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($%+&#$%( $#'3! #%#!# ' & *%! '&$'&! # $ *' # *'! '*# '%# #!2'5&!!%/1 *+! #$3'%#$! * '$F#=%* '%# $!$&! E ' 1 $%# $!!/$" # %6* '%#!F6%#! * '$ F#=%* '%# ' %% #' '!"K(--9I #! $ F&#*%! F& '=%Y! & %#' &% '+&#%*!*#' *%2###&!! ' 1 #$ %*%$ '5&"''6%#'+ &&#!# '%# &*%&*%! &# 1!%*#' 2!% &'%&!F #$ =='! '%3 $%** #! *'% " '' %6* '%# *' F%+'# &# *%!!# %' &6=='! '%'$%#&'E&#6%# #!2'5&! * '$F#=%* '%# "'%&''!"K,)),I%#'! &''#&!1!%*#'! * '$##'%& #' #!1$'& *'E'* σ#' ' σ $ $' #'!*%!F&&!!" 1!%*#'%#' ' 1!& *&! '%# &&# '&!! * $%$#'5& %&! *%!$&!!F#%6 #K'%&'3,)),M'%&''#'3,))7I" * '$F#=%* '%#%&! '%#$%*%$%**! %**8* '$ F#=%* '%#!*#' > KΨ3Ξ I = 8 = ( KΨ3ξ I!%'%$%!%&! '%#'=#!F#*+! %'%$%!!*#' $ $&# #1& Ξ = {, }!!&'F$ M %&" F Q q= 1 ξ,ξ N 1 "&+#3 ( Ψ, Ξ) = N M ( Ψ, ξ )!%'%$%!%&! '%#'=# < q F q %&1!%%#!F6%#! * '$F#=%* '%#%&&#%'%$%!!*#' ξ K!F#$ ' #' &* #! ' $'' $'%# =# F!!!F$'&I" %&!%#5&!1$'&$%1 +!'%## D" * '$' #=#, 3 KΨM I. 3%Y 3 KΨM I'!!%/1 *+! #$&1$'&%+1 '%# y %& I Ψ Ψ! 1$'& *' %&! '%# Ψ " %&!# %#! *%! K Kβ33II3 ξi # ==$'& #'&#1!%*#' 2!% &*%M$1!%*#''==$'& &'%&! *%2## =='! '% 3 1!& 5& ' %#$! E )" =# '#! $%*!6' $!$&!3 #%& #F!5&%# $''!# '%# & *%! F&M $'' '$'%# ' F!!& *! E $!! ==$'& =&' # " *%! ' ''5&F$'!%> JK Kβ33II3 ξi K Kβ3)3II3 ξ I + K )I + εkσ #' + σ K Kβ3)3II3 ξii = ) 20

($%+&#$%( '!!%/1 *+! #$3' %$ > (, 3 KΨM I!#, π +!# 1 + K. I1 K. I %Y.'1%#'!F #$* #!'! 1 #$* #!%## >. K I I =. K Kβ3)I3 ξi, JK Kβ33II3 ξi K Kβ3I3 ξi 1 K II 1 = Ω + σ + σ β ξ #' K K 3)3II3 I J = ) = ) 1 '%#!!%/1 *+! #$ %' &6 *' %&! '%# $%#&#'!%E!F6%#! * '$F#=%* '%#!*#' #+!%$> K. 31 I 2 K. 31 I ( KΨ3 ξi, 2 K. 31 I ) K. 31 I %Y K.1I)K.1I ' 2K.1I %#' +!%$* '$! # #'!F #$. '! 1 #$1%+1 '%#> ( ( ( ( K. 31 I ), 1 1 1.. 1 1 = + β β β β 1$' = (3 3 *Kβ I 1 1 ) K. 31 I = 1 1 λ λ ( ( ( ) 1 1 2 K. 31 I = 1 1 λ β ( ( ( ) 1$' = (3 3 *Kλ I 1$ m = 1,, dim( β ) ' = (3 3 *Kλ I #!' 1 &6 $' $**#'3! +!%$ 2 K. 31 I! * '$F#=%* '%# ' &% #&!3 "! # #$! 1 #$ %+1 '%# #! *' *%2##F' #$%*'"''*'% #! &'! '!DE" F6%#! * '$ $%*!' KD JI '!& $%*!6 & ='!F#'%&$'%# 1$%#&*%!" $%* %#$&6*'%#F 1! &$&#==#$ *%' #' #!F1!& '%#&66*!K'%&''#'3,))7I" F6%#! * '$ F#=%* '%# ' '#& E *%!'# #' $%*'!F#=!&#$$%1 +!&! * $%$#'5&'F&#1 +!'#' /%$$ %# &$%&==#'%' '*#'K'%&''#'3,))7I" #$*O*' 1!3!%#' &%%&#!# '%#'23$F'E&#!# '%#&*%! &'%& 21

