Tensor SVD I Matrix Ranh EfR - Statistical Sciencepdh10/Teaching/790/notes-tsvd.pdf · Tensor SVD I...
Transcript of Tensor SVD I Matrix Ranh EfR - Statistical Sciencepdh10/Teaching/790/notes-tsvd.pdf · Tensor SVD I...
Tensor SVD I
Matrix Ranh l f. cot definition ),
and extension
M , XML
M E IR m ,> m2
Det in c- Rm ' ' n '
is rank - l it M -- abt =
aob,
some a e MY{ 03 b EIR"
-1203.
Oct
rank.gritRis the smallest number for whichRM = I anbnt
.Ef=L
The Svo recovers the rank :
me nor's
= £2!dnrlnvnt,
d . is das.
- -- 3 due 30 ,
so rank I M ) - # nonzero dais
Extension to Tensors
Det : Merman ' im, is rank - I if I a c- AT'll 03
,bElR%of
,
C e 112^3/503,
St.
M = a oboe
recent :(cab on a ) firm" 'm )
Moja : aibjcn
Oed : M is
rank.grit
Ris the smallest number for which
R
M =
,
ar O Br Ocr,
thiju= Eairbje Car
,Vec ( M ) : E @Gbr Ga
, )
②One notion of a tensor Sro a based on such a decomposition
M " ¥2,
dry , of . ow,
Curia .- vive sw.tw . =L )
W ' WN
= d,
"t
. . -
t d .
w
Uy Us
This representation is culled the "CP
"
decomposition
( PARAFAC ' . Harshman ( 1970 )
CANOE COMP ! Carol & Chang C 1970 ))
ate , x Rank C 2nd def ) and Extension
Det : The column tanh of a Mattix M is din Ispan c.mall) I
Out : The row cant is dim Cspan ( Man .. . .
Mom;D)
Thy : row rank = Col rank .
Sro recovers bases of coward spaces ,and ranks .
M = NOV"
i Me ,j3= Ug,d. v. j t - - - t
Ucgmisdmzvnsj
⇒ columns of form oithoy .basis to '
M I, . - -
M ML.
Similarly ,
'volume of V fam oefhnz .basis Ki Mais . - - msn.is
Deane SVO from bases
① obtain V = ortho y basis ten span ( Mcm - . . .nl ,nil )
✓ = n Span ( Me , ,s . - .
MI min
)
② let D= UTMV C works it U,
V ace oedeeod interns of
events of MMT and MTM )
Extension to tensors
Xt
Idea : Obtain SVO fo , MEIN' " "
as follows ,
① let Ui ,= oethrg basis for cols of Me ,
= Sro ( Ma , ) $ u
R ,= dim C span C cats of Mi , ) ) : matrix tank of My ,
:I
Ye,-
01 thug basis do , cols of Mes )
s SVO ( M$17 $1 u
Rs " mate , x rank of My )
Def.
The multihull tank of M is ( R, ,Rr
,Ks )
④
They ace not eyuw in general ! !
Higher ode , SVO
wt s -
- mxlui.ui.us }
s- Luisauioui ) in
Then S c- item '× m " "
is the "core array
"
of M
Also,
Cuz On Ur GU , ) s = th
S " { U . .lk ,Nhs ) = M
This is called the ' '
higher order SVO"
o , HOSVD ,
The core allay S satisfy some properties !
u" all . orthogonality
"
u occludingu mode - specific singular valve recovery .
Truncated is VO
Model : Y = Mt E,
rank I m ) = ( R, . .
ikn )
-
Oy ! I = acy on , -
It Y - MITM ! C huh ? R
, -- Rn
⑤M hus n - rank I R
, ,R , ,Rs ) it
Ms S x { U, ,Uz,U , } Unearth""
,
setter ' 'm ^R3
U3
u . FITUr
Estates : ①"
truncated"
Hosvo - not 0251M LE
② "
Hoo I' '
stealthily Update Mi,
Us,
Us ,S
.
→
conveyorto MLEIOLS ,