Support Vector Machines (SVM) are a type of supervised learning algorithm introduced by Vapnik in 1995, widely used for classification and regression tasks including object detection and text recognition. SVM works by finding the hyperplane that maximizes the margin between different classes, with two types being hard margin and soft margin SVMs. The algorithm uses support vectors, which are the closest data points to the hyperplane, to define the decision boundary.

1. Support vector machines(svm)
Presenting by:
Eisa Jafari Amirbandi
Eisa.Jafari.Amirbandi@gmail.com

2. What is Support vector machines(svm)?
SVM‫کننده‬ ‫بندی‬ ‫دسته‬‫ای‬(classifier)‫شاخه‬ ‫جزو‬ ‫که‬ ‫است‬
Kernel Methods‫یادگیری‬ ‫در‬‫میشود‬ ‫محسوب‬ ‫ماشین‬.
SVM‫سال‬ ‫در‬1995‫توسط‬Vapnik‫معرفی‬‫شده‬‫است‬
http://link.springer.com/article/10.1007/BF00994018
Cited by 13557
‫شهرت‬SVM‫دست‬ ‫حروف‬ ‫تشخیص‬ ‫در‬ ‫آن‬ ‫موفقیت‬ ‫بخاطر‬‫نویس‬
‫برابری‬ ‫شده‬ ‫تنظیم‬ ‫بدقت‬ ‫عصبی‬ ‫های‬ ‫شبکه‬ ‫با‬ ‫که‬ ‫است‬‫میک‬‫ند‬
2
Introduction
what’s svm?
application
Flashback
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

3. ‫کاربرد‬
 SVM is widely used in object detection & recognition,
content-based
 image retrieval, text recognition, biometrics, speech
recognition, etc.
 Also used for regression (SVR), (will not cover today)
‫برای‬ ‫خطی‬ ‫غیر‬ ‫و‬ ‫خطی‬ ‫های‬ ‫ساز‬ ‫جدا‬ ‫که‬ ‫مساله‬ ‫هر‬ ‫در‬
‫از‬ ‫توان‬ ‫می‬ ‫شود‬ ‫استفاده‬ ‫کالسبندی‬svm‫کرد‬ ‫استفاده‬
3
Introduction
what’s svm?
application
Flashback
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

4. Flashback (ml)
SVM‫ای‬ ‫کننده‬ ‫بندی‬ ‫دسته‬(classifier)‫شاخه‬ ‫جزو‬ ‫که‬ ‫است‬
Kernel Methods‫یادگیری‬ ‫در‬‫میشود‬ ‫محسوب‬ ‫ماشین‬.
‫چست؟‬ ‫یادگیری‬
‫کالسیفایر‬‫چیست؟‬
4
Introduction
what’s svm?
application
Flashback
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

5. Flashback (ml)
 Supervised classification
perceptron, support vector machine, loss functions, kernels,random
forests
 Supervised regression
ridge regression, lasso regression, SVM regression
 Unsupervised learning
graphical models, sequential Monte Carlo, PCA, Gaussian Mixture
Models, probabilistic PCA, hidden Markov models
5
Refrenc:Hilary Term 2014
A. Zisserman
Introduction
what’s svm?
application
Flashback
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

6. Flashback (ml)
1-Regression - supervised
estimate parameters, e.g. of weight vs height
2-Classification - supervised
estimate class, e.g. handwritten digit classification
6
Refrenc:Hilary Term 2014
A. Zisserman
Introduction
what’s svm?
application
Flashback
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

7. Flashback (ml)
3- Unsupervised learning – model the data
clustering
dimensionality reduction
7
Introduction
what’s svm?
application
Flashback
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

8. Intuition
‫سوال‬:‫کنیم؟‬ ‫انتخاب‬ ‫را‬ ‫ها‬ ‫ساز‬ ‫جدا‬ ‫از‬ ‫یک‬ ‫کدام‬
‫سوال‬:‫کدام‬‫است؟‬ ‫بهتر‬ ‫یک‬
8
Introduction
Reminder
Discriminant Function
Margin
Types of svm’s
Issue
Summary
Conclusion
Example

