ứng dụng SVM cho bài toán phân lớp nhận dạng.
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Proceedings of ICT.rda'06. Hanoi May. 20-21.2006
Ky ylu HQi thto ICT.rda'06
IJNG DVNG SVM CHO BAI TOAN PHAN L 6 P N H ^ DANG
Phpm Anh Phuomg, Ngd Quoc Tao, Liromg Chi Mai
Tdm tat:
Trong bdi bdo ndy, chung Idi gi&i thiiu mgt hu&ng tiip can cho bdi lodn phdn l&p nhdn
dgng mdu dua tren md hinh SVM. Phuang phdp hgc mdy ndy dd dugc Vapnick nghiin dm
tir cudi thgp ky 70 vd hiin na^ dang dugc dp dung rdng rdi Irong ITnh vuc nhdn dgng. Cdc
kit qud thuc nghiim cho thdy, SVM cho kit qud phdn l&p khd chinh xdc vd dugc coi Id
phuang phdp c6 hiiu qud cd the so sdnh v&i cdc phuang phdp hgc mdy khdc nhu mgng na
ron, HMM...
Tit khda: Suppori Vector Machines, margin, feature space, kernelfunction.
2. SVM CHO BAI TOAN PHAN LdP NHf
PHAN
1. Gidl THIEU
Support Vector Machines (SVM) dupc
nghifn ciiru tir nhumg nim 60 vdi nhQng cdng
trinh ciia Vapnik vi Lemer (1963), Vapnik vi
Chervonenkis (1964). Co sd ciia SVM dya
fren nin ting ciia ly thuylt hpc thing kf, ly
tiiuylt vl so chilu VC (Vapnik-Chervonenkis)
di dupe phat triin qua 3 thpp ky bdi VapnikChervonenkis (1974) vi Vapnik (1982,1995)
[1,2,3]. Mii cho din din nhiing nim gin day
thi ly thuylt niy mdi cd nhihig bude phit triin
mpnh mg (Burges, 1996 [4]; Osuma, 1997
[6,7]; Piatt, 1998 [9]) vi nd frd thanh mpt
cdng cy khi mpnhfrongnhilu ung dyng nhu:
nh$n dpng chii vilt (Joachims, 1999 [13];
Nguyen Dire Dung, 2005 [25]), nhpn dpng
m§t ngudi (Osuna, 1997 [7])...
Khac vdi miy hpc tuyln tinh, y tudng
chinh ciia phuang phap niy li tim mpt sif u
phing phan cich sao cho khoing each
(margin) giiia hai tpp dpt c\fc dpi. Dk giii
bii toin niy, chiing ta cin nim mpt sd khii
mifm: margin, phan ldp mem (soft
classifier), vector hd trp (support vector),
khdng gian djc trung (feature space), him
nhan (kernel function).
Trong bii bio niy, chiing tdi chi tpp trung
vio bii toin phan ldp nhj phan, cii dpt thu
nghifm, sau do dl xuit hudng cii tiln dl nang
cao tic dp huin luyfn.
2.1. SVM tuyen tinh
Cho N mlu {(xi,yi),...,(xN,yN)} trong
dd xieR"^ vi yj€{±l}. Tim mpt sifu phing
phan cich:
f(x) = sgn(w.x + b)
dk phan tich tpp mau trfn thinh hai ldp
sao cho khoing cich (margin) phan chia giiia
hai ldp dpt cyc dpi. Tire li, ta mulntimmdt
sifu phing H: y=w.x + b = 0 vi hai sifu
phing song song cich diu:
H,:y=w.x + b = +l
Hj: y=w.x + b = -1
sao cho khdng cd dilm nio nim giiia H|
vi Hj, ding thdi khoing cich giiia H| vi H2
dpt eye dpi (hinh 1).
393
Ol^
Hinh 2.1. Sieu phdng phdn cdch tuyen tinh,
cdc "vector ho trg" dugc khoanh tron.
Kyyiu HQI thto ICT.rda'06
Proceedings of ICTroauB. Hanoi May, zu-zi, z w
Doi vdi mpi mit phing phan each H vi
cic mit phing H,, H2 tuomg irng, ta ludn ludn
cd thi "diiu chinh" vector w sao cho Hi se la
y=w.x + b = +1 va H2 sf la y=w.x + b = -1.