($%+&#$%( 1!&#1&!!=='! '%"&$&# *!% '%##F '*%#' 1$$'' # %$35&%##&!' '*! E$!!1!%$**#'* %& &#'* $!$&!!&!1 & =' *&! '%# %#'!%" # =='3 #! $ F&# #!2%##3!%##$ 5&#1&%#'#$%##&'&#'* '%#*5& + 2###&'%#$ O'==$'&"#1!& '%# ( ( Ψ ξ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

($%+&#$%(!& '' 1!& %! * '$ F#=%* '%# "!& &'! # * $%$#'5& ' $!& /%'*!'3 5& *' *#*!F#1 & '*# #'! * '$ F#=%* '%# %&! '%#" # '5&3! $!$&! $ $' # #$' F#1%#* '$!! &5&*#*!F#1 & '*# #'! * '$ F#=%* '%#1#'E* 6*!'*# #'! * '$"!%'* F%'* '%# %#' #$ " #%*+&6!%'* %#' ' 1!%3 $%#& #' %' E!F%'* '%# F&# %'%$%! ' #'3 %' E!F%'* '%# F&# %'%$%!'' ''5&K%&$%#'#&I" %** $' $**#'3 &# %'%$%! Ξ %& &# '& %&! '%# # * $%$#'5& &' F$ %&! =%* Ξ = { } ξ,, 1 ξn %Y$ 5& ξi $%%# &6 %'%$%!!1*#'==$'&&!&B'3 i = 1,, N "! *O*=P%#3#%& 1%# *%#'5&F!&'F6*%&! =%* Q%& Nq &B' 2 #'!*O*%'%$%! { ξ, 1 N ; 1 ξ, ; ;, 2 N2 ξ Q N Q } Ξ =!*#' > [ ] [ ] &' # ''+&E$ 5&%&&#% '%##! m &B' mq 2 #'!*O*%'%$%! ξq''!5& {[ ξ, 1 m1 ] ; [ ξ, 2 m2] ; ; ξ, Q mq} " #!!% F&# %'%$%! #'" # Q i= 1 m q N q q = 3$%%# #'E! %%'%# N = 1" Ξ &'%#$F$%&! =%* Ξ = '=# #&#%'%$%!' ''5&"#&''#$ %'%$%!' ''5&E* 5&!$%#5&$%*#')'('1= #''%&B%& Q i= 1 m = 1" %'%$%! ' *#'! +! &5&#%&#F 1%# &##%*+ mq q #'"! =&'!% *# E&# %'%$%! #' # %$ #'! %%'%# E!F#' &&!!&%$" F&#%'%$%!' ''5& &%'%$%!#'##&# 'F==$ $'&%'%$%!5&'##!''=+!" N #!%'* 'E$'' %$'!F!%'*&%%1/^ 2##K%%13(-9,M ^2##3(-9,I31!%#'!*#' #&#$%#'6'%##%#!# $! 5&& '#& &6 *%! #%#!# E ==' *6' #' '!" K(--9I"! *'! $%#'&$'%#%'%$%!$%#'#&/%'* &6"'!%'*$%#11&#%'%$%!/ %'*!5&!5&%'!%'%$%!#'!"!&'$# #'O'!%#%& ''#! $%#1#$ $!&'!!F#*+!%'%$%!!*#' %+! &$%&' '%#" %'%# ζ 3!!' $ %'%$%!!*#' 5& &1#' O' $%*% F&##%*+!1*#' ==#'!%#!$%#' #'/=#" F!%'*&%%1/^2##'' '="%'3 Ξk!%'%$%!%+'#&E!F' '%#B"! * F 'E!F' '%#BKE$%#'&!%'%$%! Ξk + 1'!5& Ξ = ( 1 α ) Ξ + α ξ 1$ α k + 1 k + 1 k + 1 k k + 1 23

