XXx0=XX1

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane: Preliminary

max w margin(w)

subject to every y n w ^T x _n > 0 margin(w) = min

n=1,...,N distance(x _n , w)

‘shorten’ x and w

distance

needs

w ₀

and

(w ₁ , . . . , w _d )

differently (to be derived)

b

=

w ₀







| w

|







=





 w ₁

.. . w _d







;

  

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane: Preliminary

max w margin(w)

subject to every y n w ^T x _n > 0 margin(w) = min

n=1,...,N distance(x _n , w)

‘shorten’ x and w

distance

needs

w ₀

and

(w ₁ , . . . , w _d )

differently (to be derived)

b

=

w ₀







| w

|







=





 w ₁

.. . w _d







;

   XX x ₀ = X X 1







| x

|







=





 x ₁

.. . x _d







for this part: h(x) = sign(w

^T x

+

b)

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane: Preliminary

max w margin(w)

subject to every y n w ^T x _n > 0 margin(w) = min

n=1,...,N distance(x _n , w)

‘shorten’ x and w

distance

needs

w ₀

and

(w ₁ , . . . , w _d )

differently (to be derived)

b

=

w ₀







| w

|







=





 w ₁

.. . w _d







;

   XX x ₀ = X X 1







| x

|







=





 x ₁

.. . x _d







for this part: h(x) = sign(w

^T x

+

b)

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 11/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane: Preliminary

max w margin(w)

subject to every y n w ^T x _n > 0 margin(w) = min

n=1,...,N distance(x _n , w)

‘shorten’ x and w

distance

needs

w ₀

and

(w ₁ , . . . , w _d )

differently (to be derived)

b

=

w ₀







| w

|







=





 w ₁

.. . w _d







;

   XX x ₀ = X X 1







| x

|







=





 x ₁

.. . x _d







for this part: h(x) = sign(w

^T x

+

b)

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

=

−

b

,

w ^T x ⁰⁰

=

−

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′ w x

distance(x,

b, w) =

   

w ^T

k

w

k

(x−

x ⁰

)    

=

1

1 k

w

k|

w ^T x

+

b

|

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 12/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

=

−

b

,

w ^T x ⁰⁰

=

−

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=

0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′

w x

distance(x,

b, w) =

   

w ^T

k

w

k

(x−

x ⁰

)    

=

1

1 k

w

k|

w ^T x

+

b

|

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 12/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

=

−

b

,

w ^T x ⁰⁰

=

−

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=

0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′

w x

distance(x,

b, w) =

   

w ^T

k

w

k

(x−

x ⁰

)    

=

1

1 k

w

k|

w ^T x

+

b

|

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 12/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

= −

b, w ^T x ⁰⁰

= −

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=

0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′

w x

distance(x,

b, w) =

   

w ^T

k

w

k

(x−

x ⁰

)    

=

1

1 k

w

k|

w ^T x

+

b

|

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 12/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

= −

b, w ^T x ⁰⁰

= −

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=

0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′

w x

distance(x,

b, w) =

   

w ^T

k

w

k

(x−

x ⁰

)    

=

1

1 k

w

k|

w ^T x

+

b

|

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 12/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

= −

b, w ^T x ⁰⁰

= −

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′

w x

distance(x,

b, w) =

   

w ^T

k

w

k

(x−

x ⁰

)    

=

1

1 k

w

k|

w ^T x

+

b

|

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 12/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

= −

b, w ^T x ⁰⁰

= −

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′

w x

distance(x,

b, w) =

   

w ^T

k

w

k

(x−

x ⁰

)    

=

1

1 k

w

k|

w ^T x

+

b

|

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 12/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

= −

b, w ^T x ⁰⁰

= −

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′

w x

distance(x,

b, w) =

   

w ^T

k

w

k

(x−

x ⁰

)    

=

1

1 k

w

k|

w ^T x

+

b

|

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 12/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

= −

b, w ^T x ⁰⁰

= −

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′

w x

distance(x,

b, w) =

   

w ^T

k

w

k(x−

x ⁰

)    

=

1

1 k

w

k|

w ^T x

+

b

|

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 12/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

= −

b, w ^T x ⁰⁰

= −

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′

w x

distance(x,

b, w) =

   

w ^T

k

w

k(x−

x ⁰

)    

=

1

1 k

w

k|

w ^T x

+

b

|

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Hyperplane

want: distance(x,

b, w), with hyperplane w ^T x ⁰

+

b

=0

consider

x ⁰

,

x ⁰⁰

on hyperplane

1 w ^T x ⁰

= −

b, w ^T x ⁰⁰

= −

b

2 w

⊥ hyperplane:









w ^T

(x

⁰⁰

−

x ⁰

)

| {z } vector on hyperplane









=0

3

distance = project (x−

x ⁰

)to

⊥ hyperplane

dist(x, h)

x^′ x^′′

w x

distance(x,

b, w) =

   

w ^T

k

w

k(x−

x ⁰

)    

