%&
! "
#
$ !! '( )
* *
+,-
&
* * * !
! '
+
+ ./! 3 & 0 ! + 1&
0 n & 0 ! ,
! "
& * ! 2
/!( *
&* * 345 6+7( !
3 #5 67 *
&* * 35 6+7 & ! !
3-5 *
38-5 !
& 67 / * !
0 9
( !
&* !
&
67 *
!& 35 ( 0 !& !
! &* !
& - 0 !
*
!
67 " !& ( 0 : ;
& * 0 8
* & * !!( /
! ! 6'7( & * 6<7( : 67( != 67( !
& 67 *
6'7( !
67 * * * 0 !
!( 0 * & !
;
8
!
& 0 !( 0 * 0
>
( & 0 ! n !
>
! ! ( &* !
!
!!
!
* *
&* ! & 0 !
2
( !
! !
! * * !
! !( &
0 ! ? & : @
* * ! 6+7 ( *
&* ! !
! !
* *
&* ! !
! !
& % 67 2( ! 0 * & 0
! !!
= 0 * *
* * !
! ! *
&* ! * * !
! & ' !
( 0 & 0 : !
*
!
!
&
!
' & ./!
* / + 0
8* ;
<
% A = {a1 , . . . , am } ⊂ d & : d * * ! 34..5( 0 * ! :
& A( E ⊂ d @ !
! !* (1 + η)!!
/( 0
η > 0( 4.. 6<7A
(A) ⊆ E, 4(E) ≤ (1 + η)4(4..(A)),
0
4(E) * ! E @*
? 6)7 !
! : @B * ! = : *
C*
!
4.. %
( ? 6+7
!
! : 6)7 :
!* !!
/ * ! ! 6<7 6)7
! E d : & d × d ! :
/ E c ∈ d 0
E = {x ∈ d |(x − c)T E(x − c) ≤ 1}.
35
* ! 6+7
4(E) = ρ (E − ),
0
ρ * * & d ( * ! 6+7
4(E) = (E − ).
A ( D D n ( 0
n = d + 1 6+7A
1
2
1
2
1
A = {±q , . . . , ±q },
m
0
q
i
=
ai
1
,
i = 1, . . . , m
0 A 6,7( 4..3A5 4..3A 5
A
M V EE(A) × {1} = M V EE(A ) ∩ Π
0
Π = {x ∈ n |xn = 1}.
( η > 0( E ⊂ n (1 + η)!!
/ 4..3A5( E = E ∩ Π (1 + η)!!
/ 4..3A5 6,7
*( A & * & ( & !: & !
'
4..3A 5 ( A *( !* 4..3A 5 & * / !=
!
& 6+7A
(P(A )) M − log det M
(q i )T M q i ≤ 1,
s.t.
M ∈
i = 1, . . . , m,
! :,
n×n
0
M ∈ n×n &
%
* (P(A )) C* 6+7
(D(A )) /p Φ(p) = log det Λ(p)
eT p = 1,
s.t.
p ≥ 0,
0
p ∈ m & &
m
Λ(p) =
Λ : m → n×n
!
pi q i (q i )T .
i=1
*E ! p (D(A ))
& 6+7
wi (p) + si = λ,
i = 1, . . . , m,
T
e p = 1,
pi si = 0,
0 p ≤ 0 s ≤ 0(
35
i = 1, . . . , m
0
wi (p) = (q i )T Λ(p)−1 q i ,
i = 1, . . . , m.
2
& * p ∈ m (D(A )) 0 Φ(p) > −∞( m
pi wi (p) = n.
i=1
*( λ = n & & & *! & 35 & pi * *! i = 1, . . . , m ( (D(A )) & 0
6+7
wi (p) ≤ n,
i = 1, . . . , m,
T
e p=1
0 p ≤ 0
? 6+7 !
!( & * p̂ : !!
