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 ,