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Lagrange Multipliers in PCA

1. Principal Component AnalysisPrincipal Component Analysis (PCA) l mt kthut c dng : Gim chiu dliu Nn mt mt thng tin Rt trch c trng Trc quan ha dliu tng:Chiu vung gc dliu ban u xung mt khng gian t chiu hn sao cho

phng sai ca dliu chiu l cc i.

2. Cc i ha phng sai dliu chiuCho trc dliu

1

N

nx , vi mixnc dchiu.

Mc ch: Chiu dliu trn xung khng gian mchiu, m

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Cho ta kt qu (*)

R rng biu thc (*) trn cho thy, u1phi l vector ring ca S.

Nhn 2 vca (*) bn tri vi u1Tv sdng rng buc u1

Tu1= 1 ta c :

Theo nh thy th phng sai c cc i ha khi u1bng vi vector ring c trring ln nht l 1

Tm 2rd PC :

Gisu2l vector ring th2 (u2phi cng di v vung gc vi u1).

tm PC th2 ta trli vi bi ton ti u nh sau:

Li p dng PP nhn tLagrange ta xt :

(1

,2

l cc nhn tLagrange)

Tng tgii bi ton trn ta c :

Kt qutrn hm rng u2ng l mt vector ring ca S vi trring ln thhai 2 . Nhng chiu

khc c a ra theo kt qugim dn ca gi trring.

Phng php nhn tLagrange l mt trong nhng kthut mnh gii quyt bi ton ti u.Lagrange kp ( Dual Lagrange) c cho l mnh hn cQuadratic Programming(QP). Tm quan

trng ca phng php nhn tLagrange trong Khoa hc my tnh ni chung v trong CVR ni ringl khng thphnhn.

References :

https://www.sec.in.tum.de/assets/lehre/ws0910/ml/slideslecture4.pdf

http://spider.ipac.caltech.edu/staff/fmasci/home/statistics_refs/PrincipalComponentAnalysis.pdf

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