Chung minh: xem [8].
D I cyc dpi khoang each giiia Hi va H2, ta
phai dya vao cac mlu nim tren Hi va H2. Cac
mau nay dupc gpi la vector hi trgf (support
vector) bdi vi chi cd chiing mdi tham gia vao
vifc xac djnh sieu phin^ phan each, cdn cic
mau khac chiing ta cd the bd qua.
Ta cd khoing cich ciia mpt diem nam
^L _
aiv ~ °
N
^ "^ T^^^ii-Vi^i) = ^
isl
N
ow=5]a.y.Xi
(3)
i=l
—=0
^b
yay=0
T\ ' '
.
,
Thay (3) va (4) vao (2). ta cd:
N
1 V^ V^
M ' 2"^^
'
tren Hi tai H la
=
suy ra
II ^11
II ^ II
,..
' u - u 'ui'c
2
khoang cach giua Hi va H2 la 5 =
.
(4)
' '
vTa cd L(w, b, a) > e(a) nfn tiiay^i
giai bai toan (*), ta se giai bai toan doi ngli
^ ^ • ui £./ wu
--.*- 1 ^
cue dai him 8(a) theo or, vai dieu kien Oi >
IIY^II
N
Do dd, df cyc dai 5, ta se cyc tieu
||w||=<w.w> vdi dieu kifn yj(w.Xi + b) >1.
Tir dd, bai toan cd the phat bieu lai nhu
sau:
J
min— < w.w > sao cho yi(w.Xi + b) >1
"• ^
^ ''
2.2. Bii toan dii ngiu
va y.ajyj =0.
i=i
O day chung ta cd cac hf sd Largrin
0^ tuang irng vdi mdi mau hpc. Sau khi d
huan luyfn xong, nhirng mau ed 04 > 0 dup
gpi la "vecta hd tra", tat ca cac mlu hpc kha
cd 05 = 0 thi nam ve hai phia ciia hai sif
phang Hi va H2.
^ ' ^^ ^^"^ ^^^^ ^^^ °i' *» cd till tin
Xft him Lagrange:
^=^^yiXi
L(w, b, a) = f(w, b) + ^a.gj(w,b)
'='
(1)
frong dd
va ngu&ng b.
Mpt doi tupng mdi x se dupc phan ldp be
ham myc tieu:
f(x)= sgn(w.x + b)
f(w,6)=l<w.w>
2
=sgn((Y^a,y,.J..
+ b)
N
vagi(w,6)=l-yi(<w.Xi> + 6).
Xet tiep ham:
e(a)
r1
= minL(w,b,a)=
*ã'ã
ô
rTiinl-<w.w>+ 2]a,(l-y,(<w.x, > + b))
, ^^^
Lay dpo ham theo hai bien w va b ta co-
=sgn(Y,a,y., <x,.x> + b)
i=i
\
(6)
Trudc khi di vio chi tilt dl giii bi
toin qui hopch loi nay, chung ta md rpng n
theo hai hudng: SVM phi tuyln vi phan Id
mlm.
2.3. SVM phi tuyen
x
Neu mit phan cich khdng phii li tuyl
tinh, ta cd the anh xp cac diem diJ lifu vi
394
Proceedings of ICT.rda'06. Hanoi May. 20-21.2006
Ky ylu HQi thto ICT.rda'06
mdt khdng gian khic vdi so chilu cao hom sao 2.4. Phan ldp mem
cho cic dilm diT lifu niy s€ tich dupc tuyln
tinh. Cho inh xp biln doi la <!>(.),frongkhdng
gian mdi vdi so chieu cao ban, thi:
N
I
N
0 ( a ) = X « i -TZ«.«jyiyj*^(*i)*^(''j) (^)
i-l
2 ,j
Gii sir 0(Xi).0(Xj) = K(Xi,Xj). Nghia
li, tich vd hudng trong khdng gian mdi tuong
ducmg vdi mpt ham nhan (kernel) ciia khdng
gian diu vio. Vi vpy, ta khdng cin phii tfnh
tryc tilp tich vd hudng a>(xi).<I>(Xj) mi chi cin
tinh giin tilp thdng qua him nhan K(xj,Xj).
