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

Alan Edelman and Po-Ru LohMIT Applied Mathematics

Random Matrices

October 10, 2010

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)

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Empirical smallest singular value distribution: n x n Gaussian entries

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Empirical smallest singular value distribution: n x n ±1 entries

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Extending to non-Gaussians: How?

• 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

• History of σmin universality

• Proof idea of Tao-Vu (in NLA language)• What the proof doesn't prove• Do stochastic differential operators say

more?

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Tao-Vu ('09) “the rigorous proof”!• 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

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

Note: Q2T M2 = R22

R11 R12 n-s

R22 s

n-s s n-s s

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Basic idea part 1: σs√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

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Basic idea part 2: R22 ≈ 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

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Basic idea (recap)

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

• History of σmin universality

• Proof idea of Tao-Vu (in NLA language)• What the proof doesn't prove• Do stochastic differential operators say

more?

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How good is the s x s estimator?

Fluctuates around ±10% of truth(at least for this matrix)

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How good is the s x s estimator?

A few more tries...

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More s x s estimator experimentsGaussian entries, ±1 entries

A lot more tries... again, accurate to say 10%1515 1515

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

A lot more tries, now comparing matrix sizess = 10 to 50% (of n)... n = 200 a bit better

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How good is the s x s estimator?

• 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

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Bounds from the proof

• “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??

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Reality check: n x n Gaussian entries, revisited

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Reality check: n x n ±1 entries, revisited

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What the proof doesn't prove

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

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

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?

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Can another approach get us closer to the truth?

• 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)

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Bidiagonalization process for n x n Gaussian matrix

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A stochastic operator connection

• 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

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A stochastic operator connection

0 0.05 0.1 0.15 0.2

0.9

1

1.1

1.2

1.3

1.4

1.5

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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],endendhold offv=v*sqrt(n);

k=1

k=2

k=3

k=4k=5k=6k=7..10k=inf

k=0

n=100n=200

Area of Detail

(Probably as n→inf, there is still a little upswing for finite k?)1 Million Trials in each experiment

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A stochastic operator connection

• 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.

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Extending to non-Gaussians: How?

• 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

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SDO mechanism

• 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?

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Some grand questions

• 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?)