9. ‫کنیم‬ ‫انتخاب‬ ‫را‬ ‫ها‬ ‫ساز‬ ‫جدا‬ ‫از‬ ‫یک‬ ‫کدام‬
‫داشته‬ ‫وجود‬ ‫دسته‬ ‫دو‬ ‫اگر‬
‫که‬ ‫باشند‬‫بصورت‬‫از‬ ‫خطی‬
‫هم‬‫جداپذیر‬‫بهترین‬ ،‫باشند‬
‫دسته‬ ‫دو‬ ‫این‬ ‫کننده‬ ‫جدا‬
‫چیست؟‬
‫الگوریتم‬‫از‬ ‫مختلفی‬ ‫های‬
‫جمله‬‫پرسپترون‬‫میتوانند‬‫این‬
‫دهند‬ ‫انجام‬ ‫را‬ ‫جداسازی‬.
‫این‬ ‫همه‬ ‫آیا‬‫الگوریتمها‬
‫بر‬ ‫اینکار‬ ‫عهده‬ ‫از‬ ‫بخوبی‬
‫میآیند؟‬
9
Introduction
Reminder
Discriminant Function
Margin
Types of svm’s
Issue
Summary
Conclusion
Example
Refrence image:wikipedia

10. ‫کنیم‬ ‫انتخاب‬ ‫را‬ ‫ها‬ ‫ساز‬ ‫جدا‬ ‫از‬ ‫یک‬ ‫کدام‬
H2‫یا‬ ‫است‬ ‫بهتر‬H3‫؟‬
10
Introduction
Reminder
Discriminant Function
Margin
Types of svm’s
Issue
Summary
Conclusion
Example
Refrence image:wikipedia

11. ‫کنیم‬ ‫انتخاب‬ ‫را‬ ‫ها‬ ‫ساز‬ ‫جدا‬ ‫از‬ ‫یک‬ ‫کدام‬
H2‫یا‬ ‫است‬ ‫بهتر‬H3‫؟‬
‫که‬ ‫است‬ ‫بهتر‬ ‫سازی‬ ‫جدا‬
‫حاشیه‬(margin)‫بزرگتری‬
‫داشته‬‫باشدو‬‫بزرگت‬ ‫حاشیه‬‫ر‬
‫اطمینان‬ ‫ضریب‬ ‫معنی‬ ‫به‬
‫است‬ ‫بیشتر‬
11
Introduction
Reminder
Discriminant Function
Margin
Types of svm’s
Issue
Summary
Conclusion
Example
Refrence image:wikipedia

12. ‫حاشیه‬(margin)
• The hyperplane with the largest margin has equal
distances to the nearest sample of both classes
• SVM finds the solution with maximum margin
12
Introduction
Reminder
Discriminant Function
Margin
Types of svm’s
Issue
Summary
Conclusion
Example

13. What is Support vector(‫پشتیبان‬ ‫?)بردار‬
‫جدا‬ ‫های‬ ‫صفحه‬ ‫ابر‬ ‫به‬ ‫آموزشی‬ ‫های‬ ‫داده‬ ‫نزدیکترین‬‫کننده‬,‫بردار‬
‫میشوند‬ ‫نامیده‬ ‫پشتیبان‬
Support vectors
Maximizes
marginNarrower
margin
13
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

14. ‫یادآوری‬(‫صفحه‬ ‫در‬ ‫خط‬ ‫معادله‬)14
‫خط‬ ‫ضمنی‬ ‫معادله‬‫فضای‬ ‫در‬2‫بعدی‬:
Ax+By+C=0
‫تغییر‬‫نوتیشن‬:
𝑤1 𝑥1 + 𝑤2 𝑥2 + 𝑏 = 0
W=
𝑤1
𝑤2
X=
𝑥1
𝑥2
𝑊 𝑇
𝑋 + b = 0
x
y
Ax+By+C=0
‫فرم‬‫ماتریسی‬:
W‫خط‬ ‫بر‬ ‫عمود‬ ‫است‬ ‫خطی‬‫جداساز‬‫آن‬ ‫جهت‬ ‫و‬
‫مثبت‬ ‫جهت‬ ‫در‬‫جداساز‬‫باشد‬ ‫می‬

15. ‫یادآوری‬(Linear Discriminant Function)15
‫فضای‬ ‫دهیم‬ ‫تعمیم‬ ‫توانیم‬ ‫می‬2‫فضای‬ ‫به‬ ‫را‬ ‫بعدی‬n‫بعدی‬
‫داریم‬ ‫پس‬.
𝑖
𝑤𝑖 𝑥𝑖 + 𝑏 = 0 𝑊 𝑇 𝑋 + b = 0
Refrence image:wikipedia

16. ‫انواع‬svm
Hard margin linear SVM
 Soft margin linear SVM
 Non-linear SVM
16
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

