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Al ParkerOctober 29, 2009
Using linear solvers to sample large Gaussians
• Colin Fox, Physics, University of Otago
• New Zealand Institute of Mathematics, University of Auckland
• Center for Biofilm Engineering, , Bozeman
Acknowledgements
2
22
2 )(2
1exp
2
1),(
yN
The normal or Gaussian distribution
• y = (σ2)1/2 z + µ ~ N(µ,σ2)
How to sample from a Gaussian N(µ,σ2)?
• Sample z ~ N(0,1)
)()(
2
1exp
)det(2
1),( 1
2/1
yyN T
n
The multivariate Gaussian distribution
T
T
T
T
n
WW
yyN
22
22
22
22
20/10
02/1
22
22
22
22
119
911
]200100[
)()(2
1exp
)det(2
1),( 1
2/1
• y = Σ1/2 z+ µ ~ N(µ,Σ)
How to sample from a Gaussian N(µ,Σ)?
• Sample z ~ N(0,I)
(eg y = WΛ1/2z + µ)
Example: From 64 faces, we can model “face space” with a Gaussian Process N(μ,Σ)
Pixel intensity at the ith row and jth column is y(s(i,j)), y(s) є R112 x R112
μ(s) є R112 x R112
Σ(s,s) є R12544 x R12544
~N( ,Σ)
How to estimate μ,Σ for N(μ,Σ)?
• MLE/BLUE (least squares)• MVQUE• Use a Bayesian Posterior via MCMC
Another example: Interpolation
One can assume a covariance function which has some parameters θ
I used a Bayesian posterior for θ|data to construct μ|data
Simulating the process:samples from N(μ,Σ|data)
y = Σ1/2 z + µ
~N(μ, )
Gaussian Processes modeling global ozone
Cressie and Johannesson, Fixed rank kriging for very large spatial datasets, 2006
Gaussian Processes modeling global ozone
The problem• To generate a sample y = Σ1/2 z+ µ ~ N(µ,Σ), how to calculate the factorization Σ =Σ1/2(Σ1/2)T ?
• Σ1/2 = WΛ1/2 by eigen-decomposition, 10/3n3 flops• Σ1/2 = C by Cholesky factorization, 1/3n3 flops
For LARGE Gaussians (n>105, eg in image analysis and global data sets), these approaches are not possible• n3 is computationally TOO EXPENSIVE • storing an n x n matrix requires TOO MUCH MEMORY
Some solutions
Work with sparse precision matrix Σ-1 models (Rue, 2001)
Circulant embeddings (Gneiting et al, 2005)
Iterative methods:• Advantages:
– COST: n2 flops per iteration– MEMORY: Only vectors of size n x 1 need be stored
• Disadvantages:– If the method runs for n iterations, then there is no cost savings over
a direct method
Gibbs: an iterative sampler of N(0,A) and N(0, A-1 )
Let A=Σ or A= Σ-1
1. Decompose A by D=diag(A), L=lower(A)2. Sample z ~ N(0,I) 3. Take conditional samples in each coordinate
direction, so that a full sweep of all n coordinates is yk =-D-1 L yk - D-1 LT yk-1 + D-1/2 z
yk converges in distribution geometrically to N(0, A-1)
Ayk converges in distribution geometrically to N(0,A)
Gibbs: an iterative sampler• Gibbs sampling from N(0,Σ) starting from (0,0)
Gibbs: an iterative sampler• Gibbs sampling from N(0,Σ) starting from (0,0)
What’s the link to Ax=b?Solving Ax=b is equivalent to minimizing an n-
dimensional quadratic (when A is pd)
)()(
2
1exp
)det(2
1),( 1
2/1
yyN T
n
Axbxf
xbAxxxf TT
)(2
1)(
A Gaussian is sufficiently specified by the same quadratic (with A= Σ-1and b=Aμ):
Gauss-Siedel Linear Solve of Ax=b
1. Decompose A by D=diag(A), L=lower (A)2. Minimize the quadratic f(x) in each coordinate
direction, so that a full sweep of all n coordinates is
xk =-D-1 L xk - D-1 LT xk-1 + D-1 b
xk converges geometrically A-1b
Gauss-Siedel Linear Solve of Ax=b
Gauss-Siedel Linear Solve of Ax=b
xk converges geometrically A-1b, (xk - A-1b) = Gk( x0 - A-1b) where ρ(G) < 1
Theorem: A Gibbs sampler is a Gauss Siedel linear solver
Proof: • A Gibbs sampler is yk =-D-1 L yk - D-1 LT yk-1 + D-1/2 z
• A Gauss-Siedel linear solve of Ax=b is xk =-D-1 L xk - D-1 LT xk-1 + D-1 b
Gauss Siedel is a Stationary Linear Solver
• A Gauss-Siedel linear solve of Ax=b is xk =-D-1 L xk - D-1 LT xk-1 + D-1 b• Gauss Siedel can be written as Mxk = N xk-1 + b where M = D + L and N = D - LT , A = M – N, the general form of a stationary linear solver
Stationary linear solvers of Ax=b
1. Split A=M-N 2. Iterate Mxk = N xk-1 + b
1. Split A=M-N 2. Iterate xk = M-1Nxk-1 + M-1b = Gxk-1 + M-1b
xk converges geometrically A-1b, (xk - A-1b) = Gk( x0 - A-1b) when ρ(G) = ρ(M-1N)< 1
Stationary Samplers from Stationary Solvers
Solving Ax=b:1. Split A=M-N 2. Iterate Mxk = N xk-1 + b xk A-1b if ρ(M-1N)< 1
Sampling from N(0,A) and N(0,A-1):1. Split A=M-N2. Iterate Myk = N yk-1 + bk-1
where bk-1 ~ N(0, MA-1MT – NA-1NT ) = N(0,M+N) when M is symmetric yk N(0,A-1) if ρ(M-1N)< 1 Ayk N(0,A) if ρ(M-1N)< 1
How to sample bk-1 ~ N(0, MA-1MT – NA-1NT ) ?
