<p>Based on this repository:</p>
<p><a href="https://github.com/arkm97/svm-from-scratch/blob/master/SVM_from_scratch.ipynb" rel="nofollow noreferrer">https://github.com/arkm97/svm-from-scratch/blob/master/SVM_from_scratch.ipynb</a></p>
<p>I will try to reverse engineering that concept:</p>
<p>So firstly there is issue for <strong>DataCleaning</strong>(removing 0,values, serialize, normalize)</p>
<p>normalizing dataset (replaced by its difference from the mean)</p>
<pre><code>credit_df_norm = (credit_df - credit_df.mean())/(credit_df.std())
</code></pre>
<p>Divide dataset into Train &amp; Test:</p>
<pre><code>train_df = credit_df_norm.drop(target, axis=1).loc[:training_points]
train_target = credit_df_norm[target].replace(0, -1).loc[:training_points]
</code></pre>
<p>And here is starting the clue of that algorithms (with maths etc.)</p>
<blockquote>
<p>given data points <span class="math-container">$\vec x_j \in \mathbb{R}^{1 \times N}$</span> and targets
<span class="math-container">$y_j = \pm 1$</span>, where <span class="math-container">$j = 1, \dots, M$</span>, find the maximum-margin
hyperplane that separates the two classes (<span class="math-container">$y_j = 1$</span> and <span class="math-container">$y_j = -1$</span>).</p>
<p>Let <span class="math-container">$\vec w$</span> be the vector normal to the hyperplane. We want to find
<span class="math-container">$\vec w$</span> that satisfies</p>
<p><span class="math-container">$$ y_j (\vec w \cdot \vec x_j + b) \geq 1 $$</span> The dual formulation of
the above is equivalent to maximizing the following over the
multipliers <span class="math-container">$\vec \alpha$</span>:</p>
<p><span class="math-container">$$ L(\vec \alpha) = \vec y \cdot \vec \alpha  - \frac 1 2  \vec \alpha
&gt; K \vec \alpha^T$$</span> subject to the constraints <span class="math-container">$\sum_{j=1}^M \alpha_j =
&gt; 0$</span> and <span class="math-container">$y_j \alpha_j \geq 0$</span>. The matrix <span class="math-container">$K$</span> defines the kernel of the
SVM; I've chosen <span class="math-container">$K_{jk} = k(\vec x_j, \vec x_k) = \vec x_j \cdot \vec
&gt; x_k$</span>. The parameters of the plane are recovered from <span class="math-container">$\vec w = \vec
&gt; \alpha \cdot \vec x$</span> and <span class="math-container">$b = y_j = \vec w \cdot \alpha_j$</span> for <span class="math-container">$j$</span>
such that <span class="math-container">$\alpha_j \neq 0$</span></p>
</blockquote>
<p>Let's divide it to the single factors:</p>
<p>we need to find <a href="https://www.quora.com/What-is-the-maximum-margin-hyperplane-in-SVM" rel="nofollow noreferrer">maximum-margin-hyperplane</a> for separate into two classes:
Separation cause SVN is classification - so we need to classify elements into categories.</p>
<p> = 1</p>
<p> = −1</p>
<p>This pattern allow to find hyperplane(?)</p>
<p>(⃗ ⋅⃗ +)≥1</p>
<p>The next pattern:</p>
<p>(⃗ )=⃗ ⋅⃗ −12⃗ &gt;⃗ </p>
<p>The author for kernel SVN choose those pattern:</p>
<p>=(⃗ ,⃗ )=⃗ ⋅&gt;⃗ </p>
<p>And the parameters for plane are here:</p>
<p>⃗ =&gt;⃗ ⋅⃗  and ==⃗ ⋅ for  such that ≠0</p>
<p>Reference:
<a href="https://arxiv.org/pdf/1307.0471.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/1307.0471.pdf</a></p>
<hr />
<p>In this line are added the kernel</p>
<p>Popular kernels are:</p>
<ul>
<li><p>Polynomial Kernel,</p>
</li>
<li><p>Gaussian Kernel,</p>
</li>
<li><p>Radial Basis Function (RBF),</p>
</li>
<li><p>Laplace RBF Kernel,</p>
</li>
<li><p>Sigmoid Kernel,</p>
</li>
<li><p>Anove RBF Kernel</p>
<p>k_value = np.array(train_df @ train_df.T + np.identity(len(train_target))*1e-12)</p>
</li>
</ul>
<p>The structure of k_value (kernel matrix) is an array.</p>
<p><a href="https://i.sstatic.net/1j4fN.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1j4fN.png" alt="enter image description here" /></a></p>
<p>Next there is use of <a href="https://www.geeksforgeeks.org/cholesky-decomposition-matrix-decomposition/" rel="nofollow noreferrer">cholesky decomposition—cholesky</a> - which is next complicated concept.</p>
<pre><code>np.linalg.cholesky(k_value)
</code></pre>
<p>Next some bits of data converting:</p>
<p>alpha = cp.Variable(shape=train_target.shape)</p>
<p>beta = cp.multiply(alpha, train_target) # to simplify notation</p>
<pre><code>K = cp.Parameter(shape=k_value.shape, PSD=True, value=k_value)