17. Hard Margin SVM17
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
‫فرض‬‫آموزشی‬ ‫های‬ ‫داده‬ ‫کنید‬‫صورت‬ ‫به‬< X 𝑖, Y𝑖 >‫است‬‫در‬ ‫که‬
‫داریم‬ ‫آن‬:
𝑊 𝑇 𝑋 + b < −1
𝑊 𝑇 𝑋 + b > 1
{X 𝑖, Y𝑖} i=1,2,…,n
X 𝑖 ∈ 𝑅 𝑑
Y𝑖∈ {1, −1}
𝑊 𝑇 𝑋𝑖 + b > 1 if Y𝑖 = 1
𝑊 𝑇
𝑋𝑖 + b < −1 if Y𝑖 = −1
For i=1,2,…,n

18. Hard Margin SVM18
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
‫تعریف‬ ‫طبق‬svm‫حاشیه‬ ‫که‬ ‫هستیم‬ ‫جداسازی‬ ‫بدنبال‬ ‫ما‬
‫به‬ ‫داریم‬ ‫احتیاج‬ ‫اینکار‬ ‫برای‬ ‫باشد‬ ‫داشته‬ ‫بزرگتری‬‫فرمولی‬‫ب‬‫رسیم‬
‫دهد‬ ‫ما‬ ‫به‬ ‫ریاضی‬ ‫پارامتر‬ ‫حسب‬ ‫بر‬ ‫را‬ ‫حاشیه‬ ‫مقدار‬ ‫که‬
‫فاصله‬ ‫همان‬ ‫حاشیه‬ ‫دانیم‬ ‫می‬ ‫طرفی‬ ‫از‬2‫اطراف‬ ‫موازی‬ ‫خط‬
‫جداساز‬‫پ‬ ‫است‬‫س‬‫داریم‬:
𝑊 𝑇 𝑋 + b < −1
𝑊 𝑇 𝑋 + b > 1
𝑑 =
2
| 𝑤 |

19. Hard Margin SVM19
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
‫مساله‬ ‫تعریف‬:
‫در‬svm‫حاشیه‬ ‫مقدار‬ ‫خواهیم‬ ‫می‬ ‫ما‬(𝑑 =
2
| 𝑤 |
)‫دهیم‬ ‫افزایش‬ ‫را‬
‫که‬ ‫این‬ ‫شرط‬ ‫به‬‫باشد‬ ‫برقرار‬ ‫زیر‬ ‫رابطه‬:
𝑊 𝑇 𝑋 + b < −1
𝑊 𝑇 𝑋 + b > 1
For i=1,2,…,n
𝑊 𝑇
𝑋𝑖 + b > 1 if Y𝑖 = 1
𝑊 𝑇 𝑋𝑖 + b < −1 if Y𝑖 = −1

20. Hard Margin SVM20
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
‫حاشیه‬ ‫افزایش‬ ‫برای‬(𝑑 =
2
| 𝑤 |
)‫مقدار‬ ‫باید‬| 𝑤 |(‫کسر‬ ‫مخرج‬)‫را‬
‫مینیم‬‫بنویسی‬ ‫گونه‬ ‫این‬ ‫توانیم‬ ‫می‬ ‫را‬ ‫مساله‬ ‫شرط‬ ‫و‬ ‫کنیم‬‫م‬:
Y𝑖(𝑊 𝑇 𝑋𝑖 + b) > 1 For i=1,2,…,n𝑊 𝑇 𝑋𝑖 + b > 1 if Y𝑖 = 1
𝑊 𝑇
𝑋𝑖 + b < −1 if Y𝑖 = −1

21. Hard Margin SVM21
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
‫جای‬ ‫به‬ ‫خطی‬ ‫جبر‬ ‫از‬ ‫استفاده‬ ‫و‬ ‫کار‬ ‫راحتی‬ ‫برای‬‫مینیم‬‫کردن‬| 𝑤 |
‫تابع‬
1
2
| 𝑤 |2
‫را‬‫مینمم‬‫داریم‬ ‫پس‬ ‫کنیم‬ ‫می‬:
For i=1,2,…,n
Minimum
1
2
| 𝑤 |2
Such that:
Y𝑖(𝑊 𝑇
𝑋𝑖 + b) > 1

22. ‫توابع‬ ‫یادآوری‬‫الگرانژ‬‫شروط‬ ‫و‬KKT
‫تابع‬ ‫بخواهیم‬ ‫اگر‬F(x)‫را‬‫نسبت‬‫به‬x‫مینمم‬‫با‬ ‫کنیم‬
‫که‬ ‫این‬ ‫شرط‬g(x)≥0‫تابع‬ ‫از‬ ‫توانیم‬ ‫می‬‫الگرانژ‬
‫کنیم‬ ‫استفاده‬
L(x,u)=F(x)-uG(x) ; u≥0
‫بایست‬ ‫می‬ ‫تابع‬ ‫این‬ ‫در‬ ‫که‬x‫را‬‫مینمم‬‫و‬ ‫کرده‬u‫را‬
‫ماکسیمم‬‫سازبم‬
‫شروط‬KKT:
22
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
∂L(x,u)
∂x
=0
uG(x)=0

