Proteins, Particles, and Pseudo-Max- Marginals: A Submodular Approach Jason Pacheco Erik Sudderth
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1 Proteins, Particles, and Pseudo-Max- Marginals: A Submodular Approach Jason Pacheco Erik Sudderth Department of Computer Science Brown University, Providence RI
2 Protein Side Chain Prediction Estimate side chains from backbone. x 1 x 4 x 2 x 3 x 7 x 8 x 5 x 6 1D 4D Continuous state. [ Image: Harder et al., BMC Informatics 2010 ]
3 Reweighted Max-Product (RMP) Message passing on discrete side chains Max-marginal: Pseudo-max-marginal: Edge Appearance Probability
4 Rotamer discretization Fit to side chain marginal statistics Rotamers Truth Rotamers 60 o 180 o 300 o Fails to capture side chain placement Penicillin Acylase Complex, Trp154 [Shapovalov & Dunbrack 2007]
5 Particle Max-Product (PMP) Latent space is continuous particle approximation of continuous RMP messages.
6 Particle Max-Product (PMP) 1 Augment Particles Sample new particles from proposals: ( Random Walk, Likelihood, Neighbor, )
7 Particle Max-Product (PMP) 1 Augment Particles 2 RMP Update Update RMP messages on augmented particles. Edge Appearance Probability
8 Particle Max-Product (PMP) 1 Augment Particles 2 RMP Update 3 Select Particles Select subset of good particles Need particle selection method.
9 Greedy PMP (G-PMP) Select best particle, sample from random walk Gaussian. [ Trinh 09, Peng 11 ] Wrong Truth Particles Naïve proposals do not exploit model.
10 Top-N PMP (T-PMP) Select N-best particles ranked by pseudomax-marginal values. [ Besse 12, Pacheco 14 ] Wrong Truth Particles Particles collapse to single solution.
11 Diverse PMP (D-PMP) Select particles to preserve messages. Diverse States Diverse States Truth Particles Encourages particle diversity Robust to initialization
12 Diverse Particle Selection At node select particles to minimize maximum outgoing message error: Binary Selection Vector RMP message over subset: Approximate IP with greedy algorithm.
13 Diverse Particle Selection Pacheco et al. ICML 2014 Pose Estimation Good empirical results Difficult to analyze Limited to tree-structured MRFs
14 Diverse Particle Selection Equivalent to minimizing norm. Consider other norms, e.g. : Easier to analyze Property 1: Message error upper bounds pseudo-max-marginal error:
15 Submodular Particle Selection Property 2: Selection IP equivalent to submodular maximization. Set function iff diminishing marginal gains. is submodular Margin P. 3: Efficient LAZYGREEDY selection within factor of optimal value.
16 LAZYGREEDY Selection Selection Objective: Joint Probability Message Margin
17 LAZYGREEDY Selection Selection Objective: Joint Probability Message Margin
18 LAZYGREEDY Selection Selection Objective: Joint Probability Message Margin
19 LAZYGREEDY Selection Selection Objective: Joint Probability Message Margin
20 LAZYGREEDY Selection Selection Objective: Joint Probability Message Margin
21 LAZYGREEDY Selection Selection Objective: Joint Probability Message Margin
22 Protein Side Chain Prediction Pairwise Markov random field (MRF): Gaussian Mixture Lennard-Jones Estimate side chain for fixed backbone 1D to 4D continuous states. [ Image: Harder et al., BMC Informatics 2010 ]
23 Protein Side Chain Prediction
24 Protein Side Chain Prediction
25 Protein Side Chain Prediction Log-probability of MAP estimate for 20 Proteins (11 Runs) 370 Proteins G-PMP, T-PMP, D-PMP, D-PMP Rosetta simulated annealing [Rohl et al., 2004]
26 Protein Side Chain Prediction Root mean square deviation (RMSD) from x-ray structure. Rosetta G-PMP T-PMP D-PMP D-PMP Oracle selects best configuration in current particle set.
27 Optical Flow Estimate 2D motion for every superpixel. Middlebury optical flow benchmark [Baker et al. 2011]
28 Optical Flow Estimate 2D motion for every superpixel. Middlebury optical flow benchmark [Baker et al. 2011]
29 Optical Flow Estimate 2D motion for every superpixel. Middlebury optical flow benchmark [Baker et al. 2011]
30 Optical Flow Flow ambiguity near object boundaries D-PMP Particles D-PMP Estimate D-PMP particles reflect this.
31 Optical Flow Training Test D-PMP accuracy equivalent to Classic-C [Sun et al. 2014]
32 Summary General purpose particle-based maxproduct for continuous graphical models with cycles. Code Available: cs.brown.edu/~pachecoj
33
34 Diverse Particle Selection Minimize sum of errors ( ): Augmented Messages Subset Messages Selection Vector Easier to analyze than selection P1: Message error upper bounds pseudomax-marginal error:
35 Submodularity A function is submodular iff diminishing marginal gains: Diverse particle selection is submodular maximization with cardinality constraint Efficient greedy approximation algorithm
36 Resolving Ties Particle diversity leads to more conflicts: Side Chain Particles T-PMP D-PMP
37 Submodular Particle Selection Augmented Messages Subset Messages Selection Vector Property 1: Message reconstruction error bounds pseudo-max-marginal error: Property 2: IP is equivalent to submodular maximization subject to cardinality constraints
38 Submodular Particle Selection Select particles to minimize sum of errors: Augmented Messages Subset Messages Selection Vector Good empirical results and we can analyze! Property 1: Message error bounds pseudo-max-marginal: Property 2: Equivalent to submodular maximization subject to cardinality constraints
39 Reweighted Max-Product (RMP) Message passing on discrete side chain states. But latent space is continuous RMP Messages: Edge Appearance Probability Pseudo-max-marginal:
40 Optical Flow Estimate motion vector for every pixel. Diverse Select Particle Top-N Particles Selection (T-PMP) (D-PMP)
41 Diverse Particle Selection Minimize maximum message error ( ): Pose Estimation Augmented Messages Subset Messages Selection Vector [ Pacheco et al., ICML 2014 ] Good empirical results No analysis/guarantees Limited to tree MRFs
Applied Bayesian Nonparametrics 5. Spatial Models via Gaussian Processes, not MRFs Tutorial at CVPR 2012 Erik Sudderth Brown University
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