typos

Author: mhjensen

doc/pub/week14/html/week14-bs.html

@@ -92,8 +92,6 @@
                2,
                None,
                'kernels-and-non-linearity'),
-              ('The equations', 2, None, 'the-equations'),
-              ('Defining the kernel', 2, None, 'defining-the-kernel'),
               ('Kernel trick', 2, None, 'kernel-trick'),
               ('The problem to solve', 2, None, 'the-problem-to-solve'),
               ('Convex optimization', 2, None, 'convex-optimization'),
@@ -265,8 +263,6 @@
      <!-- navigation toc: --> <li><a href="#derivatives-with-respect-to-b-and-boldsymbol-w" style="font-size: 80%;">Derivatives with respect to \( b \) and \( \boldsymbol{w} \)</a></li>
      <!-- navigation toc: --> <li><a href="#new-constraints" style="font-size: 80%;">New constraints</a></li>
      <!-- navigation toc: --> <li><a href="#kernels-and-non-linearity" style="font-size: 80%;">Kernels and non-linearity</a></li>
-     <!-- navigation toc: --> <li><a href="#the-equations" style="font-size: 80%;">The equations</a></li>
-     <!-- navigation toc: --> <li><a href="#defining-the-kernel" style="font-size: 80%;">Defining the kernel</a></li>
      <!-- navigation toc: --> <li><a href="#kernel-trick" style="font-size: 80%;">Kernel trick</a></li>
      <!-- navigation toc: --> <li><a href="#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
      <!-- navigation toc: --> <li><a href="#convex-optimization" style="font-size: 80%;">Convex optimization</a></li>
@@ -947,42 +943,6 @@ <h2 id="kernels-and-non-linearity" class="anchor">Kernels and non-linearity </h2
 </div>
 
 
-<!-- !split -->
-<h2 id="the-equations" class="anchor">The equations </h2>
-
-<p>Suppose we define a polynomial transformation of degree two only (we
-continue to live in a plane with \( x_i \) and \( y_i \) as variables)
-</p>
-$$
-z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right).
-$$
-
-<p>With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)</p>
-$$
-{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j,
-$$
-
-<p>subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors</p>
-$$
-y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i,
-$$
-
-<p>from which we also find \( b \).</p>
-
-<!-- !split -->
-<h2 id="defining-the-kernel" class="anchor">Defining the kernel </h2>
-
-<p>To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as</p>
-$$
-K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j).
-$$
-
-<p>For the above example, the kernel reads</p>
-$$
-K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2.
-$$
-
-
 <!-- !split -->
 <h2 id="kernel-trick" class="anchor">Kernel trick </h2>
 
@@ -1002,7 +962,7 @@ <h2 id="kernel-trick" class="anchor">Kernel trick </h2>
 <h2 id="the-problem-to-solve" class="anchor">The problem to solve </h2>
 <p>Using our definition of the kernel We can rewrite again the Lagrangian</p>
 $$
-{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
+{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j,
 $$
 
 <p>subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) in terms of a convex optimization problem</p>
@@ -1040,10 +1000,10 @@ <h2 id="different-kernels" class="anchor">Different kernels </h2>
 
 <p>There are several popular kernels being used. These are</p>
 <ol>
-<li> Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),</li>
-<li> Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),</li>
-<li> Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),</li>
-<li> Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),</li>
+<li> Linear: \( K(\boldsymbol{v},\boldsymbol{w})=\boldsymbol{v}^T\boldsymbol{w} \),</li>
+<li> Polynomial: \( K(\boldsymbol{v},\boldsymbol{w})=(\boldsymbol{v}^T\boldsymbol{w}+\gamma)^d \),</li>
+<li> Gaussian Radial Basis Function: \( K(\boldsymbol{v},\boldsymbol{w})=\exp{\left(-\gamma\vert\vert\boldsymbol{v}-\boldsymbol{w}\vert\vert^2\right)} \),</li>
+<li> Tanh: \( K(\boldsymbol{v},\boldsymbol{w})=\tanh{(\boldsymbol{v}^T\boldsymbol{w}+\gamma)} \),</li>
 </ol>
 <p>and many other ones.</p>
 
@@ -1327,7 +1287,7 @@ <h2 id="input-dependence" class="anchor">Input dependence </h2>
 </p>
 
 $$
-k(\boldsymbol{x},\boldsymbol{x}{\prime}) = \bigl\vert \langle \phi(\boldsymbol{x}) \mid \phi(\boldsymbol{x}{\prime}) \rangle\bigr\vert ^2.
+K(\boldsymbol{x},\boldsymbol{x}{\prime}) = \bigl\vert \langle \phi(\boldsymbol{x}) \mid \phi(\boldsymbol{x}{\prime}) \rangle\bigr\vert ^2.
 $$
 