23. Hard Margin SVM23
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
Minimum
1
2
| 𝑤 |2
Such that: Y𝑖(𝑊 𝑇
𝑋𝑖 + b) − 1 > 0
‫قضیه‬ ‫طبق‬‫الگرانژ‬:

24. Solving the Optimization Problem
0]1)([  by iii wx
only SVs will have non-zero ai
24
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
Karush-Kuhn-Tucker (KKT) conditions:
𝑎𝑖 ≥0



25. 0]1)([:  bycondKKT iii wx
6=1.4
Class 1
Class 2
1=0.8
2=0
3=0
4=0
5=0
7=0
8=0.6
9=0
10=0
25
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

26. Solving the Optimization Problem
‫تابع‬ ‫در‬ ‫گذاری‬ ‫جا‬ ‫با‬‫الگرانژ‬‫داریم‬:
‫قضیه‬ ‫طبق‬dual‫برای‬‫ماکسیمم‬‫حسب‬ ‫بر‬ ‫تابع‬ ‫کردن‬𝑎𝑖‫توانیم‬ ‫می‬
‫مساله‬ ‫و‬ ‫کرده‬ ‫ضرب‬ ‫منفی‬ ‫در‬ ‫را‬ ‫باال‬ ‫رابطه‬‫مینیمایزیشن‬‫شود‬
26
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
Q(α)

27. Solving the Optimization Problem
‫به‬ ‫تبدیل‬ ‫ما‬ ‫مساله‬‫یک‬quadratic program‫که‬ ‫است‬ ‫شده‬
‫در‬ ‫توان‬ ‫می‬‫متلب‬‫دستور‬ ‫با‬quadprog‫مقدار‬ ‫و‬ ‫کرد‬ ‫حل‬a‫را‬ ‫ها‬
‫آورد‬ ‫بدست‬
27
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
‫دستور‬ ‫از‬ ‫استفاده‬ ‫برای‬quadprog‫دو‬ ‫باید‬‫ماتریس‬‫کنیم‬ ‫تعریف‬ ‫را‬
H=𝑦𝑖 𝑦𝑗 𝑥𝑖
𝑇
𝑥𝑗
f=-1
a = quadprog(H,f)

28. Hard Margin SVM
‫مقدار‬ ‫وقتی‬a‫مقدار‬ ‫توانیم‬ ‫می‬ ‫آوریم‬ ‫بدست‬ ‫را‬w‫زیر‬ ‫رابطه‬ ‫از‬ ‫را‬
‫مقدار‬ ‫فقط‬ ‫و‬ ‫آوریم‬ ‫بدست‬b‫دوم‬ ‫شرط‬ ‫باید‬ ‫که‬ ‫ماند‬ ‫می‬kkt‫برقرار‬
‫شود‬
28
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

29. Hard Margin SVM29
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
0]1)([  by iii wxKKT cond: or
01)(  by ii wx
0i NSV
iiyb wx
SVs
only SVs will have non-zero ai

30. 0]1)([:  bycondKKT iii wx
6=1.4
Class 1
Class 2
1=0.8
2=0
3=0
4=0
5=0
7=0
8=0.6
9=0
10=0
30
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

31. Summary Solving the Optimization Problem
‫گام‬1:‫دیتاست‬{X 𝑖, Y𝑖}‫مقدار‬ ‫که‬ ‫است‬ ‫این‬ ‫هدف‬ ‫گیریم‬ ‫می‬ ‫را‬ ‫اولیه‬w,b
‫بیاوریم‬ ‫بدست‬ ‫را‬
‫گام‬2:‫ماتریس‬H ,f‫تابع‬ ‫از‬ ‫و‬ ‫سازیم‬ ‫می‬ ‫را‬‫متلب‬a = quadprog(H,f)
‫کنیم‬ ‫می‬ ‫استفاده‬‫مقدار‬ ‫تا‬a‫آید‬ ‫بدست‬
h𝑖𝑗=𝑦𝑖 𝑦𝑗 𝑥𝑖
𝑇
𝑥𝑗 , f=-1 , H=[h𝑖𝑗]
‫گام‬3:‫مقدار‬w ,b‫آوریم‬ ‫می‬ ‫بدست‬ ‫زیر‬ ‫فرمول‬ ‫از‬ ‫را‬
‫گام‬4:‫شود‬ ‫می‬ ‫زیر‬ ‫رابطه‬ ‫ما‬ ‫خطی‬ ‫ساز‬ ‫جدا‬:
31
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
‫خروجی‬ = 𝑠𝑖𝑔𝑛(𝑊 𝑇 𝑋𝑖 + b)