/ ! ∈ (0, 1) ≤
wi (p̂)
(1 + η)n,
i = 1, . . . , m,
3'5
! wi(p̂) ≥ (1 − )n, i = 1, . . . , m.
90( ! * p∗ (D(A)) & 4..3A5
0 6)7 % A & / 0 ith * & ai( i = 1, . . . , m
(
4..(A) = E = {w ∈ d|(x − c)T E ∗(x − c) ≤ 1},
0
1
p̂ > 0
E∗ =
(
d
(AP ∗ AT − Ap∗ (Ap∗ )T )−1 ,
log 4(M V EE(A)) =
c∗ = Ap∗ .
1
d
log d + log det Λ(p∗ ).
2
2
*( &0 p∗ 4..3A5 2
( !
& !!
/ 4..3X0 5 4..3A5 &
!!
1 : 0 6+7A
#
!* A = {a1 , . . . , am } ⊂ d( *!*
X0
m ≤ 2d( X0 = A 1
0 !
*A
X0
*
*!* . Ψ = {0}
X0 = ∅
k = 0.
' d\Ψ = ∅( !
%! + k = k + 1
# &
bk ∈ d ! Ψ
< α = maxi=1,...,m (bk )T ai
X0 = X0 ∪ {aα }
+
,
β = mini=1,...,m (bk )T ai
X0 = X0 ∪ {aβ }
Ψ = !(Ψ, {aβ − aα})
) d\Ψ = ∅( ! ! 1
0 !
%! "*
X0 *!*
@*
? 6)7 !
X0 ⊆ A 1 |X0 | = min{2d, m}(
O(md2 )
4(M V EE(A)) ≤ d2d 4(M V EE(X0 )).
!!
/ X0 1 * 2(
0 * (1 + η)!!
/ * *
! E ? !
0 :
!* & * p∗ (D(A )) 3'5 6+7A
!* A = {a1 , . . . , am } ⊂ d > 0( 0 *!* pk , Ek Xk #
: *
!!
/ !
! E "* A *!* X0
% p0 ∈ m & * p0i = 1/|X | ai ∈ X0 p0i = 0 0
' *A
#
k=0
n=d+1
q i = ((ai )T , 1)T , i = 1, . . . , m
X0 = {y ∈ n |yT Λ(p0)−1 y ≤ 1}
+ pk 3'5( !
< %! , j+ = arg max{(qi )T Λ(pk )−1 qi|i = 1, . . . , m}
κ+ = (q j+ )T Λ(pk )−1 q j+
j− = arg min{(q)T Λ(pk )−1 q i |i = 1, . . . , m, pki > 0}
κ− = (q j− )T Λ(pk )−1 q j−
)
+ = (κ+ /n) − 1
− = 1 − (κ− /n)
<
k = max{+ , − }
k = + ( κ −n
βk = n(κ
−1)
'
pk+1 = (1 − βk )pk + βk ej
+
+
+
k =k+1
.
p
n−κ
+
βk = min n(κ
−1) , 1−p
−
−
<
k
j−
k
j−
pk+1 = (1 + βk )pk − βk ej−
k =k+1
, Ek = {y ∈ n |yT Λ(pk )−1 y ≤ 1}
) Xk = {ai ∈ A|pki > 0}
pk : 3'5( ! ! 1
0 !
%! "*
pk , Ek Xk *!*
? !
(1+η)!!
/ 4..3A5
!* & 2 2
O(md2 {[(1 + η) d+1 − 1]−1 + log d})
6+7
2
A = {aq , . . . , am } ⊂ d( !
* *!* X0 ⊆
A & * p0 (D(A )) * & X0 * ! pj ! * & Ek ! qj 0
pkj ! Ek ⊆ Q = {±q 1 , . . . , ±q m }( pk *! !