 
@@ -1338,7 +1298,7 @@ <h2 id="quantum-kernels" class="anchor">Quantum kernels  </h2>
 the two quantum states.  Another common (unnormalized) version is
 </p>
 $$
-k’(\boldsymbol{x},\boldsymbol{x}’) = \langle \phi(\boldsymbol{x}) \vert\phi(\boldsymbol{x}’) \rangle,
+K’(\boldsymbol{x},\boldsymbol{x}’) = \langle \phi(\boldsymbol{x}) \vert\phi(\boldsymbol{x}’) \rangle,
 $$
 
 <p>but measuring this amplitude directly can
@@ -1360,7 +1320,7 @@ <h2 id="what-is-a-quantum-kernel" class="anchor">What is a quantum kernel? </h2>
 these states .  Concretely, one may write
 </p>
 $$
-K_{ij} \;=\; k(\boldsymbol{x}_i,\boldsymbol{x}_j) \;=\; \bigl\vert \langle \phi(\boldsymbol{x}_i)\mid\phi(\boldsymbol{x}_j)\rangle\bigr\vert ^2.
+K_{ij} = K(\boldsymbol{x}_i,\boldsymbol{x}_j) \;=\; \bigl\vert \langle \phi(\boldsymbol{x}_i)\mid\phi(\boldsymbol{x}_j)\rangle\bigr\vert ^2.
 $$
 
 <p>This forms a positive semidefinite kernel matrix \( K \) on the dataset,
@@ -1373,8 +1333,8 @@ <h2 id="what-will-we-need-in-the-case-of-a-quantum-computer" class="anchor">What
 <div class="panel panel-default">
 <div class="panel-body">
 <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>We will have to translate the classical data point \(\vec{x}\)
-into a quantum datapoint \(\vert \Phi{(\vec{x})} \rangle\). This can
+<p>We will have to translate the classical data point \( \vec{x} \)
+into a quantum datapoint \( \vert \Phi{(\vec{x})} \rangle \). This can
 be achieved by a circuit \( \mathcal{U}_{\Phi(\vec{x})} \vert 0\rangle \).
 </p>
 
@@ -1389,8 +1349,8 @@ <h2 id="what-will-we-need-in-the-case-of-a-quantum-computer" class="anchor">What
 <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
 <p>We need a parameterized quantum circuit \( W(\theta) \) that
 processes the data in a way that in the end we
-can apply a measurement that returns a classical value \(-1\) or
-\(1\) for each classical input \(\vec{x}\) that indentifies the label
+can apply a measurement that returns a classical value \( -1 \) or
+\( 1 \) for each classical input \( \vec{x} \) that indentifies the label
 of the classical data.
 </p>
 </div>
@@ -1511,6 +1471,7 @@ <h2 id="estimating-quantum-kernels" class="anchor">Estimating quantum kernels </
 
 <!-- !split -->
 <h2 id="code-example" class="anchor">Code example </h2>
+
 <p>For example, using PennyLane&#8217;s AngleEmbedding template, we can write:</p>
 
 

doc/pub/week14/html/week14-reveal.html

@@ -858,53 +858,6 @@ <h2 id="kernels-and-non-linearity">Kernels and non-linearity </h2>
 </div>
 </section>
 
-<section>
-<h2 id="the-equations">The equations </h2>
-
-<p>Suppose we define a polynomial transformation of degree two only (we
-continue to live in a plane with \( x_i \) and \( y_i \) as variables)
-</p>
-<p>&nbsp;<br>
-$$
-z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right).
-$$
-<p>&nbsp;<br>
-
-<p>With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)</p>
-<p>&nbsp;<br>
-$$
-{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j,
-$$
-<p>&nbsp;<br>
-
-<p>subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors</p>
-<p>&nbsp;<br>
-$$
-y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i,
-$$
-<p>&nbsp;<br>
-
-<p>from which we also find \( b \).</p>
-</section>
-
-<section>
-<h2 id="defining-the-kernel">Defining the kernel </h2>
-
-<p>To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as</p>
-<p>&nbsp;<br>
-$$
-K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j).
-$$
-<p>&nbsp;<br>
-
-<p>For the above example, the kernel reads</p>
-<p>&nbsp;<br>
-$$
-K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2.
-$$
-<p>&nbsp;<br>
-</section>
-
 <section>
 <h2 id="kernel-trick">Kernel trick </h2>
 
@@ -926,7 +879,7 @@ <h2 id="the-problem-to-solve">The problem to solve </h2>
 <p>Using our definition of the kernel We can rewrite again the Lagrangian</p>
 <p>&nbsp;<br>
 $$
-{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
+{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j,
 $$
 <p>&nbsp;<br>
 