32. The Quadratic Programming Problem32
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

33. Soft Margin SVM
 What if data is not linear separable?
(noisy data, outliers, etc.)
 Slack variables ξi can be added to allow miss-classification of
difficult or noisy data points
33
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
𝑊 𝑇 𝑋 + b > 1
yi (<w,xi> + b) ≥1 -xi , xi ≥ 0

34. Soft Margin SVM
‫متغیر‬ ‫معرفی‬ ‫با‬xi, i=1, 2, …, N,‫رابطه‬ ‫و‬ ‫شده‬ ‫تر‬ ‫ساده‬ ‫قبلی‬ ‫های‬ ‫محدودیت‬
yi (<w,xi> + b) ≥1
‫بصورت‬‫میکند‬ ‫تغییر‬ ‫زیر‬:
yi (<w,xi> + b) ≥1 -xi , xi ≥ 0
‫در‬‫متغیر‬ ‫این‬ ‫همه‬ ‫آل‬ ‫ایده‬ ‫حالت‬‫ها‬xi‫باشند‬ ‫صفر‬ ‫باید‬.
34
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

35. Soft Margin SVM
‫مساله‬hard margin‫که‬ ‫کند‬ ‫می‬ ‫تغییر‬ ‫گونه‬ ‫این‬ ‫ما‬‫را‬ ‫زیر‬ ‫تابع‬‫مینمم‬‫کنیم‬.
Such that
35
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
𝑊 𝑇 𝑋 + b > 1

36. Soft Margin SVM
C trades-off margin width and misclassifications
SVM tries to maintain to zero while maximizing margin.
36
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
Parametr c can be viewed as a way to control over-fiting
Such that:

37. Soft Margin SVM37
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
ST:
‫قضیه‬ ‫طبق‬‫الگرانژ‬:

38. Soft Margin SVM38
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
KKT conditions:
‫قضیه‬‫الگرانژ‬

39. Soft Margin SVM39
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
KKT conditions:
0≤𝑎𝑖 ≤ 𝐶
0≤𝐵𝑖 ≤ 𝐶

40. Soft Margin SVM
‫تابع‬ ‫در‬ ‫گذاری‬ ‫جا‬ ‫با‬‫الگرانژ‬‫و‬ ‫سازی‬ ‫ساده‬ ‫و‬
‫قضیه‬ ‫از‬ ‫استفاده‬daul(‫مثل‬hard margin)‫داریم‬:
‫همان‬ ‫این‬ ‫که‬quadratic program‫قسمت‬ ‫در‬ ‫که‬ ‫است‬hard
margin‫دستور‬ ‫با‬ ‫و‬ ‫است‬ ‫شده‬ ‫گفته‬‫متلب‬quadprog‫مقدار‬
a‫آید‬ ‫می‬ ‫بدست‬
40
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

41. Soft Margin SVM
‫با‬‫مقدار‬ ‫داشتن‬a‫مقدار‬w‫آید‬ ‫می‬ ‫بدست‬ ‫زیر‬ ‫رابطه‬ ‫از‬
41
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

42. Soft Margin SVM
‫آوردن‬ ‫بدست‬ ‫برای‬b‫آخر‬ ‫شرط‬kkt‫کنیم‬ ‫می‬ ‫بررسی‬ ‫را‬:
‫داریم‬ ‫باال‬ ‫شروط‬ ‫از‬:
1)
42
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
(𝐶-𝑎𝑖) xi =0
Not Support Vector

43. Soft Margin SVM
2)
3)
43
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
(𝐶-𝑎𝑖) xi =0
Where U is the set of unbounded support vector

44. Summary Soft Margin SVM44
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
‫گام‬1:‫دیتاست‬{X 𝑖, Y𝑖}‫پارامتر‬ ‫برای‬ ‫و‬ ‫گیریم‬ ‫می‬ ‫را‬ ‫اولیه‬c‫تعیین‬ ‫مقدار‬ ‫یک‬
‫مقدار‬ ‫که‬ ‫است‬ ‫این‬ ‫هدف‬ ‫کنیم‬ ‫می‬w,b‫بیاوریم‬ ‫بدست‬ ‫را‬
‫گام‬2:‫ماتریس‬H ,f‫تابع‬ ‫از‬ ‫و‬ ‫سازیم‬ ‫می‬ ‫را‬‫متلب‬
a = quadprog(H,f,1,c)‫کنیم‬ ‫می‬ ‫استفاده‬‫مقدار‬ ‫تا‬a‫آید‬ ‫بدست‬
H=𝑦𝑖 𝑦𝑗 𝑥𝑖
𝑇
𝑥𝑗 , f=-1
‫گام‬3:‫مقدار‬w ,b‫آوریم‬ ‫می‬ ‫بدست‬ ‫زیر‬ ‫فرمول‬ ‫از‬ ‫را‬
‫گام‬4:‫شود‬ ‫می‬ ‫زیر‬ ‫رابطه‬ ‫ما‬ ‫خطی‬ ‫ساز‬ ‫جدا‬:
‫خروجی‬ = 𝑠𝑖𝑔𝑛(𝑊 𝑇 𝑋𝑖 + b)

