarithmic maps, exponential maps, etc.), we rather refer the reader to the paper for further details&nbsp;<?php echo $references->cite("BalazevicAH19"); ?>.</p>
 			<p>As discussed in Section&nbsp;<?php echo ref("ssec:embeddings"); ?>, tensor decompositions are used for many embeddings, and at the heart of such decompositions is the tensor product, which is often used to reconstruct (an approximation of) the original tensor.</p>