Cuoi eiing 6(a) se trd thinh:
0(a) = X a i - ^ Z a i a j y j j K ( x , . x , ) (8)
i=l
2 jj
Dinh ly Mercer, mpt ham K(x,y) cd thi
dupc sir dyng nhu mpt him nhan nlu:
- K phii doi xirng: K(x,y)=K(y,x) vdi
mpi x,yeR°.
- Ton tpi mpt anh xp O vi mdt khai
friln K(x,y) = ^^MMvX^ "^" ^^ chi neu
i
Hinh 2.2. Phdn lop mem.
Mpt hudng khic df md rpng SVM cho
vifc phan Idp la cho phep cd nhifu. NghTa li,
ta khdng hoin toan ep bupc dir lifu phii nim
vl hai phia ciia H| vi H2, nhung ta muon hpn
chf toi da cac diem di^ lifu nim giii'a H| vi
H2.
Ta ndi Idng diiu kifn phan Idp bing cich
them cac biln tri (slack variables) ^i>0 sao
cho
wjc,-i-b>-i-l-(Jj vdiyi = +l
wJCi+h<-l + 4i v<5riyi = -l
vdi mdi him g(x) sao cho \g(x) Mx hihi hpn
till |K(x,y)g(x)g(y)dxdy > 0.
Sau day li 4 him nhan thudng dupc siir
dyng:
^i>OV/
va ta cpng thfm vio him myc tifu mpt
dpi lupng phpt (penalizing term):
min-<w.w>+C(2]^i)"'
+ Him nhan da thirc (Polynominal):
K(x,y) = ((x.y)+^)'*
d e N, ^ € R
+ Him nhan Gauss (Gaussian RBFRadial Basic Function):
(9)
frong dd m thudng dupc ehpn li 1, nhu
vpy bai toin (*)frdthinh:
min < w.w > + C y Ê
ô.^.4 2
K(Xi,Xi)=e "ã'
tr'
(10)
+ Him nhan sigma (Sigmoidal):
K(x, y) = tanh(/cx . y - S)
K,S
eR
+ Him nhan tuyln tfnh (Linear):
K(x, y) = X. y
395
sacicho
yi(w.Xi + b) + ^ i ^ . SO
I SO,
1 1 ^^N
Proceedings of ICT.rda'06. Hanoi May. 20-21,200(
Ky y^u HQi thto ICT.rda'06
Tuy nhien, cic ky thupt niy chi phu hp|
doi vdi cic bii toin nhd hoic trong mpt s<
Lagrange sf la:
trudng hpp d$c bift, cdn doi v<5ri vifc huii
luyfn SVM thi s\r cin trd Idn nhit chinh 1;
L(w,b,^i;a) =
kich thudc ciia t|p huin luyfn khong ll. V
1 < w.w > +C J^, -Z".[yi(*^ + '')"^^' ~ 'l-£>*i^i ~ vifc luufriiima trpn Gram ddi hdi mpt khdnj
2
i.l
i.l
i-l
gian nhd bing binh phuang kich thudc ciia tp|
huan luyfn nen rit dl vupt qui khi ning liri
- < WW
. >+£(c-a.-^.)^,-(Xa^y.^ )w-(Z**iy.)*''*'Z«.
tru' ciia may tfnh khi tap huin luyfn qui Idn.
.1
i-i
Vdi cic hf so Lagrange a, da thirc
I
N
N
N
N
(**)
Ci ^ i va ^li dfu khdng xuit hifn trong bai
toan doi ngau:
N
I N
max0(a) = X ! « i — Z « * « , y i y j ^ ^ (^^)
i.l
2
i,j
sao cho:
N
Trong so nhihig thupt toin thdng dyn
dupc thiet kl cho vifc huin luyfn SVM
chiing tdi se md ti tdm tit cic phuang phip d
dupc eung cip trong hiu hit umg dyng SVM
thuat toan chpt khuc (chunking), phan tic
(decomposition) [6] vi day toi uu cyc til
(SMO) [9].