45. ‫بین‬ ‫مقایسه‬soft margin‫و‬hard margin
 Soft-Margin is more robust to outliers
 Hard-Margin does not require to guess the cost
parameter (requires noparameters at all)
45
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

46. Nonlinear SVMs46
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

47. Nonlinear SVMs
‫ها‬ ‫نمونه‬ ‫که‬ ‫است‬ ‫این‬ ‫اصلی‬ ‫ایده‬ ‫نباشد‬ ‫خطی‬ ‫صورت‬ ‫به‬ ‫پذیری‬ ‫جدا‬ ‫که‬ ‫حالتی‬ ‫در‬
‫باال‬ ‫بعد‬ ‫با‬ ‫فضای‬ ‫یک‬ ‫به‬ ‫را‬(feature space)‫در‬ ‫که‬ ‫دهیم‬ ‫نگاشت‬ ‫مشخصه‬ ‫فضای‬
‫شوند‬ ‫جدا‬ ‫هم‬ ‫از‬ ‫خطی‬ ‫صورت‬ ‫به‬ ‫توانند‬ ‫می‬ ‫ها‬ ‫نمونه‬ ‫؛‬ ‫ها‬ ‫مشخصه‬ ‫جدید‬ ‫فضای‬.
47
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

48. Nonlinear SVMs
‫نگاشت‬ ‫برای‬(maping)‫هسته‬ ‫تابع‬ ‫یک‬ ‫اعمال‬ ‫به‬ ‫نیاز‬(‫کرنل‬)‫می‬
‫باشد‬
48
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

49. Nonlinear SVMs49
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example

50. ‫ها‬ ‫کرنل‬ ‫معرفی‬50
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
 Linear Kernels
 Polynomial Kernels
 Radial Basis Function Kernels
 Three-Layer Neural Network Kernels
 Normalizing Kernels

51. Nonlinear SVMs
‫اولیه‬ ‫تابع‬
 G(x)=w 𝑇 𝑋 + 𝑏
 G(x)=w 𝑇
φ(X)+ 𝑏
quadratic program
‫کردن‬ ‫خطی‬ ‫غیر‬
51
Kernel mode

52. Nonlinear SVMs
‫اولیه‬ ‫تابع‬
 G(X)=w 𝑇 𝑋 + 𝑏
 G(X)=w 𝑇
φ(X)+𝑏
‫نهایی‬ ‫تابع‬(‫نمونه‬ ‫خروجی‬ ‫مقدار‬X)
‫کردن‬ ‫خطی‬ ‫غیر‬
52

53. Summary Nonlinear SVMs53
Introduction
Reminder
Types of svm’s
What’s SV?
Hard Margin
Soft Margin
Nonlinear Margin
Issue
Summary
Conclusion
Example
‫گام‬1:‫دیتاست‬{X 𝑖, Y𝑖}‫پارامتر‬ ‫برای‬ ‫و‬ ‫گیریم‬ ‫می‬ ‫را‬ ‫اولیه‬c‫تعیین‬ ‫مقدار‬ ‫یک‬
‫مقدار‬ ‫که‬ ‫است‬ ‫این‬ ‫هدف‬ ‫کنیم‬ ‫می‬b,a‫بیاوریم‬ ‫بدست‬ ‫را‬
‫گام‬2:‫ماتریس‬H ,f‫تابع‬ ‫از‬ ‫و‬ ‫سازیم‬ ‫می‬ ‫را‬‫متلب‬
a = quadprog(H,f,1,c)‫کنیم‬ ‫می‬ ‫استفاده‬‫مقدار‬ ‫تا‬a‫آید‬ ‫بدست‬
H=𝑦𝑖 𝑦𝑗 𝑥𝑖
𝑇
𝑥𝑗 , f=-1
‫گام‬4:‫شود‬ ‫می‬ ‫زیر‬ ‫رابطه‬ ‫ما‬ ‫ساز‬ ‫جدا‬:

