Inference by Learning: Speeding-up Graphical Model Optimization via a Coarse-to-Fine Cascade of Pruning Classifiers
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This document describes a method called Inference by Learning (IbyL) that speeds up inference in Markov random fields (MRFs) while maintaining solution accuracy. IbyL builds a coarse-to-fine hierarchy of MRFs and learns pruning classifiers at each level to iteratively reduce the solution space. It exploits the piecewise smooth structure of MAP solutions. On stereo matching and optical flow problems, IbyL achieves significant speedups over direct optimization while agreeing highly with the best solutions and maintaining low energy.
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Inference by Learning: Speeding-up Graphical Model Optimization via a Coarse-to-Fine Cascade of Pruning Classifiers
- 1. On-‐line resources: h/p://imagine.enpc.fr/~conejob/ibyl/ TBD, X/X/2014 Bruno Conejo, Phd student Ecole des Ponts ParisTech / Research Analyst GPS, Caltech Nikos Komodakis, Associate Professor Ecole des Ponts ParisTech with S. Leprince & JP. Avouac (GPS, Caltech) Inference by Learning: Speeding-‐up Graphical Model OpZmizaZon via a Coarse-‐to-‐Fine Cascade of Pruning Classifiers (NIPS 2014)
- 2. B.Conejo N.Komodakis – Inference by Learning 2 Introduc=on: Associated materials: Paper, code, slides and poster are available on-‐line : h/p://imagine.enpc.fr/~conejob/ibyl/ MoZvaZons: 1. Speed-‐up the inference of MRFs that have a piecewise smooth MAP. 2. While maintaining the accuracy of the inference soluZon of MRFs. Approach: 1. Exploit the piecewise smooth structure of the MAP by iteraZvely opZmizing a coarse to fine representaZon of the MRF. 2. Rely on learning to progressively reduce the soluZon space.
- 3. B.Conejo N.Komodakis – Inference by Learning 3 Nota=ons: Let’s consider the following discrete MRF: : the unary terms of vertex i : the pairwise terms of edge ij support graph set of ver1ces set of edges discrete label set set of unary terms set of pairwise terms i j : the configuraZon of vertex i : the configuraZon of the MRF
- 4. B.Conejo N.Komodakis – Inference by Learning 4 Nota=ons: The Energy: i.e., the total cost is given by The MAP: Maximum At Posteriori The pruning matrix: is defined by: and its associated soluZon space:
- 5. With such that: B.Conejo N.Komodakis – Inference by Learning 5 Approach: To obtain an inference speed-‐up we solve: belongs to most elements of are 0 (i.e., the labels are pruned) The IbyL framework iteraZvely esZmates by: 1. Building a coarse to fine set of MRFs from 2. Learning at each scale pruning classifiers to refine
- 6. B.Conejo N.Komodakis – Inference by Learning 6 Model coarsening: Given and a grouping funcZon , we create a coarsen MRF (slight abuse of the notaZon): The unary and pairwise potenZals of are given by The verZces and edges of are given by
- 7. B.Conejo N.Komodakis – Inference by Learning 7 Coarse to fine op=miza=on and label pruning We iteraZvely apply the previous coarsening to build a coarse to fine set of N+1 progressively coarser MRFs: with: We set all elements of the coarsest pruning matrix to 1 (no pruning) At each scale (s) we apply the following steps: 1. OpZmize the current MRF 2. Update the next scale pruning matrix 3. Up-‐sample the current soluZon for next scale.
- 8. B.Conejo N.Komodakis – Inference by Learning 8 Coarse to fine op=miza=on and label pruning Step 1: OpZmize the current MRF: Step 2: Update the next scale pruning matrix: i. Compute feature map: ii. Update pruning matrix from off-‐line trained classifier: Step 3: Up-‐sample the current soluZon:
- 9. B.Conejo N.Komodakis – Inference by Learning 9 Inference by learning framework We sZll need to define: 1. How to compute the feature map 2. How to train the classifiers
- 10. B.Conejo N.Komodakis – Inference by Learning 10 Feature map We stack K individual scalar features defined on Presence of strong disconZnuity: Local energy variaZon: Unary coarsening:
- 11. B.Conejo N.Komodakis – Inference by Learning 11 Learning pruning classifiers On a training set of MRFs, we run the IbyL framework without any pruning, i.e., . We keep track of features and compute: such that: where denotes the binary OR operator.
- 12. B.Conejo N.Komodakis – Inference by Learning 12 Learning pruning classifiers We split the features in two groups w.r.t the PSD feature. For each group: 1. We have two classes (c0=to prune and c1=to remain acZve) defined from 2. We weight c0 to 1, and c1 to 3. We train a linear C-‐SVM classifier During tesZng, the classifier apply the trained classifier of the corresponding group (w.r.t. the PSD feature).
- 13. B.Conejo N.Komodakis – Inference by Learning 13 Experiments: Setup We experiment with two problems: 1. Stereo-‐matching. 2. OpZcal flow. We evaluate different pruning aggressiveness factor , and compute: 1. The speed-‐up w.r.t. the direct opZmizaZon. 2. The raZo of acZve labels. 3. The energy raZo w.r.t. the direct opZmizaZon. 4. The MAP agreement w.r.t. the best computed soluZon. As an opZmizaZon sub-‐rouZne we use Fast-‐PD.
- 14. B.Conejo N.Komodakis – Inference by Learning 14 Experiments: Results Stereo matching: OpZcal flow:
- 15. B.Conejo N.Komodakis – Inference by Learning 15 Experiments: Stereo matching
- 16. B.Conejo N.Komodakis – Inference by Learning 16 Experiments: Op=cal Flow
- 17. B.Conejo N.Komodakis – Inference by Learning 17 Conclusions The IbyL framework: 1. Gives an important speed-‐up while maintaining excellent accuracy of the soluZon. 2. Can be easily adapted to any MRF task by compuZng task dependent features. 3. Can be easily adapted to high order MRFs. 4. Is available to download at: h/p://imagine.enpc.fr/~conejob/ibyl/
- 18. B.Conejo -‐ Fast global Matching via Energy Pyramid 18