• Gauss Siedel M = D + L, bk-1 ~ N(0, D)• SOR (successive over-relaxation) M = 1/wD + L, bk-1 ~ N(0, (2-w)/w D)• Richardson M = I, bk-1 ~ N(0, 2I-A)• Jacobi M = D, bk-1 ~ N(0, 2D-A)
Theorem: Stat Linear Solver converges iff Stat Sampler converges
andthe convergence is geometric
• Proof: They have the same iteration operator: For linear solves: xk = Gxk-1 + M-1 b so that (xk - A-1b) = Gk( x0 - A-1b)
For sampling: yk = Gyk-1 + M-1 bk-1 E(yk)= Gk E(y0) Var(yk) = A-1 - Gk A-1 GkT
Proof of convergence for Gaussians by Barone and Frigessi, 1990. For arbitrary distributions by Duflo, 1997
An algorithm to make journal papers
1. Look up your favorite optimization algorithm2. Turn it into a sampler
For Example: Acceleration schemes for Stationary Linear Solvers can work for
Stationary SamplersPolynomial acceleration of a stationary solver of
Ax=b is 1. Split A = M - N 2. xk+1 = (1- vk) xk-1 + vk xk + vk uk M-1 (b-A xk)
which replaces (xk - A-1b) = Gk( x0 - A-1b)with a kth order polynomial (xk - A-1b) = p(G)( x0 - A-1b)
Chebyshev Acceleration• xk+1 = (1- vk) xk-1 + vk xk + vk uk M-1 (b-A xk)
where vk , uk are functions of the 2 extreme eigenvalues of G (not very expensive to get estimates of these eigenvalues)
Gauss-Siedel converged like this …
• xk+1 = (1- vk) xk-1 + vk xk + vk uk M-1 (b-A xk)
where vk , uk are functions of the 2 extreme eigenvalues of G (not very expensive to get estimates of these eigenvalues)
… convergence (geometric-like) with Chebyshev acceleration
Chebyshev Acceleration
• xk+1 = (1- vk) xk-1 + vk xk + vk uk M-1 (b-A xk)
where vk , uk are functions of the residuals b-Axk
… convergence guaranteed in n finite steps with CG acceleration
Conjugate Gradient (CG) Acceleration
Polynomial Accelerated Stationary Sampler from N(0,A) and N(0,A-1)
1. Split A = M - N 2. yk+1 = (1- vk) yk-1 + vk yk + vk uk M-1 (bk -A yk)
where bk ~ N(0, (2-vk)/vk ( (2 – uk)/ uk M + N)
TheoremA polynomial accelerated sampler converges if
vk , uk are independent of the iterates yk ,bk.
Gibbs Sampler Chebyshev Accelerated Gibbs
• The theorem does not apply since the parameters vk , uk are functions of the residuals bk - A yk
• We have devised an approach called a CD sampler to construct samples with covariance
Var(yk) = VkDk-1 Vk
T A-1
where Vk is a matrix of unit length residuals b - Axk from the standard CG algorithm.
Conjugate Gradient (CG) Acceleration
• A GOOD THING: The CG algorithm is a great linear solver! If the eigenvalues of A are in c clusters, then a solution to Ax=b is found in c << n steps.
• A PROBLEM: When the CG residuals get small , the CD sampler is forced to stop after only c << n steps. Thus, covariances with well separated eigenvalues work well.
• The covariance of the CD samples yk ~ N(0,A-1) and Ayk ~ N(0,A)
have the correct covariances if A’s eigenvectors in the Krylov space spanned by the residuals have small/large eigenvalues.
CD sampler (CG accelerated Gibbs)
Lanczos sampler• Fix the problem of small residuals is easy:
hijack the iterative Lanczos eigen-solver to produce samples yk ~ N(0,A-1) with
Var(yk) = WkDk-1 Wk
T A-1
where Wk is a matrix
of “Lanczos vectors”
The real issue is the spread of the eigenvalues of A. If the large ones are well separated …
… then the CD sampler and Lanczos sampler can produce samples extremely quickly from a large Gaussian N(0,A) (n= 106) with the correct moments.
One extremely effective sampler for LARGE Gaussians
Use a combination of the ideas presented:• Generate samples with the CD or Lanczos
sampler while at the same time cheaply estimating the extreme eigenvalues of G.
• Seed these samples and extreme eigenvalues into a Chebyshev accelerated SOR sampler
ConclusionsGaussian Processes are cool!
Common techniques from numerical linear algebra can be used to sample from Gaussians• Cholesky factorization (precise but expensive)
• Any stationary linear solver can be used as a stationary sampler (inexpensive but with geometric convergence)
• Polynomial accelerated Samplers– Chebyshev – Conjugate Gradients
• Lanczos Sampler
Estimation of Σ(θ,r) from the data using a a Markov Chain
Marginal Posteriors
Simulating the process:samples from N(μ,Σ|data)
x = Σ1/2 z + µ
~N(μ, )
TOO COURSE OF A GRID
Why is CG so fast?
Gauss Siedel’s Coordinate directions CG’s conjugate directions
I used a Bayesian posterior for θ|data to construct μ(s|θ)
Simulating the process:samples from N(μ,Σ)
y = Σ1/2 z + µ
~N(μ, )