54. C‫اس‬ ‫کننده‬ ‫تنظیم‬ ‫پارامتر‬ ‫یک‬‫ت‬
C‫گرفته‬ ‫نادیده‬ ‫راحتی‬ ‫به‬ ‫که‬ ‫محدودیت‬ ‫تا‬ ‫دهد‬ ‫می‬ ‫اجازه‬ ‫کوچک‬‫شود‬
C‫باعث‬ ‫بزرگ‬‫محدودیت‬ ‫شود‬ ‫می‬‫را‬ ‫ها‬‫سخت‬‫نادیده‬‫بگیرد‬
54
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

55. RBF Kernel SVM Example
data is not linearly separable in original feature space
http://www.robots.ox.ac.uk/~az/lectures/ml/
55
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

56. RBF Kernel SVM Example56
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

57. RBF Kernel SVM Example57
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

58. RBF Kernel SVM Example58
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

59. RBF Kernel SVM Example
‫تغیرات‬ ‫بررسی‬‫سیگما‬
59
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

60. RBF Kernel SVM Example60
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

61. RBF Kernel SVM Example61
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

62. Multi-Class Classification Build from binary
classifiers
62
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

63. Some Issues(Choice of kernel)
 Gaussian or polynomial kernel is default
 If ineffective, more elaborate kernels are needed
 Domain experts can give assistance in formulating
appropriate similarity measure.
63
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

64. Some Issues(Optimization criterion)
 Hard margin v.s. Soft margin
 a lengthy series of experiments in which various
parameters are tested
64
Introduction
Reminder
Types of svm’s
Issue
parameter c
Multi-Class Classification
Choice of kernel
Optimization criterion
Summary
Conclusion
Example

65. ‫الگوریتم‬svm
(1‫کرنل‬ ‫تابع‬ ‫انتخاب‬
(2‫مقدار‬ ‫کردن‬ ‫مشخص‬C
(3‫مساله‬ ‫کردن‬ ‫حل‬‫کوا‬‫د‬‫راتیک‬
Solve the quadratic programming problem (many software packages available)
(4‫تابع‬ ‫ساختن‬svm‫آمده‬ ‫بدست‬ ‫پارامترهای‬ ‫از‬
65
Introduction
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

66. Summary: Support Vector Machine
 Soft Margin Classifier :
Better generalization ability & less over-fitting
 The Kernel Trick:
Map data points to higher dimensional space in order to
make them
 linearly separable:
Since only dot product is used, we do not need to
represent the mapping explicitly
66
Introduction
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

67. Strengths
 No local minimal
 Robustness to outliers
 Training is relatively easy
 Good generalization in theory and practice
 Work well with few training instances
 Find globally best model, No local optimal, unlike in
neural networks
 It scales relatively well to high dimensional data
 Tradeoff between classifier complexity and error can be
controlled explicitly
 Non-traditional data like strings and trees can be used as
input to SVM, instead of feature vectors
 Notice: SVM does not minimize the number of
misclassifications (NP-complete problem) but the sum of
distancesfrom the margin hyperplanes.
67
Introduction
Reminder
Types of svm’s
Issue
Summary
Strengths
Weakness
SVMs vs. Neural Networks
Conclusion
Example

68. Weakness
 Selection of parameters)Need to choose a “ good”
kernel function.(
 Extension to multiclass problems
Refrenc:Advances in Pattern Recognition
Chapter 2 part 2.5 “Advantages and Disadvantages”
Shigeo Abe
68
Introduction
Reminder
Types of svm’s
Issue
Summary
Strengths
Weakness
SVMs vs. Neural Networks
Conclusion
Example

69. SVMs vs. Neural Networks
 Kernel maps to a high
dimensional spaces
 Search space has a
unique minimum
 Training is extremely
efficient
 Classification extremely
efficient
 Kernel and cost are the
two parameters to select
 Very good accuracy in
typical domains
 Extremely robust
 Hidden Layers map to
lower dimensional
spaces
 Search space has
multiple local minimum
 Training is expensive
 Classification extremely
efficient
 Requires number of
hidden units and layers
 Very good accuracy in
typical domains
 Could be robust
Refrenc:Advances in Pattern Recognition
Chapter 2 part 2.6 “Characteristics of Solutions”
Shigeo Abe
69
Introduction
Reminder
Types of svm’s
Issue
Summary
Strengths
Weakness
SVMs vs. Neural Networks
Conclusion
Example