3.1. Thuat toan chunking
Phuang phap Chunking bit dau vdi mj
tap con bit ky (chunk) cua tpp dir lieu hui
De huan luyen SVM, chiing ta can tim luyfn, sau do huan luyfn SVM theo mi
cac Q!i thdng qua mien xac djnh cua bai toan phuang an toi uu tren chunk du lifu vira chpi
doi ngau de cue dai ham muc tieu. Giai phap Tifp din, thuat toan giii' Ipi cac vector hd tr
toi uu nay cd the dupc kiem tra bang each sir (cac mau co o^ > 0) tir chunk sau khi da loi
dyng dieu kifn KKT.
bd cac phin tiir khac (tuang iirng vdi 04 = 0) v
diing cac vector hd trp nay de kiem tra ci
2.5. Dieu kien KKT (Karush-Kuhn-Tuckcr) phan tiir trong phan cdn lai cua tpp dii lifi
Dieu kifn toi uu KKT cua bai toan (*"•) Phan tir nao vi pham dieu kifn KKT thi duq
bd sung vao tpp cac vector hd trp de tpo 1
la:
chunk mai. Cong viec nay dupc lip di l^p I9
• Dao ham ciia L(w,b,^;a) theo cac bien vifc khdi tao Ipi a cho mdi bai toan con mi
w,b,^ phai trift tieu.
phu thupc vao gia trj diu ra ciia trpng th.
trudc dd va tiep tyc toi uu bai toan con mi
• Vdi l < i < N ,
vdi cac tham so toi uu da dupc lya chp
a.(yi(w.x.+b)+^-l) = 0
(i2) Thupt toin se dimg Ipi khi thda man diiu kic
tdi uu. Chunk ciia du lifu tpi thdi dilm dar
Mi4i=0
(13) xet thudng dupc hieu nhu tap lam vi\
(working set). Kich thudc cua tpp lim vii
ludn thay ddi, nhung cuoi cimg nd bing ;
3. CAC PHUONG PHAP T 6 I l/U
lupng 0^ khac khdng (bing so lupng vector 1
AP DVNG CHO SVM
trp). Phuang phap nay dupc sir dyng vdi g
Trong phan trudc, chiing ta thay ring thiet ring ma tran Gram dung dl luu tich ^
huan luyen mpt SVM tuomg duang vdi vifc hudng cua tirng cip cac vector hd trp phii h<
giai bai toan qui hoach loi thda man cac rang vdi kfch thudc bp nhd (chiing ta cd the tfnh I
bupc tuyen tinh. Bai toan Max/Min cua mpt ma frpn Gram bat ciir luc nao khi thiy c;
ham nhieu bien da dupc nhilu ngudi nghien thiet, nhung dieu nay se lam giim toe dp hu;
ciru va hiu het cac ky thupt chuan cd thf ap luyfn). Trong thyc nghifm, cd thi xiy
dung tryc tiep cho vifc huan luyen SVM.
trudng hpp so lupng vector ho trp qui Id
0 ^
^va
X^iy* ^ ^ •
•XQf^
Ky yCu HQi thto ICT.rda'06
Proceedings of ICT.rda'06. Hanoi May. 20-21.2006
lim cho ma trpn kernel vupt qui khi ning luu
trii ciia miy tinh.
Thupt toin niy li trudng hpp d|u; bift ciia
thu|t toin Decomposition, tiirc li nd giii bii
toin qui hopch loi vdi kfch thudc t|p lim vifc
la 2 frong mdi bude lip. Uu dilm eiia thu|il
toan niy li cd thi giii bii toan tii uu bing
phuang phip giii tich.
3.2. Thuft toin decomposition
Phuong phip decomposition khic phyc
dupc khd khin ciia phuang phip Chunking
bing cich co djnh kich thudc cua bai toan con
(kich thudc ciia ma trpn Gram). Vi vpy tpi mpi
thdi dilm, mdt phin tiJr mdi dupc bo sung vio
/^p lorn vifc thi mpt phan tiir khic bj lopi ra.
Diiu niy cho phep SVM cd khi ning huin
luyfn vdi tpp dii' lifu Idn. Tuy nhifn, th\rc
nghifm cho thiy phuang phip niy hpi ty rit
chpm.