=

1

1

k

w

k|

w ^T x + b

|

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 12/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Separating Hyperplane

distance(x,

b, w) =

1

k

w

k|

w ^T x + b

|

• separating

hyperplane: for every n

y n (w ^T x _n + b) > 0

•

distance to

separating

hyperplane: distance(x

_n

,

b, w) =

1

k

w

k

y _n

(w

^T x _n

+

b)

max

b,w

margin(b,

w)

subject to every

y n (w ^T x _n + b) > 0

margin(b,

w) =

min

n=1,...,N

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Separating Hyperplane

distance(x,

b, w) =

1

k

w

k|

w ^T x + b

|

• separating

hyperplane: for every n

y n (w ^T x n + b) > 0

•

distance to

separating

hyperplane: distance(x

_n

,

b, w) =

1

k

w

k

y _n

(w

^T x _n

+

b)

max

b,w

margin(b,

w)

subject to every

y n (w ^T x _n + b) > 0

margin(b,

w) =

min

n=1,...,N

distance(x

_n

,

b, w)

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 13/28

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Separating Hyperplane

distance(x,

b, w) =

1

k

w

k|

w ^T x + b

|

• separating

hyperplane: for every n

y n (w ^T x n + b) > 0

•

distance to

separating

hyperplane:

distance(x

_n

,

b, w) =

1

k

w

k

y _n

(w

^T x _n

+

b)

max

b,w

margin(b,

w)

subject to every

y n (w ^T x _n + b) > 0

margin(b,

w) =

min

n=1,...,N

distance(x

_n

,

b, w)

Linear Support Vector Machine Standard Large-Margin Problem

Distance to Separating Hyperplane

distance(x,

b, w) =

1

k

w

k|

w ^T x + b

|

• separating

hyperplane: for every n

y n (w ^T x n + b) > 0

•

distance to

separating

hyperplane:

distance(x

_n

,

b, w) =

1

k

w

k

y _n

(w

^T x _n

+

b)

max

b,w

margin(b,

w)

subject to every

y n (w ^T x _n + b) > 0

margin(b,

w) =

min

n=1,...,N 1

kwk y _n

(w

^T x _n

+

b)

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 13/28

Linear Support Vector Machine Standard Large-Margin Problem

Margin of Special Separating Hyperplane

max

b,w

margin(b,

w)

subject to every y

_n

(w

^T x _n

+

b)

> 0 margin(b,

w) =

min

n=1,...,N 1

kwk

y

_n

(w

^T x _n

+

b)

• w ^T x + b

=0 same as 3w

^T x + 3b

=0: scaling does not matter

• special

scaling: only consider separating (b,

w)

such that

min

n=1,...,N y _n (w ^T x _n + b) = 1

=⇒

margin(b,

w) = _kwk ¹

max

b,w 1 kwk

subject to every y

n

(w

^T x _n

+b)> 0

min

n=1,...,N y _n (w ^T x _n + b) = 1

Linear Support Vector Machine Standard Large-Margin Problem

Margin of Special Separating Hyperplane

max

b,w

margin(b,

w)

subject to every y

_n

(w

^T x _n

+

b)

> 0 margin(b,

w) =

min

n=1,...,N 1

kwk

y

_n

(w

^T x _n

+

b)

• w ^T x + b

=0 same as 3w

^T x + 3b

=0: scaling does not matter

• special

scaling: only consider separating (b,

w)

such that

min

n=1,...,N y _n (w ^T x _n + b) = 1

=⇒

margin(b,

w) = _kwk ¹

max

b,w 1 kwk

subject to every y

n

(w

^T x _n

+b)> 0

min

n=1,...,N y _n (w ^T x _n + b) = 1

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 14/28

Linear Support Vector Machine Standard Large-Margin Problem

Margin of Special Separating Hyperplane

max

b,w

margin(b,

w)

subject to every y

_n

(w

^T x _n

+

b)

> 0 margin(b,

w) =

min

n=1,...,N 1

kwk

y

_n

(w

^T x _n

+

b)

• w ^T x + b

=0 same as 3w

^T x + 3b

=0: scaling does not matter

• special

scaling: only consider separating (b,

w)

such that

n=1,...,N min y n (w ^T x _n + b) = 1

=⇒

margin(b,

w) = _kwk ¹

max

b,w 1 kwk

subject to every y

n

(w

^T x _n

+b)> 0

min

n=1,...,N y _n (w ^T x _n + b) = 1

Linear Support Vector Machine Standard Large-Margin Problem

Margin of Special Separating Hyperplane

max

b,w

margin(b,

w)

subject to every y

_n

(w

^T x _n

+

b)

