update

Author: mhjensen

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

@@ -93,6 +93,10 @@
                None,
                'kernels-and-non-linearity'),
               ('Kernel trick', 2, None, 'kernel-trick'),
+              ('Visualization of the Kernel trick',
+               2,
+               None,
+               'visualization-of-the-kernel-trick'),
               ('The problem to solve', 2, None, 'the-problem-to-solve'),
               ('Convex optimization', 2, None, 'convex-optimization'),
               ('Different kernels', 2, None, 'different-kernels'),
@@ -283,6 +287,7 @@
      <!-- 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="#kernel-trick" style="font-size: 80%;">Kernel trick</a></li>
+     <!-- navigation toc: --> <li><a href="#visualization-of-the-kernel-trick" style="font-size: 80%;">Visualization of the 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>
      <!-- navigation toc: --> <li><a href="#different-kernels" style="font-size: 80%;">Different kernels</a></li>
@@ -986,6 +991,15 @@ <h2 id="kernel-trick" class="anchor">Kernel trick </h2>
 \( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.
 </p>
 
+<!-- !split -->
+<h2 id="visualization-of-the-kernel-trick" class="anchor">Visualization of the Kernel trick </h2>
+
+<br/><br/>
+<center>
+<p><img src="figures/kerneltrick.png" width="900" align="bottom"></p>
+</center>
+<br/><br/>
+
 <!-- !split -->
 <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>
@@ -1472,7 +1486,7 @@ <h2 id="quantum-kernels" class="anchor">Quantum kernels  </h2>
 (and theoretical definitions) use the squared overlap.  In any case,
 the kernel measures similarity: if \( \vert \phi(\boldsymbol{x})\rangle \) and
 \( \vert \phi(\boldsymbol{x}&#8217;)\rangle \) are close in Hilbert space,
-\( k(\boldsymbol{x},\boldsymbol{x}&#8217;) \) is large.
+\( K(\boldsymbol{x},\boldsymbol{x}&#8217;) \) is large.
 </p>
 
 <!-- !split -->
@@ -1902,7 +1916,8 @@ <h2 id="and-nisq-quantum-kernels" class="anchor">And NISQ Quantum Kernels </h2>
 possible quality or feature-based advantage: the quantum feature map
 might separate data better than any known classical kernel.  This
 approach has been demonstrated on small datasets (e.g. Iris) and
-studied theoretically.  For example, Havlicek et al. showed on a toy
+studied theoretically.  For example, Havlicek <em>et al.</em>, see <a href="https://www.nature.com/articles/s41586-019-0980-2" target="_self"><tt>https://www.nature.com/articles/s41586-019-0980-2</tt></a>
+showed on a toy
 problem that a quantum kernel can correctly classify points that a
 simple classical kernel cannot.  However, other studies have found
 that for random data classical kernels often suffice, so the advantage
@@ -2230,7 +2245,6 @@ <h2 id="computing-quantum-kernel-matrices" class="anchor">Computing Quantum Kern
   </div>
 </div>
 
-<p>This is equivalent.</p>
 
 <!-- !split -->
 <h2 id="training-svm-with-precomputed-quantum-kernels" class="anchor">Training SVM with Precomputed Quantum Kernels </h2>

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

@@ -876,6 +876,16 @@ <h2 id="kernel-trick">Kernel trick </h2>
 </p>
 </section>
 
+<section>
+<h2 id="visualization-of-the-kernel-trick">Visualization of the Kernel trick </h2>
+
+<br/><br/>
+<center>
+<p><img src="figures/kerneltrick.png" width="900" align="bottom"></p>
+</center>
+<br/><br/>
+</section>
+
 <section>
 <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>
@@ -1379,7 +1389,7 @@ <h2 id="quantum-kernels">Quantum kernels  </h2>
 (and theoretical definitions) use the squared overlap.  In any case,
 the kernel measures similarity: if \( \vert \phi(\boldsymbol{x})\rangle \) and
 \( \vert \phi(\boldsymbol{x}&#8217;)\rangle \) are close in Hilbert space,
-\( k(\boldsymbol{x},\boldsymbol{x}&#8217;) \) is large.
+\( K(\boldsymbol{x},\boldsymbol{x}&#8217;) \) is large.
 </p>
 </section>
 
@@ -1841,7 +1851,8 @@ <h2 id="and-nisq-quantum-kernels">And NISQ Quantum Kernels </h2>
 possible quality or feature-based advantage: the quantum feature map
 might separate data better than any known classical kernel.  This
 approach has been demonstrated on small datasets (e.g. Iris) and
-studied theoretically.  For example, Havlicek et al. showed on a toy
+studied theoretically.  For example, Havlicek <em>et al.</em>, see <a href="https://www.nature.com/articles/s41586-019-0980-2" target="_blank"><tt>https://www.nature.com/articles/s41586-019-0980-2</tt></a>
+showed on a toy
 problem that a quantum kernel can correctly classify points that a
 simple classical kernel cannot.  However, other studies have found
 that for random data classical kernels often suffice, so the advantage
@@ -2177,8 +2188,6 @@ <h2 id="computing-quantum-kernel-matrices">Computing Quantum Kernel Matrices </h
     </div>
   </div>
 </div>
-
-<p>This is equivalent.</p>
 </section>
 
