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Efficient Variational Inference in Large-Scale

Bayesian Compressed Sensing

George Papandreou and Alan Yuille

Department of StatisticsUniversity of California, Los Angeles

ICCV Workshop on Information Theory in Computer Vision

November 13, 2011, Barcelona, Spain

Inverse Image Problems

Denoising Deblurring Inpainting

2 / 22

The Sparse Linear Model

A hidden vector x ∈ RN and noisy measurements y ∈ R

M .

Sparse linear model

P(x;θ) ∝K∏

k=1

t(gTk x)

P(y|x;θ) = N (y;Hx, σ2I)

g1 g2 g3 gK

x1 x2 x3 x4 xN

h1 h2 h3 hM

◮ Sparsity directions: s = Gx, with G = [gT1 ; . . . ;g

TK ]

◮ Measurement directions: H = [hT1 ; . . . ;h

TM ]

◮ Sparse potential: t(s), e.g., Laplacian t(s) = e−τk |sk |

◮ Model parameters: θ = (G,H, σ2)

3 / 22

Deterministic or Probabilistic Modeling?

Deterministic modeling: Standard Compressive Sensing

◮ Find minimum energy configuration

◮ Same as finding the posterior MAP

Probabilistic modeling: Bayesian Compressive Sensing

◮ Try to capture the full posterior distribution

◮ Suitable for learning parameters by maximum likelihood

(ML)

◮ Harder than just point estimate

4 / 22

Deterministic Modeling

MAP estimate as an optimization problem

Estimate is xMAP = argminφMAP(x), where

φMAP(x) = σ−2‖y − Hx‖2 − 2

K∑

k=1

log t(sk ) , sk = gTk x .

Properties

◮ Modern optimization techniques allow us find xMAP

efficiently for large-scale problems.

5 / 22

Deterministic Modeling

MAP estimate as an optimization problem

Estimate is xMAP = argminφMAP(x), where

φMAP(x) = σ−2‖y − Hx‖2 − 2

K∑

k=1

log t(sk ) , sk = gTk x .

Properties

◮ Modern optimization techniques allow us find xMAP

efficiently for large-scale problems.

◮ How much do we trust the solution? What about error

bars?

◮ Is the MAP best in terms of PSNR performance?

5 / 22

Probabilistic Modeling

Work with the full posterior distribution

P(x|y) ∝ N (y;Hx, σ2I)K∏

k=1

t(gTk x) .

Pri

or/

Me

asu

reP

oste

rio

r

(Figure from Seeger & Wipf, ’10)6 / 22

Probabilistic ModelingMarkov Chain Monte-Carlo vs. Variational Bayes

Markov Chain Monte-Carlo

◮ Draw samples from the posterior

◮ Typically model prior with Gaussian mixtures and perform

block Gibbs sampling.

◮ Very general, but can be slow and difficult to monitor

convergence

◮ [Schmidt, Rao & Roth ’10], [Papandreou & Yuille, ’10], ...

Variational Bayes

◮ Approximate the posterior distribution with a tractable

parametric form

◮ Systematic error but often guaranteed convergence

◮ [Attias, ’99], [Girolami, ’01], [Lewicki & Sejnowski, ’00], [Palmer et al., ’05], [Levin

et al., ’11], [Seeger & Nickisch, ’11], ...7 / 22

Variational Bounding

◮ Approximate the posterior distribution with a Gaussian

Q(x|y) ∝ N (y;Hx, σ2I)e− 12

sTΓ−1s = N (x; xQ,A

−1) ,

with xQ = A−1b , A = σ−2HT H + GTΓ−1G ,

Γ = diag(γ) , and b = σ−2HT y .

◮ Suitable for super-Gaussian priors

t(sk ) = supγk>0

e−s2k/(2γk )−hk (γk )/2

◮ Optimization problem: Find the variational parameters γ

that give the tightest fit.

8 / 22

Variational Bounding: Double-Loop Algorithm

Outer Loop: Variance Computation

Compute z = diag(GA−1GT ), i.e. the vector of variances

zk = VarQ(sk |y) along the sparsity directions sk = gTk x.

Inner Loop: Smoothed Estimation

◮ Obtain the variational mean xQ = argminx φQ(x; z), where

φQ(x; z) = σ−2‖y − Hx‖2 − 2

K∑

k=1

log t(

(s2k + zk )

1/2)

◮ Update the variational parameters

γ−1k = −2

d log t(√

v)

dv

∣

∣

∣

v=s2k+zk

Convex if standard MAP is convex. See [Seeger & Nickisch, ’11].9 / 22

Variance Computation

Goal: Estimate elements of Σ = A−1, where

A = σ−2HT H + GTΓ−1G

◮ Direct inversion is hopeless (N ≈ 106).

◮ Accurate and fast techniques for problems of special

structure [Malioutov et al., ’08].

◮ Lanczos iteration (only MVM required) [Schneider & Willsky, ’01],

[Seeger & Nickisch, ’11].

10 / 22

Variance Computation

Goal: Estimate elements of Σ = A−1, where

A = σ−2HT H + GTΓ−1G

◮ Direct inversion is hopeless (N ≈ 106).

◮ Accurate and fast techniques for problems of special

structure [Malioutov et al., ’08].

