updating week 46
This commit is contained in:
mhjensen
@@ -42,39 +42,41 @@ Automatically generated HTML file from DocOnce source
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Overview of week 46', 2, None, '___sec0'),
|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
|
||||
('Hyperplanes and all that', 2, None, '___sec2'),
|
||||
('What is a hyperplane?', 2, None, '___sec3'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec4'),
|
||||
('The two-dimensional case', 2, None, '___sec5'),
|
||||
('Getting into the details', 2, None, '___sec6'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec7'),
|
||||
('Solving the equations', 2, None, '___sec8'),
|
||||
('Code Example', 2, None, '___sec9'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec10'),
|
||||
('A better approach', 2, None, '___sec11'),
|
||||
('Thursday', 2, None, '___sec1'),
|
||||
('Friday', 2, None, '___sec2'),
|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec3'),
|
||||
('Hyperplanes and all that', 2, None, '___sec4'),
|
||||
('What is a hyperplane?', 2, None, '___sec5'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec6'),
|
||||
('The two-dimensional case', 2, None, '___sec7'),
|
||||
('Getting into the details', 2, None, '___sec8'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec9'),
|
||||
('Solving the equations', 2, None, '___sec10'),
|
||||
('Code Example', 2, None, '___sec11'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec12'),
|
||||
('A better approach', 2, None, '___sec13'),
|
||||
('A quick Reminder on Lagrangian Multipliers',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Adding the Multiplier', 2, None, '___sec13'),
|
||||
('Setting up the Problem', 2, None, '___sec14'),
|
||||
('The problem to solve', 2, None, '___sec15'),
|
||||
('The last steps', 2, None, '___sec16'),
|
||||
('A soft classifier', 2, None, '___sec17'),
|
||||
('Soft optmization problem', 2, None, '___sec18'),
|
||||
('Kernels and non-linearity', 2, None, '___sec19'),
|
||||
('The equations', 2, None, '___sec20'),
|
||||
('The problem to solve', 2, None, '___sec21'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
|
||||
('The moons example', 2, None, '___sec23'),
|
||||
'___sec14'),
|
||||
('Adding the Multiplier', 2, None, '___sec15'),
|
||||
('Setting up the Problem', 2, None, '___sec16'),
|
||||
('The problem to solve', 2, None, '___sec17'),
|
||||
('The last steps', 2, None, '___sec18'),
|
||||
('A soft classifier', 2, None, '___sec19'),
|
||||
('Soft optmization problem', 2, None, '___sec20'),
|
||||
('Kernels and non-linearity', 2, None, '___sec21'),
|
||||
('The equations', 2, None, '___sec22'),
|
||||
('The problem to solve', 2, None, '___sec23'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec24'),
|
||||
('The moons example', 2, None, '___sec25'),
|
||||
('Mathematical optimization of convex functions',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('How do we solve these problems?', 2, None, '___sec25'),
|
||||
('A simple example', 2, None, '___sec26'),
|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
'___sec26'),
|
||||
('How do we solve these problems?', 2, None, '___sec27'),
|
||||
('A simple example', 2, None, '___sec28'),
|
||||
('Back to the more realistic cases', 2, None, '___sec29')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -113,33 +115,35 @@ MathJax.Hub.Config({
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Thursday</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Friday</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs029.html#___sec28" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs030.html#___sec29" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -155,20 +159,24 @@ MathJax.Hub.Config({
|
||||
<a name="part0009"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec8" class="anchor">Solving the equations </h2>
|
||||
<h2 id="___sec8" class="anchor">Getting into the details </h2>
|
||||
|
||||
<p>
|
||||
We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations
|
||||
Let us define the function
|
||||
$$
|
||||
b \leftarrow b +\eta \frac{\partial C}{\partial b},
|
||||
f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\boldsymbol{w} \leftarrow \boldsymbol{w} +\eta \frac{\partial C}{\partial \boldsymbol{w}},
|
||||
$$
|
||||
as the function that determines the line \( L \) that separates two classes (our two features), see the figure here.
|
||||
|
||||
where \( \eta \) is our by now well-known learning rate.
|
||||
<p>
|
||||
Any point defined by \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_2 \) on the line \( L \) will satisfy \( \boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0 \).
|
||||
|
||||
<p>
|
||||
The signed distance \( \delta \) from any point defined by a vector \( \boldsymbol{x} \) and a point \( \boldsymbol{x}_0 \) on the line \( L \) is then
|
||||
$$
|
||||
\delta = \frac{1}{\vert\vert \boldsymbol{w}\vert\vert}(\boldsymbol{w}^T\boldsymbol{x}+b).
|
||||
$$
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -195,7 +203,7 @@ where \( \eta \) is our by now well-known learning rate.
|
||||
<li><a href="._week46-bs017.html">18</a></li>
|
||||
<li><a href="._week46-bs018.html">19</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs030.html">31</a></li>
|
||||
<li><a href="._week46-bs010.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
Reference in New Issue
Block a user