Bayesian Experimental Design for Large Scale Signal Acquisition - PowerPoint PPT Presentation

Bayesian Experimental Design for Large Scale Signal Acquisition Optimization Matthias Seeger Laboratory for Probabilistic Machine Learning Ecole Polytechnique Fédérale de Lausanne http://lapmal.epfl.ch/ 9/12/2013 Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 1 / 20

Motivation Magnetic Resonance Imaging ⊕ Extremely versatile ⊕ Noninvasive, no ionizing radiation ⊖ Very expensive ⊖ Long scan times: Major limiting factor Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 2 / 20

Motivation Magnetic Resonance Imaging Faster scans by undersampled reconstruction Which fast designs give best images? Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 2 / 20

Motivation Magnetic Resonance Imaging Faster scans by undersampled reconstruction Which fast designs give best images? Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 2 / 20

Motivation Image Reconstruction y X u Measurement Design Ideal Image u Data y Reconstruction Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 3 / 20

Motivation Sampling Optimization y X u Reconstruction Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 4 / 20

Bayesian Experimental Design Bayesian Experimental Design Posterior: Uncertainty in reconstruction Experimental design: Find poorly determined directions Sequential search with interjacent partial measurements Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 5 / 20

Bayesian Experimental Design Bayesian Experimental Design Posterior: Uncertainty in reconstruction Experimental design: Find poorly determined directions Sequential search with interjacent partial measurements Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 5 / 20

Bayesian Experimental Design Maximizing Information Gain Score design extension X ∗ by information gain: I ( X ∗ ) = I ( y ∗ , u | y ) = H [ P ( u | y )] − H [ P ( u | y ∗ , y )] � �� � � �� � Before After Most work: Combinatorial aspects Assume: I ( X ∗ ) tractable to compute Assume: I ( X ∗ ) cheap to compute (many X ∗ ) Simple greedy forward works well in practice . . . Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 6 / 20

Bayesian Experimental Design Maximizing Information Gain Score design extension X ∗ by information gain: I ( X ∗ ) = I ( y ∗ , u | y ) = H [ P ( u | y )] − H [ P ( u | y ∗ , y )] � �� � � �� � Before After Most work: Combinatorial aspects Assume: I ( X ∗ ) tractable to compute Assume: I ( X ∗ ) cheap to compute (many X ∗ ) Simple greedy forward works well in practice . . . Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 6 / 20

Bayesian Experimental Design Maximizing Information Gain Score design extension X ∗ by information gain: I ( X ∗ ) = I ( y ∗ , u | y ) = H [ P ( u | y )] − H [ P ( u | y ∗ , y )] � �� � � �� � Before After Most work: Combinatorial aspects Assume: I ( X ∗ ) tractable to compute Assume: I ( X ∗ ) cheap to compute (many X ∗ ) Simple greedy forward works well in practice . . . So is it . . . ? Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 6 / 20

Bayesian Experimental Design Challenges I ( X ∗ ) = H [ P ( u | y )] − H [ P ( u | y ∗ , y ) Assume: I ( X ∗ ) tractable to compute. Only if P ( u | y ) Gaussian . . . Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 7 / 20

Bayesian Experimental Design Image Statistics Whatever images are . . . they are not Gaussian! Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 8 / 20

Bayesian Experimental Design Challenges I ( X ∗ ) = H [ P ( u | y )] − H [ P ( u | y ∗ , y ) Assume: I ( X ∗ ) tractable to compute? No: Needs approximate inference Assume: I ( X ∗ ) cheap to compute (many X ∗ ). Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 9 / 20

Bayesian Experimental Design Size Does Matter Global covariances 1 Scores I ( X ∗ ) need full Cov P [ u | y ] Massive scale 2 R 131072 (just one slice). Many times 3 Posterior after each design extension Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 10 / 20

Bayesian Experimental Design Challenges I ( X ∗ ) = H [ P ( u | y )] − H [ P ( u | y ∗ , y ) Assume: I ( X ∗ ) tractable to compute? No: Needs approximate inference Assume: I ( X ∗ ) cheap to compute (many X ∗ )? No: Needs new algorithms and high performance computing Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 11 / 20

Variational Bayesian Inference Variational Bayesian Inference Approximate inference for non-Gaussian models Computations driven by Gaussian inference Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 12 / 20