($%+&#$%( * $%*#')'(' ξ # ζ 3&#%'%$%!!*#' E B%&' &%'%$%!3 &5&!&# * α k + 1' ''+&#' #'&#= $'%#* ''+& &6 &'%'%$%!!*#' $%*% #' Ξ k " %'%$%!!*#' ξ E = #' #! %'%$%! * $%%# #''$!&%&!5&!! ' #$ d ( Ξk, ξ ) '* 6*!3$5&1#'E$$ ξ * '!5& ξ arg max * d ( k, ξ ) ( ( )) log det M F ( Ψ, Ξk + 1 ) =# α = Ξ " ξ ζ k + 1 αk+ 1= 0 F!%'* & %%1/^ 2## ' + &! '%* F5&1!#$ R= ' ^%!=%C'?K(-8-I5&' +!'!F5&1!#$#'! '%#&1 #'> -!%'%$%! Ξ'/%'*! - max ξ ζ d ( Ξ, ξ ) = P 1$ P!#%*+ *'E'*&*%! - Ξ*#* max ξ ζ d ( Ξ, ξ ) #3! *%#' 5&F&# %'%$%! %'*! %&!F'* '%# *' & %'5&1!#'%&! 1 #$! $'%#" '&$'&' '1!F!%'*'!%! &1 #'> (" %&#%'%$%!' ''5&#'! Ξ0 #%##","!F' B3 1$!%'%$%! Ξk 3'%&1 ξ = arg max ξ ζ d ( Ξk, ξ ) O' max ξ ζ d ( Ξk, ξ ) P + ε 1$ ε 13&#&!'%! #$'*#" * 7" #%#3 $'&!!%'%$%! Ξ k + 1 = ( 1 αk + 1 ) Ξ k + αk + 1ξ 3%Y α k + 1'$%& ] 0,1[ '5& * 6* det ( M F ( Ψ, Ξk + 1 )) 3$5&$%%#E α = P d * d ( Ξk, ξ ) P ( ( Ξk, ξ ) 1) k + 1 * '!%'*%'* #!#%*+<%'%$%!!*#' '!%%'%# &B'#$!& #$ 5&%'%$%!!*#' ξq&%'%$%!%&! '%#%'*! Ξ"$ ='3!*'! '*# '%#F&#%'%$%!%&! '%##!&=# #'&#'&$'&3$F' E&#+%##+! #$#'!#%*+%& Q E#$!&3!#%*+ nq!1*#' %&'#=#!#%*+&B'E#$!& #$ 5&%&" #1 #$3%&!F%'* '%#%'%$%!#'3!%'*$=5&%#'' 1!%3 5& #! $ %'%$%! /%'* &63 %#'!%'* F$ #" & #$' #' 'F&#%'%$%!#'!'!F *!%&$$1*#'#$ # #'&# '*!1*#' &# *!!& '*" 5& &' O'!*' #' 5& #! $ #*#'!&&'*#*&!' #'&#*!!&%!&'%#K%%13(-9,M^ 2##3 (-9,M'$!!3(-9;I"!'# '1%#''%%+ &!*O*#$3$# #' &$&#$!%'*# #''! $%#1#$1!%'%$%!%'*!" " 24

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