> 0 margin(b,

w) =

min

n=1,...,N 1

kwk

y

_n

(w

^T x _n

+

b)

• w ^T x + b

=0 same as 3w

^T x + 3b

=0: scaling does not matter

• special

scaling: only consider separating (b,

w)

such that

n=1,...,N min y n (w ^T x _n + b) = 1

=⇒ margin(

b, w) = _kwk ¹

max

b,w 1 kwk

subject to every y

n

(w

^T x _n

+b)> 0

min

n=1,...,N y _n (w ^T x _n + b) = 1

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 14/28

Linear Support Vector Machine Standard Large-Margin Problem

Margin of Special Separating Hyperplane

max

b,w

margin(b,

w)

subject to every y

_n

(w

^T x _n

+

b)

> 0 margin(b,

w) =

min

n=1,...,N 1

kwk

y

_n

(w

^T x _n

+

b)

• w ^T x + b

=0 same as 3w

^T x + 3b

=0: scaling does not matter

• special

scaling: only consider separating (b,

w)

such that

n=1,...,N min y n (w ^T x _n + b) = 1

=⇒ margin(

b, w) = _kwk ¹

max

b,w 1 kwk

subject to every y

n

(w

^T x _n

+b)> 0

min

n=1,...,N y _n (w ^T x _n + b) = 1

Linear Support Vector Machine Standard Large-Margin Problem

Margin of Special Separating Hyperplane

max

b,w

margin(b,

w)

subject to every y

_n

(w

^T x _n

+

b)

> 0 margin(b,

w) =

min

n=1,...,N 1

kwk

y

_n

(w

^T x _n

+

b)

• w ^T x + b

=0 same as 3w

^T x + 3b

=0: scaling does not matter

• special

scaling: only consider separating (b,

w)

such that

n=1,...,N min y n (w ^T x _n + b) = 1

=⇒ margin(

b, w) = _kwk ¹

max

b,w 1 kwk

subject to

every y n (w ^T x _n + b) > 0 min

n=1,...,N y _n (w ^T x _n + b) = 1

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 14/28

Linear Support Vector Machine Standard Large-Margin Problem

Standard Large-Margin Hyperplane Problem

max

b,w

1

k

w

k subject to

min

n=1,...,N y n (w ^T x _n + b) = 1

necessary constraints: y

_n

(w

^T x _n

+

b)

≥ 1 for all n

original constraint:

min _n=1,...,N y n (w ^T x n + b) = 1

want: optimal (b,

w) here (inside)

if optimal (b,

w)

outside, e.g. y

_n

(w

^T x _n

+

b)

>

1.126

for all n

—can scale (b,

w)

to “more optimal” (

_1.126 ^b

,

_1.126 ^w

)

(contradiction!)

final change: max =⇒ min, remove√

w

, add

¹ ₂

min

b,w

1 2 w ^T w

subject to y

_n

(w

^T x _n

+

b)

≥ 1

for all n

Linear Support Vector Machine Standard Large-Margin Problem

Standard Large-Margin Hyperplane Problem

max

b,w

1

k

w

k subject to

min

n=1,...,N y n (w ^T x _n + b) = 1

necessary constraints: y

_n

(w

^T x _n

+

b)

≥ 1 for all n original constraint:

min _n=1,...,N y n (w ^T x n + b) = 1

want: optimal (b,

w) here (inside)

if optimal (b,

w)

outside, e.g. y

_n

(w

^T x _n

+

b)

>

1.126

for all n

—can scale (b,

w)

to “more optimal” (

_1.126 ^b

,

_1.126 ^w

)

(contradiction!)

final change: max =⇒ min, remove√

w

, add

¹ ₂

min

b,w

1 2 w ^T w

subject to y

_n

(w

^T x _n

+

b)

≥ 1

for all n

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 15/28

Linear Support Vector Machine Standard Large-Margin Problem

Standard Large-Margin Hyperplane Problem

max

b,w

1

k

w

k subject to

min

n=1,...,N y n (w ^T x _n + b) = 1

necessary constraints: y

_n

(w

^T x _n

+

b)

≥ 1 for all n original constraint:

min _n=1,...,N y n (w ^T x n + b) = 1

want: optimal (b,

w) here (inside)

if optimal (b,

w)

outside, e.g. y

_n

(w

^T x _n

+

b)

>

1.126

for all n

—can scale (b,

w)

to “more optimal” (

_1.126 ^b

,

_1.126 ^w

)

(contradiction!)

final change: max =⇒ min, remove√

w

, add

¹ ₂

min

b,w

1 2 w ^T w

subject to y

_n

(w

^T x _n

+

b)