@@ -971,10 +924,10 @@ <h2 id="different-kernels">Different kernels </h2>
 
 <p>There are several popular kernels being used. These are</p>
 <ol>
-<p><li> Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),</li>
-<p><li> Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),</li>
-<p><li> Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),</li>
-<p><li> Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),</li>
+<p><li> Linear: \( K(\boldsymbol{v},\boldsymbol{w})=\boldsymbol{v}^T\boldsymbol{w} \),</li>
+<p><li> Polynomial: \( K(\boldsymbol{v},\boldsymbol{w})=(\boldsymbol{v}^T\boldsymbol{w}+\gamma)^d \),</li>
+<p><li> Gaussian Radial Basis Function: \( K(\boldsymbol{v},\boldsymbol{w})=\exp{\left(-\gamma\vert\vert\boldsymbol{v}-\boldsymbol{w}\vert\vert^2\right)} \),</li>
+<p><li> Tanh: \( K(\boldsymbol{v},\boldsymbol{w})=\tanh{(\boldsymbol{v}^T\boldsymbol{w}+\gamma)} \),</li>
 </ol>
 <p>
 <p>and many other ones.</p>
@@ -1267,7 +1220,7 @@ <h2 id="input-dependence">Input dependence </h2>
 
 <p>&nbsp;<br>
 $$
-k(\boldsymbol{x},\boldsymbol{x}{\prime}) = \bigl\vert \langle \phi(\boldsymbol{x}) \mid \phi(\boldsymbol{x}{\prime}) \rangle\bigr\vert ^2.
+K(\boldsymbol{x},\boldsymbol{x}{\prime}) = \bigl\vert \langle \phi(\boldsymbol{x}) \mid \phi(\boldsymbol{x}{\prime}) \rangle\bigr\vert ^2.
 $$
 <p>&nbsp;<br>
 </section>
@@ -1280,7 +1233,7 @@ <h2 id="quantum-kernels">Quantum kernels  </h2>
 </p>
 <p>&nbsp;<br>
 $$
-k’(\boldsymbol{x},\boldsymbol{x}’) = \langle \phi(\boldsymbol{x}) \vert\phi(\boldsymbol{x}’) \rangle,
+K’(\boldsymbol{x},\boldsymbol{x}’) = \langle \phi(\boldsymbol{x}) \vert\phi(\boldsymbol{x}’) \rangle,
 $$
 <p>&nbsp;<br>
 
@@ -1305,7 +1258,7 @@ <h2 id="what-is-a-quantum-kernel">What is a quantum kernel? </h2>
 </p>
 <p>&nbsp;<br>
 $$
-K_{ij} \;=\; k(\boldsymbol{x}_i,\boldsymbol{x}_j) \;=\; \bigl\vert \langle \phi(\boldsymbol{x}_i)\mid\phi(\boldsymbol{x}_j)\rangle\bigr\vert ^2.
+K_{ij} = K(\boldsymbol{x}_i,\boldsymbol{x}_j) \;=\; \bigl\vert \langle \phi(\boldsymbol{x}_i)\mid\phi(\boldsymbol{x}_j)\rangle\bigr\vert ^2.
 $$
 <p>&nbsp;<br>
 
@@ -1320,8 +1273,8 @@ <h2 id="what-will-we-need-in-the-case-of-a-quantum-computer">What will we need i
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
-<p>We will have to translate the classical data point \(\vec{x}\)
-into a quantum datapoint \(\vert \Phi{(\vec{x})} \rangle\). This can
+<p>We will have to translate the classical data point \( \vec{x} \)
+into a quantum datapoint \( \vert \Phi{(\vec{x})} \rangle \). This can
 be achieved by a circuit \( \mathcal{U}_{\Phi(\vec{x})} \vert 0\rangle \).
 </p>
 
@@ -1335,8 +1288,8 @@ <h2 id="what-will-we-need-in-the-case-of-a-quantum-computer">What will we need i
 <p>
 <p>We need a parameterized quantum circuit \( W(\theta) \) that
 processes the data in a way that in the end we
-can apply a measurement that returns a classical value \(-1\) or
-\(1\) for each classical input \(\vec{x}\) that indentifies the label
+can apply a measurement that returns a classical value \( -1 \) or
+\( 1 \) for each classical input \( \vec{x} \) that indentifies the label
 of the classical data.
 </p>
 </div>
@@ -1475,6 +1428,7 @@ <h2 id="estimating-quantum-kernels">Estimating quantum kernels </h2>
 
 <section>
 <h2 id="code-example">Code example </h2>
+
 <p>For example, using PennyLane&#8217;s AngleEmbedding template, we can write:</p>