Maximizing a Tree Series in the Representation Space Guillaume Rabusseau, Fran¸ cois Denis ICGI 2014 September 19, 2014
Overview Starting Point: Metropolis Procedural Modeling 1 Problem Formulation 2 Working in the Representation Space 3 Experiments 4 Conclusion 5 Guillaume Rabusseau, Fran¸ cois Denis (Qarma) Maximization in the Representation Space September 19, 2014 2 / 23
Overview Starting Point: Metropolis Procedural Modeling 1 Problem Formulation 2 Preliminaries: Tree Series and the Representation Space Motivations and Problematic Working in the Representation Space 3 Complexity Study Metropolis-Hastings in the Representation Space Experiments 4 Conclusion 5 Guillaume Rabusseau, Fran¸ cois Denis (Qarma) Maximization in the Representation Space September 19, 2014 3 / 23
Metropolis Procedural Modeling [Talton et al., 2011] PCFG G PCFG G , I = · · · t 1 ∈ T G t 2 ∈ T G ˆ t ∈ T G · · · 2 steps: Define a posterior distribution p ( t | I ) ∝ π ( t ) L ( I | t ) on T G Find ˆ t ∈ T G maximizing p ( ·| I ) ⇒ Metropolis-Hastings Guillaume Rabusseau, Fran¸ cois Denis (Qarma) Maximization in the Representation Space September 19, 2014 4 / 23
Metropolis-Hastings Algorithm � p : X → R + such that Z = X p ( x ) d x < ∞ ⇒ ˆ p : x �→ p ( x ) / Z is a probability distribution on X . Guillaume Rabusseau, Fran¸ cois Denis (Qarma) Maximization in the Representation Space September 19, 2014 5 / 23
Metropolis-Hastings Algorithm � p : X → R + such that Z = X p ( x ) d x < ∞ ⇒ ˆ p : x �→ p ( x ) / Z is a probability distribution on X . To sample from ˆ p : (i) Choose a jump distribution q x ( · ) (distribution on X for each x ∈ X ). (ii) Build a Markov chain in X : x n ∈ X Input : Returns : x n+1 ∈ X 1: Draw a candidate x ∗ ∼ q x n ( · ) 2: Accept x ∗ (i.e. x n+1 ← x ∗ , otherwise x n+1 ← x n ) with probability � 1 , p ( x ∗ ) q x ∗ ( x n ) � α ( x n , x ∗ ) = min p ( x n ) q xn ( x ∗ ) Guillaume Rabusseau, Fran¸ cois Denis (Qarma) Maximization in the Representation Space September 19, 2014 5 / 23
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