Variational Bayesian Inference Variational Bayesian Inference P ( u | y ) = P ( y | u ) × P ( u ) P ( y ) Variational Inference Approximation Write intractable integration as optimization Relax to tractable optimization problem Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 13 / 20

Variational Bayesian Inference Variational Bayesian Inference P ( u | y ) = P ( y | u ) × P ( u ) P ( y ) Variational Relaxation: Bound the master function � P ( u , y ) d u ≤ 1 − log P ( y ) = − log 2 min γ min u ∗ φ ( u ∗ , γ ) Approximate posterior P ( u | y ) by Gaussian Integration ⇒ Convex optimization Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 13 / 20

Variational Bayesian Inference Variational Bayesian Inference P ( u | y ) = P ( y | u ) × P ( u ) P ( y ) Variational Relaxation: Bound the master function � P ( u , y ) d u ≤ 1 − log P ( y ) = − log 2 min γ min u ∗ φ ( u ∗ , γ ) Approximate posterior P ( u | y ) by Gaussian Integration ⇒ Convex optimization Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 13 / 20

Variational Bayesian Inference No Inference Without . . . Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 14 / 20

Variational Bayesian Inference Double Loop Algorithm Double loop algorithm Inner loop optimization: Standard MAP Estimation Outer loop update: Gaussian Variances Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 15 / 20

Variational Bayesian Inference Double Loop Algorithm Double loop algorithm Inner loop optimization: Standard MAP Estimation Outer loop update: Gaussian Variances Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 15 / 20

Variational Bayesian Inference Tricks of the Trade Monte Carlo Gaussian variances: Perturb&MAP Papandreou, Yuille, NIPS 2010 Inner loop: Fast first-order MAP solvers Warmstarting variational optimization: Small changes after each ( X ∗ , y ∗ ) Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 16 / 20

Variational Bayesian Inference Tricks of the Trade Monte Carlo Gaussian variances: Perturb&MAP Papandreou, Yuille, NIPS 2010 Inner loop: Fast first-order MAP solvers Warmstarting variational optimization: Small changes after each ( X ∗ , y ∗ ) Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 16 / 20

Experimental Results Optimizing Cartesian MRI Bayes Optim. VD Random Low Pass Seeger et.al. , MRM 63(1), 2010 Lustig, Donoho, Pauli, MRM 58(6), 2007 Common MRI practice Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 17 / 20

Experimental Results Experimental Results: Test Set Errors sagittal short TE sagittal long TE Low Pass 7 L 2 reconstruction error VD Random 6 Bayes Optim 5 4 3 2 1 axial short TE axial long TE 7 L 2 reconstruction error 6 5 4 3 2 1 80 100 120 140 160 80 100 120 140 160 Number phase encodes Number phase encodes Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 18 / 20

Outlook Large Scale Bayesian Inference Advanced Bayesian experimental design Signal acquisition optimization Computer Vision Hierarchically structured image priors Ko, Seeger, ICML 2012 Learning Image Models (fields of experts, . . . ) Bayesian dictionary learning Intelligent user interfaces (Bayesian active learning) Advanced variational inference Speeding up expectation propagation Seeger, Nickisch, AISTATS 2011 Generic framework You can do MAP estimation efficiently? You can do variational Bayesian inference! Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 19 / 20

Outlook Large Scale Bayesian Inference Advanced Bayesian experimental design Signal acquisition optimization Computer Vision Hierarchically structured image priors Ko, Seeger, ICML 2012 Learning Image Models (fields of experts, . . . ) Bayesian dictionary learning Intelligent user interfaces (Bayesian active learning) Advanced variational inference Speeding up expectation propagation Seeger, Nickisch, AISTATS 2011 Generic framework You can do MAP estimation efficiently? You can do variational Bayesian inference! Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 19 / 20

Outlook People& Code glm-ie: Toolbox by Hannes Nickisch mloss.org/software/view/269/ Generalized sparse linear models MAP reconstruction and variational Bayesian inference (double loop algorithm for super-Gaussian bounding) Matlab 7.x, GNU Octave 3.2.x Hannes Nickisch (now Philips Research, Hamburg) Rolf Pohmann, Bernhard Schölkopf (MPI Tübingen) Young Jun Ko Emtiyaz Khan Seeger (EPFL) Large Scale Bayesian Inference 9/12/2013 20 / 20