* /* * Ek+1 k = +( pk+1 /
& pk ej 90( βk +
−
+
βk = arg max log det Λ((1 − β)pk + βej+ ) =
β∈[0,1]
κ+ − n
,
n(κ+ − 1)
log 4(Ek+1 ) log det Λ(pk+1) 1
0( = − ( pk+1 * pk & D 0D ej ( βk −
βk = arg
max
k
p
β∈
log det Λ((1−β)p
k
j−
0,
1−pk
j−
,
−βe ) = min
j−
pkj−
n − κ−
,
n(κ− − 1 1 − pkj−
.
β *
& pk+1 *( 2
*
(1 + η)!!
/ 4..3A5( &* Xk 4..3X0 5 !
& !!
/ 4..3A5 pk (
0 ! * 3D(A)5 6+7
% !
! *
&* &
!
! n !!
( : 0 67A
$
>
*&
A
x1 ∼ U (−rλmax , rλmax )
x2 ∼ U (−rλmax , rλmax )
xn ∼ U (−rλmax , rλmax )
% x = [xq , x2, . . . , xn ] & y xT x ≤ rλmin y=x
xT S −1 x ≤ r y=x
> ! 0
U (a, b) *
&* (a, b)( r * !
! E ( λmax /* * E −1 (
0 E / : 35
E −1 0( &
!
2
( 0 ( ! &
xi ∼ U (−rλi , rλi ),
0
λi , i = 1, . . . , n * E −1 2( *
&* ! & & & 67 *( : 3 !!
: 0A
#
8* V = {v1 , . . . , vn } * D = {λ1 , . . . , λn }
E −1 .
V & &( baseV >
*&
baseV A
x1 ∼ U (−rλ1 , rλ1 )
x2 ∼ U (−rλ2 , rλ2 )
xn ∼ U (−rλn , rλn )
' " x = [x1 , x2, . . . , xn ] - & y xT x ≤ rλmin y=x
xT S −1 x ≤ r y=x
> ! ! ! P1 (
0
(−rλmax , rλmax ) 3( ! P2 ( 0 * !(
: & * !
! *( * ! P2 * P1 ( 4(P1 ) = λmax λmax . . . λmax = λnmax
n
4(P2 ) = λ1λ2 . . . λn =
n
λi .
i=1
2
( λ1 & λmax P2 90 * ! P1 P2 & /!
0
λnmax
λn−1
4(P1 ) =
max
=
35
4(P2 ) λmax λ2λ3 . . . λn λ2 λ3 . . . λn > 1.
!
* 35 C* λmax ( λi 35 * λmax 1
0( !
! 0* & !
!
( & !
! ( &
3 0 !
)
!!
* 0 C*
/ A
b( : 3<5( !* !
* *
&*
! & 0 ! 0 !
!
0 !
* 0 !
*
2
( /
! & 0 ! * k & 0 ! /
! n ( k ≤ n 4.. *
n( 4.. 0* *
& n 0
( k < n( 4.. 0* n − k *
&*
! 4.. n ( & 0
! 0* & !
4.. "!( 4.. k ( & 0 ! 0 4.. *( k & 0 ! & 0 ! * 2
( &0 /
! V = {v1 , . . . , vm } :
& & V
bv1 ( bv1 * & bv2 *( 0 & & / & 0 0 & !
* !"#
$
% $ &%
' 0
K * & ( Y A ⊆ V ! &
0 ! x 0 & *( !!
/ B2 = {bv1 , . . . , bvp } /
! &
!! 0 & extbv = B2T ext,
0
extbv /
! & B2 ! ext /
0 & * ( /
! !
0 & B2 ( =
! /
! /
! ( *
!
!
!
*
* 90( /
! !* 4.. ! : & /
! * * 2 !
* / E ( &* 0 : E *
*!*
*( ! : / E * r( 0 : !
! ( E ( r m( 0 : *&
*
&* ! & 0 !( * ! !& &
0 0 *
&* ! 4.. !
* m *
&* ! & 0 ! *( ( m *
&* ! 4.. ( !( 0 & 0 !( !
