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# An NLA Look at σ min Universality (& the Stoch Diff Operator) - PowerPoint PPT Presentation

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

Random Matrices

October 10, 2010

An NLA Look at σmin Universality(& the Stoch Diff Operator)

• History of σmin universality

• Proof idea of Tao-Vu (in NLA language)

• What the proof doesn't prove

• Do stochastic differential operators say more?

• E ('89): Explicit formula (for finite iid Gaussian n x n matrices) for distribution of σn

• Analytic techniques: Integration of a joint density function, Tricomi functions, Kummer's differential equation, etc.

• Striking convergence to limiting distribution even in non-Gaussian case

• Parallel Matlab in the MIT computer room

• “Please don’t turn off” (no way to save work in the background on those old Sun workstations)

• Central limit theorem is a mathematical statement and a “way of life”

• Formally: a (series of) theorems – with assumptions (e.g., iid) – and if the assumptions are not met, the theorems don't apply

• Way of life: “When a bunch of random variables are mixed up enough, they behave as if Gaussian”

• Example from our discussions: Does the square root of a sum of squares of (almost) iid random variables go to χn? Probably an application of CLT but not precisely CLT without some tinkering (what does “go to” mean when n is changing?)

Outline entries

• History of σmin universality

• Proof idea of Tao-Vu (in NLA language)

• What the proof doesn't prove

• Do stochastic differential operators say more?

• Basic idea (NLA reformulation)...Consider a 2x2 block QR decomposition of M:

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

n-s s

n-s s

M = (M1 M2) = QR = (Q1 Q2)( )

Note: Q2T M2 = R22

R11 R12 n-s

R22 s

Basic idea part 1: σ entriess√n/s ≈ σn

• The smallest singular value of M is the reciprocal of the largest singular value of M-1

• Singular values of R22 are exactly the inverse singular values of an s-row subsample of M-1

• The largest singular value of an approximately low-rank matrix reliably shows up in a random sample (Vempala et al.; Rokhlin, Tygert et al.)

• Referred to as “property testing” in theoretical CS terminology

Basic idea part 2: R entries22 ≈ Gaussian

• Recall R22 = Q2T M2

• Note that Q1 is determined by M1 and thus independent of M2

• Q2 can be any orthogonal completion of Q1

• Thus, multiplying by Q2T “randomly stirs up” entries of the (independent) n x s matrix M2

• Any “rough edges” of M2 should be smoothed away in the s x s result R22

Basic idea (recap) entries

• 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) ≈ s x s Gaussian

• We feel comfortable (from our CLT “way of life”) that part 2 works well

• How well does part 1 work?

Outline entries

• History of σmin universality

• Proof idea of Tao-Vu (in NLA language)

• What the proof doesn't prove

• Do stochastic differential operators say more?

(at least for this matrix)

How good is the s x s estimator?

A few more tries... entries

How good is the s x s estimator?

More s x s estimator experimentsGaussian entries, ±1 entries

15

15

15

15

s = 10 to 50% (of n)... n = 200 a bit better

More s x s estimator experimentsn = 100 vs. n = 200

• On one hand, surprisingly good, especially when not expecting any such result

• “Did you know you can get the smallest singular value to within 10% just by looking at a corner of the QR?”

• On the other hand, still clearly an approximation: n would need to be huge in order to reach human-indistinguishable agreement

Bounds from the proof entries

• “C is a sufficiently large const (104 suffices)”

• Implied constants in O(...) depend on E|ξ|C

• For ξ = Gaussian, this is 9999!!

• s = n500/C

• To get s = 10, n ≈ 1020?

• Various tail bounds go as n-1/C

• To get 1% chance of failure, n ≈ 1020000??

… but the truth is far stronger than what the approximation can tell us

What the proof doesn't prove

• The s x s estimator is pretty nifty...

Outline approximation can tell us

• History of σmin universality

• Proof idea of Tao-Vu (in NLA language)

• What the proof doesn't prove

• Do stochastic differential operators say more?

Can another approach get us closer to the truth? approximation can tell us

• Recall the standard numerical SVD algorithm starting with Householder bidiagonalization

• In the case of Gaussian random matrices, each Householder step puts a χ distribution on the bidiagonal and leaves the remaining subrectangle Gaussian

• At each stage, all χ's and Gaussians in the entries are independent of each other (due to isotropy of multivariate Gaussians)

A stochastic operator connection approximation can tell us

• E ('03) argued that the bidiagonal of χ's can be viewed as a discretization of a stochastic Bessel operator

• – √x d/dx + “noise” / √2

• As n grows, the discretization becomes smoother, and the (scaled) singular value distributions of the matrices ought to converge to those of the operator

A stochastic operator connection approximation can tell us

k=1 approximation can tell us

How close are we if we use kxk chi’s in the bottom,

rest Gaussian?

n=200; t=1000000; v=zeros(t,1);

for k=1

x=sqrt(n:-1:1);

y=sqrt(n-1:-1:1);

v=zeros(t,1);

k, endx=(n-k+1:n); endy=(n-k+1:n-1);

dofx=k:-1:1;

dofy=(k-1):-1:1;

for i=1:t

yy=y+randn(1,n-1)/sqrt(2);

xx=x+randn(1,n)/sqrt(2);

xx(endx)=sqrt(chi2rnd(dofx));

yy(endy)=sqrt(chi2rnd(dofy));

v(i)=min(bidsvd(xx,yy));

if rem(i,500)==0,[i k],end

end

hold off

v=v*sqrt(n);

n=100

n=200

k=2

k=3

k=0

k=4

Area of Detail

k=5

k=6

k=7..10

k=inf

1 Million Trials in each experiment

(Probably as n→inf, there is still a little upswing for finite k?)

A stochastic operator connection approximation can tell us

• Ramírez and Rider ('09) produced a proof

• In further work with Virág, they have applied the SDO machinery to obtain similar convergence results for largest eigenvalues of beta distributions, etc.

Extending to non-Gaussians: How? approximation can tell us

• The bidiagonalization mechanism shouldn't care too much about the difference...

• Each Householder spin “stirs up” the entries of the remaining subrectangle, making them “more Gaussian” (according to Berry-Esseen, qTx is close to Gaussian as long as entries of q are evenly distributed)

• Almost-Gaussians combine into (almost-independent) almost-χ's

• Original n2 entries compress to 2n-1

SDO mechanism approximation can tell us

• Old intuition: non-Gaussian n x n matrices act like Gaussian n x n matrices (which we understand)

• New view: non-Gaussian and Gaussian n x n matrices are both discretizations of the same object

• Non-random discretizations have graininess in step size, where to take finite differences, etc.

• SDO discretizations have issues like almost-independence... but can be overcome?

Some grand questions approximation can tell us

• Could an SDO approach circumvent property testing (sampling the bottom-right s x s) and thereby get closer to the truth?

• Does the mathematics of today have enough technology for this? (If not, can someone invent the new technology we need?)