MATH 182 : HIDDEN DATAIN RANDOM MATRICES

Todd Kemp ,APM 5202

www.math.ucsd.edu/ntkemp/l82

TODAY . Spiked Covariance ModelsNEXT :

'

outliers

Labs 4- 6 Due Friday,March 13

Homework 3 Due TONIGHT

Final Lab Project Due Monday , March 16

Final Exam Friday , March 20 lithe - 2:30,this room)

Feature matrix X E Mmm C = d- XXT mxm

f- my feature,

t samples .

gw -

- fxTX AN

tf u

Pure noise : Eigenvalues of W, f l"

tho) ④€=µµu, %gmah" of C µG#ga ,

ftp.VK.hfft#IDg..pd"

eisetgnal Ej Iid. Me

.

Im)og. = g- t ?, Centered

k ; = g. t ?g - t,!f # t- bynoe by LLN( = Is this ttiijd.

↳often,we're thinking of ¥ , as Samp↳ ⇒ g.

= g,i. I. -- I . : % - I -- e .

Just have nose !

Noisetlhoisysigheoj= Gj t Bj Gj,By idp.

P T environmental noisesignal+structural noise

.

Nce.fm)

NHC)&

Gv = Gvkgjltced = CTI.

"

per!Yb§¥mItc

x.j n Nce , Itg ) expect rankle)is small

.

ko - rank perturbation

Spiked Guarani model Iof rank r)M

X -- ha.

.

. -il Me, Itc )

"""g"" """ " "

C is peseta semidefinite i. symmetric,

C - QDat QE 0cm)

D= In:*:/ → visas. an:pIt C '

- It QDQT = QQTTQDQT = QC It D)QT

Proposition If Ir Nce,ID ,

QE Olm) ,

a'I n Nk, Ind .

fgi.at : if x. iiiiiidtb Gainsaying !!:&,Pf

.Ceuta

tact : GHAZI = FAT i. If A-QI-

Gula = at QQIf Guffy. I T -

- I.

H

Theory : The distribution of eigenvalues of

*↳ x.s.sx.nl X'X X;- Nce

,Itc)

and X"X' = ELIADE Xjr Ale , ItD)

are the same.

Pf.

Construct Xj is Xj -- AZ's where Atc.

Itc = QLITDIQTA - QCVITDTQT i . AAT '-A2=QItDQT

= Itc .-

X -- f%z, .AZ#--,AZnI= AZ ItDfZATA '- Itci

. xtx -- ZTATAZ III. tag Ijaz = zJcItDaTE

i.

The most general spiked avwrnu medelis ( as far as eigenvalues of XTX are concerned)

X '- Hi,

-, Hl Xj -Ale, ItD)

Tdiagonal .6

we = txtx n 's El#DIZ

Now restart our Wishart matrix analysis?Thy : The hotel -gram of eigenvalues of Wo

( m, a →es, F-→g , r freed )is asymptotically the same as We

.

tr,( WI ) - I Lwclii

WE's CITTDZLxiii''¥. ha.!sii÷÷i÷.⇐⇒. ...

trulwo-trlwd-t.si?raholaisteer IT 'T Trtems

→ o.