*
! *
m *
&* ! & 0 ! &* ! 4.. * 0 3( !
' *( *
&* ! & 0 ! C*
!
*
!
*
0 *
&* ! / E r k *( ! !
! k !
& B2 ( C*
*
&* ! 4.. & !! & n !( 0 * & 0 !( 0 !( 0( & 0 ! C* 0 : & 0
! 2
/!( & 0 ! S : 0
S=
3
w ∈ | wi ≥ 0,
3
i=1
1 w2
w2
1
wi = 1, w1 ≤ ,
≤ 2,
≥
2 w3
w3
2
,
3+5
& * A1 w ≤ b1
wi ≥ 0
3
wi = 1
i=1
3<5
0
⎡
⎤
1 0
0
A1 = ⎣ 0 1 −2 ⎦
0 −1 12
⎤
⎡
w1
w = ⎣ w2 ⎦
w3
⎡ 1 ⎤
2
b1 = ⎣ 0 ⎦ .
0
90( ! & ' *&* 3<5 ! 3<5 *( *
&* ! & 0 ! ! 3
!
0 & 0 ! 3 3 ! / *( * & ! = *
:
& 0 ! !
3+5(
S1 =
w ∈ 3 | wi ≥ 0,
3
i=1
1 w2
w2
1
wi = 1, w1 ≤ ,
≤ 2,
≥
2 w3
w3
2
C* 0 : ! S1 ;
3<5 2
( /
! & 0 ! S1 * ⎡
ext1 = ⎣
0
1
3
2
3
0
2
3
1
3
1
2
1
6
1
3
1
2
1
3
1
6
⎤
⎦.
./
! ext1 ! 2 ! *( 2 & 0 ! /
! * & base1 !! /
! ext2d
1 ⎡
⎤
−0.6470 0.4981
base1 = ⎣ −0.1078 −0.8093 ⎦
0.7548
0.3113
2.3184 −1.0244 −1.8871 0.5931
2d
.
ext1 =
0
1.6809
0.5603 −2.2413
3,5
Extreme points
Random points in hyperellipsoid
Random points in feasible weight space
0.8
0.7
0.6
w3
0.5
0.4
0.3
0.2
0.1
−1
0
1
0.1
0.2
0.4
0.3
0.6
0.5
0.7
0.8
w2
w1
2*
A # 90 /
! 3,5 & !* *( 4.. ! : & /
! 3,5
2 × 2 ! : / E1 (
0 : 4.. base1 ( E1 =
16.4674 −6.6454
−6.6454 19.9845
.
/ E1 & * !* ' *
&* ! 4.. 2*
F'( 1000 *
&* ! 0
4.. (
& 4.. ! & 0 ! & & : * ! 2 base1( & !! &
3 2( ! *&* 3<5 *
&* ! & 0 ! "* *
2*
./
! ( ! !
! &* ! & 0 ! 0 *( 1F3 !
* !
!
(1+η)!!
/
4.. *
&* ! &
0 ! & * 4.. & 0 !
E 0 * & '
*! * ! 4.. ,
γ=
35
! S
γ ! : 0 *
&* ! 4.. & *
&*
! & 0 !
3( 1000 *
&* ! & 0
! 0
G S1( * 0.01 * 4.. 0 3.92 2710 ! 4.. 0
1000 *
&* ! & 0
! S1 *(
γ1 =
2710
= 2.710.
1000
*
*( 0 /! 3 0
& 0 ! S2 S2 =
w ∈ 3 | wi ≥ 0,
3
i=1
w1
4
1
1
− w2 ≤ , w2 + w3 ≤
wi = 1, w1 ≤ , −
5
2
5
2
/ A2 b2( 0 : & 0 !
S2 ( ⎡
⎤
1
0 0
A2 = ⎣ − 12 −1 0 ⎦
0
1 1
⎡ 4 ⎤
b2 = ⎣
5
1
5
1
2
⎦.