 <section>

doc/pub/week14/html/week14-solarized.html

@@ -120,6 +120,10 @@
                None,
                'kernels-and-non-linearity'),
               ('Kernel trick', 2, None, 'kernel-trick'),
+              ('Visualization of the Kernel trick',
+               2,
+               None,
+               'visualization-of-the-kernel-trick'),
               ('The problem to solve', 2, None, 'the-problem-to-solve'),
               ('Convex optimization', 2, None, 'convex-optimization'),
               ('Different kernels', 2, None, 'different-kernels'),
@@ -888,6 +892,15 @@ <h2 id="kernel-trick">Kernel trick </h2>
 \( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.
 </p>
 
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="visualization-of-the-kernel-trick">Visualization of the Kernel trick </h2>
+
+<br/><br/>
+<center>
+<p><img src="figures/kerneltrick.png" width="900" align="bottom"></p>
+</center>
+<br/><br/>
+
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <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>
@@ -1370,7 +1383,7 @@ <h2 id="quantum-kernels">Quantum kernels  </h2>
 (and theoretical definitions) use the squared overlap.  In any case,
 the kernel measures similarity: if \( \vert \phi(\boldsymbol{x})\rangle \) and
 \( \vert \phi(\boldsymbol{x}&#8217;)\rangle \) are close in Hilbert space,
-\( k(\boldsymbol{x},\boldsymbol{x}&#8217;) \) is large.
+\( K(\boldsymbol{x},\boldsymbol{x}&#8217;) \) is large.
 </p>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1794,7 +1807,8 @@ <h2 id="and-nisq-quantum-kernels">And NISQ Quantum Kernels </h2>
 possible quality or feature-based advantage: the quantum feature map
 might separate data better than any known classical kernel.  This
 approach has been demonstrated on small datasets (e.g. Iris) and
-studied theoretically.  For example, Havlicek et al. showed on a toy
+studied theoretically.  For example, Havlicek <em>et al.</em>, see <a href="https://www.nature.com/articles/s41586-019-0980-2" target="_blank"><tt>https://www.nature.com/articles/s41586-019-0980-2</tt></a>
+showed on a toy
 problem that a quantum kernel can correctly classify points that a
 simple classical kernel cannot.  However, other studies have found
 that for random data classical kernels often suffice, so the advantage
@@ -2122,7 +2136,6 @@ <h2 id="computing-quantum-kernel-matrices">Computing Quantum Kernel Matrices </h
   </div>
 </div>
 
-<p>This is equivalent.</p>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <h2 id="training-svm-with-precomputed-quantum-kernels">Training SVM with Precomputed Quantum Kernels </h2>

doc/pub/week14/html/week14.html

@@ -197,6 +197,10 @@
                None,
                'kernels-and-non-linearity'),
               ('Kernel trick', 2, None, 'kernel-trick'),
+              ('Visualization of the Kernel trick',
+               2,
+               None,
+               'visualization-of-the-kernel-trick'),
               ('The problem to solve', 2, None, 'the-problem-to-solve'),
               ('Convex optimization', 2, None, 'convex-optimization'),
               ('Different kernels', 2, None, 'different-kernels'),
@@ -965,6 +969,15 @@ <h2 id="kernel-trick">Kernel trick </h2>
 \( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.
 </p>
 
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="visualization-of-the-kernel-trick">Visualization of the Kernel trick </h2>
+
+<br/><br/>
+<center>
+<p><img src="figures/kerneltrick.png" width="900" align="bottom"></p>
+</center>
+<br/><br/>
+
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <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>
@@ -1447,7 +1460,7 @@ <h2 id="quantum-kernels">Quantum kernels  </h2>
 (and theoretical definitions) use the squared overlap.  In any case,
 the kernel measures similarity: if \( \vert \phi(\boldsymbol{x})\rangle \) and
 \( \vert \phi(\boldsymbol{x}&#8217;)\rangle \) are close in Hilbert space,
-\( k(\boldsymbol{x},\boldsymbol{x}&#8217;) \) is large.
+\( K(\boldsymbol{x},\boldsymbol{x}&#8217;) \) is large.
 </p>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1871,7 +1884,8 @@ <h2 id="and-nisq-quantum-kernels">And NISQ Quantum Kernels </h2>
 possible quality or feature-based advantage: the quantum feature map
 might separate data better than any known classical kernel.  This
 approach has been demonstrated on small datasets (e.g. Iris) and
-studied theoretically.  For example, Havlicek et al. showed on a toy
+studied theoretically.  For example, Havlicek <em>et al.</em>, see <a href="https://www.nature.com/articles/s41586-019-0980-2" target="_blank"><tt>https://www.nature.com/articles/s41586-019-0980-2</tt></a>
+showed on a toy
 problem that a quantum kernel can correctly classify points that a
 simple classical kernel cannot.  However, other studies have found
 that for random data classical kernels often suffice, so the advantage
@@ -2199,7 +2213,6 @@ <h2 id="computing-quantum-kernel-matrices">Computing Quantum Kernel Matrices </h
   </div>
 </div>
 
-<p>This is equivalent.</p>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <h2 id="training-svm-with-precomputed-quantum-kernels">Training SVM with Precomputed Quantum Kernels </h2>