◮ Lanczos iteration (only MVM required) [Schneider & Willsky, ’01],

[Seeger & Nickisch, ’11].

◮ This work: Monte-Carlo variance estimation.

10 / 22

Gaussian Sampling by Local Perturbations

g1 g2 g3 gK

x1 x2 x3 x4 xN

h1 h2 h3 hM

g1 g2 g3 gK

x1 x2 x3 x4 xN

h1 h2 h3 hM

Gaussian MRF sampling by local noise injection

1. Local Perturbations : y ∼ N (0, σ2I), and β ∼ N (0,Γ−1)

2. Gaussian Mode : Ax = σ−2HT y + GT β

Then x ∼ N (0,A−1), where A = σ−2HT H + GTΓ−1G.

[Papandreou & Yuille, ’10]

11 / 22

Monte-Carlo Variance Estimation

Let xi ∼ N (0,A−1), with i = 1, . . . ,Ns.

General purpose Monte-Carlo variance estimator

Σ =1

Ns

Ns∑

i=1

xi xTi , zk =

1

Ns

Ns∑

i=1

s2k ,i ,

where sk ,i = gTk xi .

Properties

◮ Marginal distribution of estimates zk/zk ∼ 1Nsχ2(Ns).

◮ Unbiased E {zk} = zk .

◮ Relative error is r = ∆(zk )/zk =√

2/Ns.

12 / 22

Monte-Carlo vs. Lanczos Variance Estimates

0 2 4 6 8

x 10−3

0

2

4

6

8x 10

−3

zk

zk

SAMPLELANCZOSEXACT

13 / 22

Application: Image Deconvolution

≈ ∗

◮ Measurement equation: y ≈ k ∗ x = Hx.

◮ Non-blind deconvolution (known blur kernel k).

◮ Blind deconvolution (unknown blur kernel k).

14 / 22

Blind Image Deconvolution

Blur kernel recovery by Maximum Likelihood

◮ ML objective: k = argmaxk P(y; k) = argmaxk

∫

P(y, x; k)dx.

◮ Variational ML: k = argmaxk Q(y; k)

◮ Contrast with argmaxk (maxx P(x, y; k)).

◮ [Fergus et al., ’06], [Levin et al., ’09].

15 / 22

Variational EM for Maximum Likelihood

Find k by maximizing Q(y; k) [Girolami, ’01], [Levin et al., ’11].

E-Step

Given current kernel estimate kt , do variational Bayesian

inference, i.e., fit Q(x|y; kt).

M-Step

Maximize w.r.t. k the expected complete log-likelihood

EQ(x|y;kt ) {log Q(x, y; k)}. Equivalently, minimize w.r.t. k

EQ(x|y;kt )

{

1

2‖y − Hx‖2

}

=1

2tr(

(HT H)(A−1 + xxT ))

− yT Hx + (const)

=1

2kT Rxxk − rT

xyk + (const)

Expected moments Rxx estimated by Gaussian sampling.

16 / 22

Summary of Computational Primitives

Smoothed estimationObtain the variational mean xQ = argminx φQ(x; z), where

φQ(x; z) = σ−2‖y − Hx‖2 − 2

K∑

k=1

log t(

(s2k + zk )

1/2)

◮ Inner loop of variational inference.

Sparse linear system

Ax = b, where A = σ−2HT H + GTΓ−1G .

◮ Estimate variances in outer loop of variational inference

and moments Rxx in blind image deconvolution.

17 / 22

Summary of Computational Primitives

Smoothed estimationObtain the variational mean xQ = argminx φQ(x; z), where

φQ(x; z) = σ−2‖y − Hx‖2 − 2

K∑

k=1

log t(

(s2k + zk )

1/2)

◮ Inner loop of variational inference.

Sparse linear system

Ax = b, where A = σ−2HT H + GTΓ−1G .

◮ Estimate variances in outer loop of variational inference

and moments Rxx in blind image deconvolution.

◮ Solve with preconditioned conjugate gradients.17 / 22

Efficient Circulant Preconditioning

Approximate

A = σ−2HT H + GTΓ−1G with P = σ−2HT H + γ−1GT G ,

with γ−1 , (1/K )∑K

k=1 γ−1k [Lefkimmiatis et al., ’12].

Properties

◮ Thanks to stationarity of P, DFT techniques apply.

◮ Optimality: P = argminX∈C‖X − A‖

18 / 22

Effect of Preconditioner

0 20 40 60 80 100 12010

−15

10−10

10−5

100

105

1010

CGPCG

19 / 22

Non-Blind Image Deblurring Example

ground truth our result (PSNR=31.93dB)

blurred (PSNR=22.57dB) VB stdev

20 / 22

Blind Image Deblurring Example

ground truth our result (PSNR=27.54dB)

blurred (PSNR=22.57dB) kernel

21 / 22

Summary

Main Points

◮ Variational Bayesian inference using standard optimization

primitives.

◮ Scalable to large-scale problems.

◮ Open question: Monte-Carlo or Variational?

Summary

Main Points

◮ Variational Bayesian inference using standard optimization

primitives.

◮ Scalable to large-scale problems.

◮ Open question: Monte-Carlo or Variational?

Our software integrated in the glm-ie open source toolbox.

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