• Results in abstract and full text:
B>F=0
delta 1<delta 2<⋯<delta L>0
pone.0211463.e021">removedGRAPHIC X^R=∑ l=1
p l=q l ⊤ q l=1
l=1
F 2 subject-{ p l ⊤ p l=1, p l ⊤ p l '=0, q l ⊤ q l=1, q l ⊤ q l '=0
q^(0) are randomly initialized;<p specific use="line">delta^(0)=0;<p specific use="line">delta^(1)=p^(0)^⊤ Xq^(0) ;<p specific use="line">s=0;<p specific use="line">while | delta^(s +1) - delta^s |<eta do<p specific use="line">p^(s +1
<p specific use="line">q^(s +1)<- normalize (X^⊤ p^(s +1
line">delta^(s +1)=p^(s +1)⊤ Xq^(s +1) ;<p specific use="line">s<- s + 1;<p specific use="line">end<p specific use="line">X^(l +1
X<p specific use="line">Define p=0 ;<p specific use="line">Define Q=0
q^(0) are randomly initialized;<p specific use="line">delta^(0)<- 0;<p specific use="line">delta^(1)<- p^(0)⊤ Xq^(0) ;<p specific use="line">s<- 0;<p specific use="line">while | delta^(s +1) - delta^s |<eta do<p specific use="line">p^(s +1
B L 2 (1) ∩ P ⊥)<p specific use="line">q^(s +1)<- proj(X^⊤ p^(s +1
Q, q^(s +1) ];<p specific use="line">end Algorithm 3
k=1
line">Define x^(0)=x ;<p specific use="line">while | x^(s +1) - x^s |<eta do<p specific use="line">x^(s +1
k=2..., K do<p specific use="line">x^(s +1)<- proj(x^(s +1
<p specific use="line">end<p specific use="line">s<- s + 1<p specific use="line">end Algorithm 1
q^(0) are randomly initialized;<p specific use="line">delta^(0)=0;<p specific use="line">delta^(1)=p^(0)⊤ Xq^(0) ;<p specific use="line">s=0;<p specific use="line">while | delta^(s +1) - delta^s |<eta do<p specific use="line">p^(s +1
p^(s +1) ‖ 1=c 1, l ;<p specific use="line">q^(s +1)<- normalize (S (X^⊤ p^(s +1), lambda 2
q^(s +1) ‖ 1=c 2, l ;<p specific use="line">delta^(s +1)=p^(s +1)⊤ Xq^(s +1) ;<p specific use="line">s<- s + 1;<p specific use="line">end<p specific use="line">X^(l +1
l=1
F 2 subject-{ p l ⊤ p l=1, q l ⊤ q l=1, C 1 p l<=c 1, l, C 2 q l<=c 2
F 2 subject-{ p l ⊤ p l=1, p l ⊤ p l '=0, q l ⊤ q l=1, q l ⊤ q l '=0, ∀ l ' !=l, and-{ C 1 p l<=c 1, l, C 2 q l<=c 2
l<1
pone.0211463.e053">removedGRAPHIC arg max p, q { p ⊤ X q } subject-{ p ⊤ p=1, q ⊤ q=1
p^⊤ p=1
q^⊤ q=1
pone.0211463.e054">removedGRAPHIC arg max p, q { p ⊤ X q } subject-{ p ⊤ p<=1, q ⊤ q<=1
X<p specific use="line">Define p=0 ;<p specific use="line">Define Q=0
q^(0) are randomly initialized;<p specific use="line">delta^(0)<- 0;<p specific use="line">delta^(1)<- p^(0)⊤ Xq^(0) ;<p specific use="line">s<- 0;<p specific use="line">while | delta^(s +1) - delta^s |<eta do<p specific use="line">p^(s +1
B 1 (c 1, l) ∩ B 2 (1) ∩ P ⊥)<p specific use="line">q^(s +1)<- proj(X^⊤ p^(s +1
Q, q^(s +1
P M an I=150*5=750 by 5
J=600*5=3, 000 by 5
c 2=11
I=2, 100 ≈ 25
J=30 ≈ 2
k<1
p=0
removedGRAPHIC 1 p + 1 q=1, then<disp formula id="pone.0211463.e096
With p=1
b-1, we obtain<disp formula id="pone.0211463.e100">removedGRAPHIC | ∑ i=1 N | x i | |<=‖ x ‖ 2 ‖ 1 ‖ 2
B L 2 (1) when rho<=1
removedGRAPHIC phi (0)=‖ x ~ ‖ 1 removedGRAPHIC phi (x ~ 1)=0
B L 1 c x<p specific use="line">1
p specific use="line">2
removedGRAPHIC phi (x ~ i)<=c<phi (x ~ i + 1) ;<p specific use="line">3
removedGRAPHIC phi (x ~ i - delta)=∑ k=1
<p specific use="line">4
pone.0211463.e129">removedGRAPHIC b i={ 1 if x i !=0, 0 if x i=0
becomes<disp formula id="pone.0211463.e130">removedGRAPHIC ∑ i=1
pone.0211463.e141">removedGRAPHIC ‖ S (x ~, lambda) ‖ 1=(∑ j=1
k (x ~ j - lambda) 2=(∑ j=1 k x ~ j 2) -2 lambda (∑ j=1 k x ~ j) + k lambda 2
pone.0211463.e143">removedGRAPHIC ψ ' lambda=1
lambda) ‖ 2 - k ‖ S (x ~, lambda) ‖ 2)=1 ‖ S (x ~, lambda) ‖ 2 (ψ lambda 2
lambda<=0
pone.0211463.e158">removedGRAPHIC ‖ S (x ~, x ~ i - delta) ‖ 1=∑ j=1 i (x ~ j - (x ~ i - delta)=∑ j=1 i (x ~ j - x ~ i) + i delta=‖ S (x ~, x ~ i) ‖ 1 + i delta=l 1
i (x ~ j - (x ~ i - delta) 2=∑ j=1 i (x ~ j - x ~ i) 2 + 2 delta (x ~ j - x ~ i) + delta 2
c^2>0
pone.0211463.e173">removedGRAPHIC delta=-2 l 1 (i - c 2) + delta 2 i (i - c 2)=-2 l 1 (i - c 2) + 2 c c 2
c 2)=-l 1
B L 1 c ∩ B L 2 (1) x<p specific use="line">1
p specific use="line">2
removedGRAPHIC ψ (x ~ i + 1)<=c<ψ (x ~ i) ;<p specific use="line">3
x ~ i) 2 i - c 2 - ψ (x ~ i) ;<p specific use="line">4
Q</disp formula>- u 1, u 2
pone.0211463.e204">removedGRAPHIC u 2 ' (q ~ q p | d 2) | q ~=p=f ' q p | [0] d 2 ], s. t. q + d 2 ∈ Q,</disp formula>- u 1
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