# objective function
obj = .5 * cp.quad_form(beta, K) - np.ones(alpha.shape).T @ alpha

# constraints
const = [np.array(train_target.T) @ alpha == 0,
        -alpha &lt;= np.zeros(alpha.shape),
        alpha &lt;= 10*np.ones(shape=alpha.shape)]
prob = cp.Problem(cp.Minimize(obj), const)
</code></pre>
<p>The next step is recreate the hyperplane:</p>
<pre><code>w = np.multiply(train_target, alpha.value).T @ train_df
S = (alpha.value &gt; 1e-4).flatten()
b = train_target[S] - train_df[S] @ w
b = b[0]
# b = np.mean(b)
</code></pre>
<p>And finally - the classification and evaluation:</p>
<pre><code>def classify(x):
    result = w @ x + b
    return np.sign(result)

correct = 0
incorrect = 0
validation_set = credit_df_norm.drop(target, axis=1)
predictions = []
for i, x in validation_set.iterrows():
    my_svm = classify(x)
    if my_svm==credit_df_norm[target].replace(0, -1)[i]: correct +=1
    else: incorrect +=1
    predictions.append(my_svm)
predictions = np.array(predictions)


print(f&quot;fraction correct: {correct/(correct + incorrect)}&quot;)
</code></pre>
<hr />
<p>That was the more detailed implementation - for shorcut there is an interesting article:</p>
<p><a href="https://towardsdatascience.com/svm-and-kernel-svm-fed02bef1200" rel="nofollow noreferrer">https://towardsdatascience.com/svm-and-kernel-svm-fed02bef1200</a></p>
<p>There are existing function from another library - that can be reused in simple manner:</p>
<pre><code># Fitting SVM to the Training set
from sklearn.svm import SVC
classifier = SVC(kernel = 'rbf', C = 0.1, gamma = 0.1)
classifier.fit(X_train, y_train)

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# use seaborn plotting defaults
import seaborn as sns; sns.set()
</code></pre>
<p>Visualized Dataset:
<a href="https://i.sstatic.net/T2eMH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/T2eMH.png" alt="enter image description here" /></a></p>
<p>Dataset after apply hyperplanes:
<a href="https://i.sstatic.net/WUzFT.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/WUzFT.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2021-05-01T14:11:04.967Z
2021-05-01T14:11:04.967Z
Piotr Żak
<p>I'm a computer scientist who's studying support vector machines (SVMs) in a machine learning course. I have some understanding of how SVMs are designed, thanks to <a href="https://www.youtube.com/watch?v=_PwhiWxHK8o" rel="nofollow noreferrer">16. Learning: Support Vector Machines - MIT</a>. However, what I'm not understanding is the transition from the optimization problem of the Lagrangian function to its implementation in any programming language. Basically, what I need to understand is how to build, from scratch, the decision function, given a training set. In particular, how do I find Lagrange multipliers in order to know which points are to be considered to define support vectors and the decision function?</p>
<p>Can anyone explain this to me?</p>

CC BY-SA 4.0
2020-02-16T05:41:50.740Z
We're you ever able to answer your question because I have arrived at a similar point?
FeedMeInformation
<machine-learning><implementation><support-vector-machine>
How to implement SVM algorithm from scratch in a programming language?
Marco
# How to implement SVM algorithm from scratch in a programming language?

(Asked by: username_0 on 2019-08-23)

I'm a computer scientist who's studying support vector machines (SVMs) in a machine learning course. I have some understanding of how SVMs are designed, thanks to [16. Learning: Support Vector Machines - MIT](https://www.youtube.com/watch?v=_PwhiWxHK8o). However, what I'm not understanding is the transition from the optimization problem of the Lagrangian function to its implementation in any programming language. Basically, what I need to understand is how to build, from scratch, the decision function, given a training set. In particular, how do I find Lagrange multipliers in order to know which points are to be considered to define support vectors and the decision function?

Can anyone explain this to me?

**Tags:** machine-learning, implementation, support-vector-machine

> **Comments:**
> **username_1**: We're you ever able to answer your question because I have arrived at a similar point?  