Trong thyc nghifm, ta cd thi chpn vii
mlu dl bo sung vi lopi bd tijr bai toan con dl
ting toe dp hpi ty. Thupt toin niy dupc trinh
biy tdm tit nhu sau:
Input:
- Tap S gdm N mlu huan luyfn
Thupt toan SMO tiiyc hifn hai cdng
vifc chinh: Giii bii toin toi uu eho hai nhin
tir Lagrange bing phuang phip giii tich vi
mpt phuang phip heuristic dl chpn hai nhin
tiir cho vifc tii uu.
4. KET QUA THirC N G H I | : M
Chiing tdi tiln hinh thyc nghifm vdi tpp
mau huin luyfn li 100 miu trong mit phing
(hinh 4.1) vi tap miu thiir nghifm li 71.
{(Xi,yi)}i=i....N
- Kich thudc cua Working Set la M.
Output:
Tap {oili^i N
1. Khdi tao
• Dit cic Oj = 0;
• Chpn Working Set B vdi kich thudc
M;
2. Tim nghifm toi iru
Repeat
• Giii bii toin toi uu cue bp tren B;
• Cpp nhpt Ipi B;
Until
Optimization)
Day li mpt thupt toin dom giin dl giii bai
toin quy hopch ldi SVM mdt cich nhanh
chdng mi khdng cin luu trir ma trpn Gram.
SMO phan tich bai toan quy hopch loi tong
quit thinh cic bii toin quy hopch loi con, sir
dyng djnh ly Osuma dl dim bio sy hpi ty.
397
(a) Tgp mdu hudn luyfn
(b) Tgp mdu thir nghifm
Hmh 4.1. Tgp da lifu thgc nghifm.
Proceedings of lCT.rda'06. Hanoi May. 20-21,2006
Ky y^u HQi thto ICT.rda'06
Qua bing I ta tiiiy dp chinh xic khi phan
ldp ciia him nhan tuyln tinh thudng thip hon
so vdi cac ham nhan da thiire vi him RBF.
4.1. Ket qui huan luyfn
Bang 4.2. Dp chinh xac phan Idp vdi eie gii
tri C khic nhau
(a) Phdn lap vai hdm tuyen tinh
Ham nhan
C=IO
C = 50
C=100
Tuyln tinh
Da thurc bac 2
74%
74%
74%
80%
82%
84%
RBF
86%
86%
87%
Bang 4.3. Dp ehinh xac phin Idp vdi bim
nhan da thirc v i him RBF
Ham da thiire
(C=10)
Bac 2
Bic 4
B|c5
80%
83%
84%
Ham RBF
(C=100)
0=10
o=15
0=20
86%
88%
91%
Bang 4.2 va 4.3 cho thay dp chfnh xic cua
vifc phan ldp phu thupc vao miirc dp ndi Idng
C cung nhu bpc cua da thirc vi hf sl gamma.
NIU bac cua da thiire vi gamma cing Idn thi
dp chfnh xac cang cao, tuy nhien thdi gian
huan luyfn se chpm hom.
(b) Phan lap vai hdm da thirc bgc 2
4.2. Ket qua nhpn dang
Bang 4.4. Ket qua nhpn dang vdi 100 mau
huan luyfn, 71 mau thir, C = 100
(c) Phdn lap vai hdm RBF
Hinh 4.2. Kit qud phdn lap vai C=10.
Ham nhan
s l vector ho
frp
DO chinh
xic
Tuyln tinh
58
80%
Da diiirc bpc 2
50
97%
RBF
47
96%
Bang 4.1. Ket qui phan ldp vdi C = 10
Ket qui d bing 4.4 eho thiy phuang phap
SVM nhpn dpng khi chfnh xac khi sii dyng
cic ham nhan RBF vi ham nhan da thurc.