≥ 1

for all n

Linear Support Vector Machine Standard Large-Margin Problem

Standard Large-Margin Hyperplane Problem

max

b,w

1

k

w

k subject to

min

n=1,...,N y n (w ^T x _n + b) = 1

necessary constraints: y

_n

(w

^T x _n

+

b)

≥ 1 for all n original constraint:

min _n=1,...,N y n (w ^T x n + b) = 1

want: optimal (b,

w) here (inside)

if optimal (b,

w)

outside,

e.g. y

_n

(w

^T x _n

+

b)

>

1.126

for all n

—can scale (b,

w)

to “more optimal” (

_1.126 ^b

,

_1.126 ^w

)

(contradiction!)

final change: max =⇒ min, remove√

w

, add

¹ ₂

min

b,w

1 2 w ^T w

subject to y

_n

(w

^T x _n

+

b)

≥ 1

for all n

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 15/28

Linear Support Vector Machine Standard Large-Margin Problem

Standard Large-Margin Hyperplane Problem

max

b,w

1

k

w

k subject to

min

n=1,...,N y n (w ^T x _n + b) = 1

necessary constraints: y

_n

(w

^T x _n

+

b)

≥ 1 for all n original constraint:

min _n=1,...,N y n (w ^T x n + b) = 1

want: optimal (b,

w) here (inside)

if optimal (b,

w)

outside, e.g. y

_n

(w

^T x _n

+

b)

>

1.126

for all n

—can scale (b,

w)

to “more optimal” (

_1.126 ^b

,

_1.126 ^w

)

(contradiction!)

final change: max =⇒ min, remove√

w

, add

¹ ₂

min

b,w

1 2 w ^T w

subject to y

_n

(w

^T x _n

+

b)

≥ 1

for all n

Linear Support Vector Machine Standard Large-Margin Problem

Standard Large-Margin Hyperplane Problem

max

b,w

1

k

w

k subject to

min

n=1,...,N y n (w ^T x _n + b) = 1

necessary constraints: y

_n

(w

^T x _n

+

b)

≥ 1 for all n original constraint:

min _n=1,...,N y n (w ^T x n + b) = 1

want: optimal (b,

w) here (inside)

if optimal (b,

w)

outside, e.g. y

_n

(w

^T x _n

+

b)

> 1.126 for all n

—can scale (b,

w)

to “more optimal” (

_1.126 ^b

,

_1.126 ^w

)

(contradiction!)

final change: max =⇒ min, remove√

w

, add

¹ ₂

min

b,w

1 2 w ^T w

subject to y

_n

(w

^T x _n

+

b)

≥ 1

for all n

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 15/28

Linear Support Vector Machine Standard Large-Margin Problem

Standard Large-Margin Hyperplane Problem

max

b,w

1

k

w

k subject to

min

n=1,...,N y n (w ^T x _n + b) = 1

necessary constraints: y

_n

(w

^T x _n

+

b)

≥ 1 for all n original constraint:

min _n=1,...,N y n (w ^T x n + b) = 1

want: optimal (b,

w) here (inside)

if optimal (b,

w)

outside, e.g. y

_n

(w

^T x _n

+

b)

> 1.126 for all n

—can scale (b,

w)

to “more optimal” (

_1.126 ^b

,

_1.126 ^w

)

(contradiction!)

final change: max =⇒ min, remove√

w

, add

¹ ₂

min

b,w

1 2 w ^T w

subject to y

_n

(w

^T x _n

+

b)

≥ 1

for all n

Linear Support Vector Machine Standard Large-Margin Problem

Standard Large-Margin Hyperplane Problem

max

b,w

1

k

w

k subject to

min

n=1,...,N y n (w ^T x _n + b) = 1

necessary constraints: y

_n

(w

^T x _n

+

b)

≥ 1 for all n original constraint:

min _n=1,...,N y n (w ^T x n + b) = 1

want: optimal (b,

w) here (inside)

if optimal (b,

w)

outside, e.g. y

_n

(w

^T x _n

+

b)

> 1.126 for all n

—can scale (b,

w)

to “more optimal” (

_1.126 ^b

,

_1.126 ^w

)

(contradiction!)

final change: max =⇒ min, remove√

w

, add

¹ ₂

min

b,w

1 2 w ^T w

subject to y

_n

(w

^T x _n

+

b)

≥ 1

for all n

Hsuan-Tien Lin (NTU CSIE) Machine Learning Techniques 15/28

Linear Support Vector Machine Standard Large-Margin Problem

Standard Large-Margin Hyperplane Problem

max

b,w

1

在文檔中 Machine Learning Techniques (ᘤᢈ) (頁 38-71)