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

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

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MIT Applied Mathematics

Random Matrices

October 10, 2010

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

- History of σmin universality
- Proof idea of Tao-Vu (in NLA language)
- What the proof doesn't prove
- Do stochastic differential operators say more?

A bit of history

- 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)

Empirical smallest singular value distribution: n x n Gaussian entries

Extending to non-Gaussians: How? entries

- 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?

Tao-Vu ('09) “the rigorous proof”! entries

- 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?

A few more tries... entries

How good is the s x s estimator?A lot more tries... again, accurate to say 10% entries

More s x s estimator experimentsGaussian entries, ±1 entries15

15

15

15

A lot more tries, now comparing matrix sizes entries

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

More s x s estimator experimentsn = 100 vs. n = 200How good is the s x s estimator? entries

- 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)

Bidiagonalization process for n x n Gaussian matrix approximation can tell us

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?)

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