---

## 1 answers

### Answer by username_2 on 2021-05-01. Score: 1

Based on this repository:
https://github.com/arkm97/svm-from-scratch/blob/master/SVM_from_scratch.ipynb
I will try to reverse engineering that concept:
So firstly there is issue for DataCleaning(removing 0,values, serialize, normalize)
normalizing dataset (replaced by its difference from the mean)
credit_df_norm = (credit_df - credit_df.mean())/(credit_df.std())

Divide dataset into Train & Test:
train_df = credit_df_norm.drop(target, axis=1).loc[:training_points]
train_target = credit_df_norm[target].replace(0, -1).loc[:training_points]

And here is starting the clue of that algorithms (with maths etc.)

given data points $\vec x_j \in \mathbb{R}^{1 \times N}$ and targets
$y_j = \pm 1$, where $j = 1, \dots, M$, find the maximum-margin
hyperplane that separates the two classes ($y_j = 1$ and $y_j = -1$).
Let $\vec w$ be the vector normal to the hyperplane. We want to find
$\vec w$ that satisfies
$$ y_j (\vec w \cdot \vec x_j + b) \geq 1 $$ The dual formulation of
the above is equivalent to maximizing the following over the
multipliers $\vec \alpha$:
$$ L(\vec \alpha) = \vec y \cdot \vec \alpha  - \frac 1 2  \vec \alpha
> K \vec \alpha^T$$ subject to the constraints $\sum_{j=1}^M \alpha_j =
> 0$ and $y_j \alpha_j \geq 0$. The matrix $K$ defines the kernel of the
SVM; I've chosen $K_{jk} = k(\vec x_j, \vec x_k) = \vec x_j \cdot \vec
> x_k$. The parameters of the plane are recovered from $\vec w = \vec
> \alpha \cdot \vec x$ and $b = y_j = \vec w \cdot \alpha_j$ for $j$
such that $\alpha_j \neq 0$

Let's divide it to the single factors:
we need to find maximum-margin-hyperplane for separate into two classes:
Separation cause SVN is classification - so we need to classify elements into categories.
 = 1
 = −1
This pattern allow to find hyperplane(?)
(⃗ ⋅⃗ +)≥1
The next pattern:
(⃗ )=⃗ ⋅⃗ −12⃗ >⃗ 
The author for kernel SVN choose those pattern:
=(⃗ ,⃗ )=⃗ ⋅>⃗ 
And the parameters for plane are here:
⃗ =>⃗ ⋅⃗  and ==⃗ ⋅ for  such that ≠0
Reference:
https://arxiv.org/pdf/1307.0471.pdf

In this line are added the kernel
Popular kernels are:

Polynomial Kernel,

Gaussian Kernel,

Radial Basis Function (RBF),

Laplace RBF Kernel,

Sigmoid Kernel,

Anove RBF Kernel
k_value = np.array(train_df @ train_df.T + np.identity(len(train_target))*1e-12)


The structure of k_value (kernel matrix) is an array.

Next there is use of cholesky decomposition—cholesky - which is next complicated concept.
np.linalg.cholesky(k_value)

Next some bits of data converting:
alpha = cp.Variable(shape=train_target.shape)
beta = cp.multiply(alpha, train_target) # to simplify notation
K = cp.Parameter(shape=k_value.shape, PSD=True, value=k_value)

# objective function
obj = .5 * cp.quad_form(beta, K) - np.ones(alpha.shape).T @ alpha

# constraints
const = [np.array(train_target.T) @ alpha == 0,
        -alpha <= np.zeros(alpha.shape),
        alpha <= 10*np.ones(shape=alpha.shape)]
prob = cp.Problem(cp.Minimize(obj), const)

The next step is recreate the hyperplane:
w = np.multiply(train_target, alpha.value).T @ train_df
S = (alpha.value > 1e-4).flatten()
b = train_target[S] - train_df[S] @ w
b = b[0]
# b = np.mean(b)

And finally - the classification and evaluation:
def classify(x):
    result = w @ x + b
    return np.sign(result)

correct = 0
incorrect = 0
validation_set = credit_df_norm.drop(target, axis=1)
predictions = []
for i, x in validation_set.iterrows():
    my_svm = classify(x)
    if my_svm==credit_df_norm[target].replace(0, -1)[i]: correct +=1
    else: incorrect +=1
    predictions.append(my_svm)
predictions = np.array(predictions)


print(f"fraction correct: {correct/(correct + incorrect)}")


That was the more detailed implementation - for shorcut there is an interesting article:
https://towardsdatascience.com/svm-and-kernel-svm-fed02bef1200
There are existing function from another library - that can be reused in simple manner:
# Fitting SVM to the Training set
from sklearn.svm import SVC
classifier = SVC(kernel = 'rbf', C = 0.1, gamma = 0.1)
classifier.fit(X_train, y_train)

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# use seaborn plotting defaults
import seaborn as sns; sns.set()

Visualized Dataset:

Dataset after apply hyperplanes:


---