Ham kernel
So vector ho
trp
DO chfnh
xac
Tuyen tinh
74%
Da thurc bic 2
61
61
80%
5. KET LUAN
RBF
50
86%
Md hinh SVM cd uu dilm la cho kit qui
nhpn dang kha chfnh xic. Tuy nhien, nd ciing
cd mpt so nhupc dilm khi tpp miu huin luyfn
Idn thi toe dp huin luyfn chpm, md hinh sau
398
Ky y^u HQI thto ICT.ixU'06
Proceedings of ICT.rda'06. Hanoi May. 20-21,2006
khi huin luyfn rit Idn, d$c bift li cd till vupt
Intemational Conference on Machine
Leaming (pp. 71-77). San Mateo, CA, 1996.
qui khi ning liru trii cua miy tfnh.
Kit qui huin luyfn phy thupc rit nhilu [5] B. Scholkopf. Support vector leaming, R.
Oldenbourg Veriag, Munich, 1997.
vio cic die trung ciia du' lifu dau vio. Trong
ci hai thupt toin phan tich vi SMO, khi kich [6] Osuma E.. Freund R.. Girosi F.. An Improved
thudc tpp mau huin luyfn Idn se dan din sy
Training Algorithm for Support Vector
gia ting sl lupng vector hd trp, neu so lupng
Machines, Proc IEEE NNSP '97, 1997.
vector ho trp vupt qui kich thudc cua tgp ldm
[7] Osuma E., Freund R., Girosi F., Support
vifc (ddi vdi thupt toin phan tach) hoic cache
Vector Machines: Training and Applicaltons,
(doi vdi thupt toin SMO) thi se inh hudng
1997.
din chit lupng ciia thupt toin huin luyfn.
[8] Christopher J. C. Burges, A Tutorial on
TIC dp hpi ty ciia thupt toin huin luyfn
Support Vector Machines for Pattern
SVM phy thupc vio phuang phip lya. chpn
Recognition, Data Mining and Knowledge
heuristic khi giii bai toan qui hopch ldi.
Discovery, 3(2), 1998.
Tdm Ipi, mau chdt ciia df tang dp chfnh [9] J. Piatt. Sequential Minimal Optimization: A
Fast Algorithm for Training Support Vector
xac hope toe dp nhpn dang theo md hinh SVM
Machines, Microsoft Research Technical
la phuang phap giai bai toan tdi uu ldi. Chiing
Report MSR-TR-98-I4. 1998.
tdi df xuat mpt so hudng nghien cuu de ting
toe dp hpi ty ho^c dp chfnh xac ciia md hinh [10] B. Scholkopf, C. Burges and A. Smola
SVM:
Introdution to support vector leaming
Advance
in kernel methods - Support vecloi
• Tim each de giam toi da so vector hd trp.
leaming, pages 1 - 22, MIT press, 1998.
• Cii tifn cache df cd thi xiir ly vdi sd lupng
[11] Steve R. Gunn, Support Vector Machines foi
mau huan luyfn Idn.
Classification and Regression, University o
• Lya chpn cic tham so vi ham nhan nhu the
Southampton,
1998.
nao dl md hinh SVM dpt hifu qui nhat
ciing la dieu dang quan tam.
[12] Schoelkopf B., Mika S., Burges C. J. C.
ã DĐc bift doi vdi bai toan nhpn dpng mlu thi
Knirsch P., Muller K., Ratsch G., & Smol:
A. J., Input space versus feature space h
khdng thi khdng ouan tam din vin dl trfch
kernel-based methods, IEEE Trans. Neura
chpn dpc trung. Neu trfch chpn die trung tdt
Networks,
10. 1000-1017, 1999.
se gdp phin nang cao chit lupng nhpn dpng.
[13] T. Joachim. Making large-scalar Svr.
leaming practical. Advances in kemt
method Support vector leaming
Cambridge, MA, 1999.
Tii lifu tham khao
[1]
V. Vapnik, Estimation of Dependences
Based on Empirical Data [in Russian].
Nauka, Moscow, 1979. (English translation:
Springer Veriag, New York, 1982).
[2]
V. Vapnik. The Nature of Statistical
Leaming Theory, Springer Verlag, 1995.
[3]
V. Vapnik, Structure of Statistical Leaming
Theory, Chapter in Uie Book: Computational
Leaming and Probabilistic Reasoning, ed. A.
Gammennan. pp. 33-41, John Wiley and
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