./
!( & 0 ! /
!( !! /
!( 4.. & 0 ! S2( *
&* ! 4.. & 0 !
* :
& 0 ! S1 Extreme points
Random points in hyperellipsoid
Random points in feasible weight space
0.6
w3
0.4
0.2
0
−0.2
−0.2
0
0.2
0.4
0.6
0.9
0.8
w2
0.7
0.6
0.5
0.4
w1
2*
A # *( ext2 ( base2( ext2d
2 / E2 ⎡
ext2 = ⎣
⎡
1
2
1
2
0
1
2
0
1
2
4
5
0
1
5
4
5
1
5
⎤
⎦
0
⎤
−0.3765 −0.7245
0.0362 ⎦
base2 = ⎣ 0.8157
−0.4392 0.6883
2.3906 −1.3741 −1.2612 0.2447
2d
ext2 =
0
1.9562 −0.5869 −1.3693
15.8677 8.4414
.
E2 =
8.4414 27.7261
2*
* /
! 1000 *
&* ! 4.. & 0 ! S2 (
/
! ( ! 4.. &*
! & 0 ! 0 1 − 3 !
* !
!
*
E * *( 0.01
0
* * 4.. S2 0 3.63 2763 ! 4.. 0
1000 *
&*
! & 0 ! S2 *(
γ2 =
2763
= 2.763.
1000
+
& 0 ! S3 S3 =
w ∈ 3 | wi ≥ 0,
3
i=1
2
1
wi = 1, w1 ≤ w2 , w1 + w3 ≤ , −w3 ≤ −
5
5
/ A3 b3( 0 : & 0 !
S3 ( ⎡
1 0
A3 = ⎣ 0 − 32
0 13
⎡ 3 ⎤
⎤
0
1 ⎦
−1
5
b3 = ⎣ 0 ⎦ .
0
./
!( & 0 ! /
!( !! /
!( 4.. & 0 ! S3( *
&* ! 4.. & 0
! * !
* /! *(
ext3 ( base3 ( ext2d
3 / E3 ⎡
ext3 = ⎣
⎡
1
2
1
2
0
1
2
0
1
2
4
5
0
1
5
4
5
1
5
⎤
⎦
0
⎤
−0.7042 −0.4133
base3 = ⎣ −0.0059 0.8165 ⎦
0.7100 −0.4032
2.1302 −1.2532 −1.7544 0.8774
2d
ext3 =
0
−1.4940 −0.6403 2.1344
15.8677 8.4414
.
E3 =
8.4414 27.7261
2*
' * /
! *
&* ! 4.. & 0 ! ( /
! ( ! 4.. &*
! & 0 ! 0 1 − 3 !
* !
!
*
G E ( 0.01 0
* * 4.. S3( 0 3.59 2823
! 4.. 0
1000 *
&* ! & 0 ! S3 *(
γ3 =
2823
= 2.823.
1000
<
Extreme points
Random points in hyperellipsoid
Random points in feasible weight space
0.7
0.6
0.5
w3
0.4
0.3
0.2
0.1
0
−1
0
1
0.1
0.2
0.4
0.3
0.5
0.6
0.7
0.8
w2
w1
2*
'A # '
( !
* *
&* ! 3 4.. 0 *
! ( 0 *
&* ! *
* γ 0 2.710 2.823 *( !!
/ 2.8 *
&* ! 4.. !
* *
&* ! & 0 !
n
/ E 0 n & 0 !( : n & 0 ! Sn
0 * A
Sn = {w ∈ | wi ≥ 0,
n
n
wi = 1, wi ≤ wi+1
∀i}.
i=1
0 * 4 7 ( 10( 100 1000 *
&* ! 0
& 0 ! Sn 9*&
*
&* ! 4.. *! &
4.. * ! * 0
* & :
* /!
Sn ( * *&
*
&*
,
& A @ :*
n - " ! " ! Sn
4..