70. Conclusion
Svm‫ترین‬ ‫بهینه‬‫جداکننده‬‫خطی‬‫راپیدا‬،‫کند‬ ‫می‬
‫دیگر‬ ‫خالف‬ ‫بر‬‫الگوریتم‬‫خطای‬ ‫که‬ ‫ها‬‫مدلسازی‬‫کنند‬ ‫می‬ ‫کمینه‬ ‫را‬
svm‫گیرد‬ ‫می‬ ‫نظر‬ ‫در‬ ‫هدف‬ ‫تابع‬ ‫عنوان‬ ‫به‬ ‫را‬ ‫عملیاتی‬ ‫ریسک‬
‫ابر‬‫که‬ ‫را‬ ‫صفحه‬‫بیشتربن‬‫داشته‬ ‫را‬ ‫حاشیه‬‫باشدانتخاب‬‫کند‬ ‫می‬
‫شوند‬ ‫می‬ ‫باعث‬ ‫ها‬ ‫کرنل‬‫الگوریتم‬svm‫شود‬ ‫خطی‬ ‫غیر‬
‫از‬ ‫مناسب‬ ‫استفاده‬ ‫صورت‬ ‫در‬SVM‫این‬‫الگوریتم‬‫خوبی‬ ‫تعمیم‬ ‫قدرت‬
‫خواهد‬‫داشت‬
‫زیاد‬ ‫ابعاد‬ ‫داشتن‬ ‫علیرغم‬high dimensionality))‫از‬overfitting
‫میکند‬ ‫پرهیز‬.‫از‬ ‫ناشی‬ ‫خاصیت‬ ‫این‬optimization‫این‬‫الگوریتم‬‫است‬
‫اطالعات‬ ‫سازی‬ ‫فشرده‬:
‫از‬ ‫آموزشی‬ ‫های‬ ‫داده‬ ‫بجای‬‫بردارهای‬‫میکند‬ ‫استفاده‬ ‫پشتیبان‬.
Refrenc:Advances in Pattern Recognition
Chapter 2 part 2.6 “Characteristics of Solutions”
Shigeo Abe
70
Introduction
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

71. Svm code
 http://www.kernel-machines.org/software.html
 http://www.csie.ntu.edu.tw/~cjlin/libsvm
 http://svmlight.joachims.org/
 A Practical Guide to Support Vector
Classification(Chih-Wei Hsu, Chih-Chung Chang, and
Chih-Jen Lin)
71
Introduction
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

72. matlab
‫از‬svmtrain‫ساختار‬ ‫کردن‬ ‫درست‬ ‫برای‬svm‫شود‬ ‫می‬ ‫استفاده‬
 Syntax
SVMStruct = svmtrain(Training,Group)
SVMStruct = svmtrain(Training,Group,Name,Value)
‫از‬svmclassify‫برای‬‫شود‬ ‫می‬ ‫استفاده‬ ‫تست‬ ‫نمونه‬ ‫کالسبندی‬
 Syntax
Group = svmclassify(SVMStruct,Sample)
Group = svmclassify(SVMStruct,Sample,'Showplot',true)
72
Introduction
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

73. matlab
 svmtrain(Training,Group, 'kernel_function',Value)
 Value:
 'linear' — Linear kernel, meaning dot product.
 'quadratic' — Quadratic kernel.
 'polynomial' — Polynomial kernel (default order 3). Specify
another order with the polyorder name-value pair.
 'rbf' — Gaussian Radial Basis Function kernel with a default
scaling factor, sigma, of 1. Specify another value for sigma
with the rbf_sigma name-value pair.
 'mlp' — Multilayer Perceptron kernel with default scale [1 –1].
Specify another scale with the mlp_params name-value pair.
 Default: 'linear
73
Introduction
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

74. ‫مثال‬1( :fisheriris)
 load fisheriris
 xdata = meas(51:end,3:4);
 group = species(51:end);
 svmStruct = svmtrain(xdata,group,'ShowPlot',true);
74
Introduction
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

75. ‫مثال‬1:‫نمونه‬ ‫یک‬ ‫تست‬
 species = svmclassify(svmStruct,[5 2],'ShowPlot',true)
 hold on;
 plot(5,2,'ro','MarkerSize',12);
 hold off
75
Introduction
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

76. ‫رفرنس‬
 Advances in Pattern Recognition(Support Vector Machines
for Pattern Classification) Professor Dr Shigeo Abe
‫اسالید‬A. Zisserman
‫فایل‬MATLAB Workshop 2‫سایت‬ ‫از‬www.david-lindsay.co.uk
‫ماشین‬ ‫یادگیری‬ ‫درس‬ ‫ویدیویی‬ ‫فایل‬Andrew Ng
76
Introduction
Reminder
Types of svm’s
Issue
Summary
Conclusion
Example

77. ‫پایان‬77