Alan Edelman and Po-Ru Loh MIT Applied Mathematics Random Matrices October 10, 2010. An NLA Look at σ min Universality (& the Stoch Diff Operator). Outline. History of σ min universality Proof idea of Tao-Vu (in NLA language) What the proof doesn't prove
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MIT Applied Mathematics
October 10, 2010An NLA Look at σmin Universality(& the Stoch Diff Operator)
1. The smallest singular value of R22, scaled by √n/s, is a good estimate for σn!
2. R22 (viewed as the product Q2T M2) is roughly s x s Gaussian
M = (M1 M2) = QR = (Q1 Q2)( )
Note: Q2T M2 = R22
R11 R12 n-s
Fluctuates around ±10% of truth entries
(at least for this matrix)How good is the s x s estimator?
A few more tries... entriesHow good is the s x s estimator?
s = 10 to 50% (of n)... n = 200 a bit betterMore s x s estimator experimentsn = 100 vs. n = 200
… but the truth is far stronger than what the approximation can tell usWhat the proof doesn't prove
k=1 approximation can tell us
How close are we if we use kxk chi’s in the bottom,
n=200; t=1000000; v=zeros(t,1);
k, endx=(n-k+1:n); endy=(n-k+1:n-1);
if rem(i,500)==0,[i k],end
Area of Detail
1 Million Trials in each experiment
(Probably as n→inf, there is still a little upswing for finite k?)