A Study on Piecewise Polynomial Smooth
Approximation to the Plus Function
Linkai Luo, Chengde Lin,Hong Peng,and Qifeng Zhou
Department of Automation,
Xiamen University, China. 361005
Tel: (86)-592-2580181
e-mail: [email protected]
Content
 A introduction to smooth support vector
machine (SSVM)
 The smooth approximation to the plus
function
 The best piecewise polynomial smooth
approximation to the plus function
 Conclusions
The end
Smooth support vector machine
(SSVM)
1
min  (1  D( Aw  1 )) 
w, 2
•
•
1 T
 (w w   2 )
2
2
2
It is a strongly convex minimization problem
without any constraints
The plus function ( x)  max{0, x} must be
approximated by some smooth functions
The smooth approximation to the plus function
The smooth
approximation
function
-1/k
The Plus function
1/k
• The smooth approximation functions
– The integral of sigmoid function
– The piecewise polynomial functions
– ……
The best piecewise polynomial smooth
approximation to the plus function
• The first-order smooth approximation
• The second-order smooth approximation
• The comparisons with previous results
The first-order smooth approximation
Result:
p1 ( x)  (kx  1) 2 ( 161 kx 2  81 x  163k ), 
11
R( p1 )  1008
k 3
1
1
x
k
k
The second-order smooth approximation
 x,

p( x, k )   L6 ( x),
 0,

Result:
x
1
k
 1k  x 
1
k
x   1k
1
1
1
( kx  1)3 ( k 2 x 3  3kx 2  x  k5 ),   x 
32
k
k
R ( p2 )  0.0064k 3
p2 ( x) 
The comparisons with previous results(1)
f1(x)
The error
under 2-norm
1
40
The error
under infinitynorm
1
4
k 3
k 1
p1(x)
p2(x)
3
11
k
1008
3
16
k 1
k
1
1 1
1
f1 ( x)  x 2  x  ,   x 
4
2 4k k
k
• Where
polynomial function in [7]
0.0064k 3
5
32
is the
k 1
The comparisons with previous results(2)
Conclusions
• We formulate the standard piecewise polynomial
smooth approximation problems to the plus function.
In the proof of the existence and uniqueness of the
solution for these problems, their analytic solutions
are obtained.
• We claim that our piecewise polynomial functions
achieve a better approximation performance than
previous results.
• The smooth approximation with higher level to the
plus function, can be carried out by our method.
Thank you！