4.. 67 S 67
,
'
,'
,
'
,'<
)
,
+
+
,
+
,<<
'+
+
'')+<
)
<
)<
<'
<
,<
'
<
,+
)
+
,
',<
)
'<
,
)
,
<
,
+)'+
'
! Sn ( * *&
*
&* ! 4..( *
* *! 4.. * : * *! *
&* ! * (
* 4.. C*
>
*
&* ! 4.. * *
4( C*
ψ4 =
27.0
= 0.27
1000
*
&* ! n ( n = 5, 6, 7 811
= 0.811
1000
5880
= 5.88
ψ6 =
1000
12300
= 12.3,
ψ7 =
1000
S4 ψ5 =
! *( * *
&*
! Sn Sn γ ( : 35( 0 * /! & * !
& :
* & Sn ( * *&
*
&* ! Sn * γ & A E γ - " ! Sn
+
+
+
<
<
<
,
,
,
γ
,
,'
+
,,
'
)<
,
',<
)
+)
4* γ ( * 0
* * 0* & ! ( γ /! 0* & 9
( & γ
0 Sn( 0 4 5 γ ;
γ4AV = 9.9
γ5AV = 25.6
γ6AV = 74.4
γ7AV = 63.1,
0
γnAV * 0
( *&
C*
*
&* ! Sn /!
* γ "! * !
! 0 /
! 0* &* H* *
&* ! Sn γ . * γ * * 6 7( *
*
&* ! Sn 7 (
& : * & *
7 6 *( * : 4..
)
" - !
.;
& !
*& * 3-5 ! *
!&
35 !
*& 0 0 !
!
& 1 !
8
( 0 C*
! !!
!
0
!
* *
&* ! & 0
! &* !!
*
&* !
& 0 ! S C*
/ A b( 0 : & 0 ! S 2
( /
! & 0 !
( & bs( 0 ! /
!( * /
! !! 0 & 4.. E & 0 ! & /
! bs (
*
&* ! 4.. ( *
&* ! !! & & 2( !( 0 &
0 !( C*
0 & 0 ! 3 * !
( !
*
*
&* ! & 0 !
Si 8* 4.. * >
*
&* ! 4.. * 9
( !
* *
&* ! & 0 ! 3 E 0 n ( 0
n = 4, . . . , 7 &
& 0 ! Sn ( 0 ( 0 * 0 * *
10( 100 1000 *
&* ! Sn * 4.. /! 9
( * *
&* ! Sn :( 0 n 2
/!( 1000 ! 4 0
* 27 ( 0 !!
/ 3 25 * *
! 7 !
* *
&*
! & 0 ! !
!
9
( : *( *
!
* *
&* !
& 0 ! & &:
67 4
*
= C* : 3)))5
67 I* != 3<5
6'7 ! !
!
!
4
( ))<
67 "#$
%& ' ( #
A 8- /!
" # $ % 35( ''J+
6+7 ) * *++ &
' $()*
+ ,-
G( ),<
6<7 ) "* !! *&
!* $
3))<5( ',J+)
6,7 ) ! " 1 !/ !!
/
/ & ! !! 3))'5( ',J+)
67 ) * C* 9-. !
& * !
*
.""" ,
/0 1%2 3)5(
+<<J+<)
6)7 ) ( * * !
# $3 ,4 !
3+5( J
67 * A *
!& *! $
/ 1%2 35( J+
67 , >
! *
&* !
! 5
.""" !
3))5(
<+J<+
67 --- * 1 *
!
!
# 4
3)),5( ')J')
6'7 . " . / . 8!*
* !
& ; *
H0 !! &
. . ,4 35
67 0 * ! *
!& " # $ 67 3,5( +J+'
6+7 " ( 1 @B !* * * ! &
!
00 3,5( ,'J,
6<7 , 1 2$ " 0 ! ! C* & * .""" .
! &
8
3,5