Instagram Cheat Sheet

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need to compute f and check the stopping criterion in line 8 of algorithm 2. This optimization saves touching an extra vector in memory and a 1-norm computation. In our numerical experiments we adopt this version of the accelerated algorithm, i.e. we take im = 1. The “power(αy + (1 − α)v)” clause in line 9 means apply the power method with αy + (1 − α)v as an initial guess, until convergence. Algorithm 3 – Inner-Outer power iterations. Input: P, α, β, τ, η, v, im Output: x 1: x ← v 2: y ← Px 3:while ∥αy+(1−α)v−x∥1 ≥τ x ← f + βy y ← Px until∥f+βy−x∥1 <η ifi≤im, x=power(αy+(1−α)v);return 5.5 ⋅ extensions 113 f ←(α−β)y+(1−α)v for i = 1, . . . repeat 4: 5: 6: 7: 8: 9: 10: end while 11: x ← αy + (1 − α)v When im = 1, then algorithm 3 and algorithm 2 produce exactly the same iterates. Once the inner iteration of algorithm 2 converges in a single iteration, then it will always converge in a single iteration. We hope we aren’t belaboring the point by reiterating that a single iteration of the inner iteration is precisely the power method. Program 8 – The inner-outer iteration for PageRank. The input to the inner-outer code is a = α, v = v, P = PT , tol = ε, and maxit, an upper bound on the number of iterations. Our PageRank solvers are quite compact but work with PT for performance reasons. On Matlab R2007a and R2007b, the code uses only three vectors of storage. 1 function [x,flag,reshist]=inoutpr(P,a,v,tol,maxit) 2 b=0.5*(a≥0.6); itol=1e-2; n=size(P,1); 3 x=zeros(n,1)+v; y=P’*x;y=y+(sum(x)-sum(y))*v; nm=1; f=a*y;f=f+(1-a)*v;f=f-x; 4 dlta=norm(f,1);x=x-b*y;reshist=[dlta;zeros(maxit-1,1)]; 5 while nmtol 6 f=(a-b)*y; f=f+(1-a)*v; x=x-f; % f=(a-b)*y+vt;x=x-b*y-f; 7 d = norm(x,1); ii=0; 8 while nm+iiitol, 9 x=b*y; x=x+f; y=P’*x; y=y+(sum(x)-sum(y))*v;ii=ii+1; % x=b*y+f;y=Pd’*x 10 x=x-b*y; x=x-f; d=norm(x,1); end % x=x-b*y-f; 11 if ii<2, x=a*y; x=x+(1-a)*v; y=y-x; break; end % no mult => no hist updte 12 x=x+f; f=a*y; f=f+(1-a)*v; f=f-x; f=f-b*y; dlta=norm(f,1); % ||a*y+(1-a)*v-x||_1 13 reshist(nm+1:nm+ii)=dlta; nm=nm+ii; end 14 while nmtol % run the power method until convergence 15 y=x./norm(x,1); x=a*(P’*y); w=1-sum(x); x=x+w*v; 16 y=y-x; dlta=norm(y,1); nm=nm+1; reshist(nm)=dlta; end 17 x=x./norm(x,1); flag=dlta>tol; reshist=reshist(1:nm);

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