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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

SLIDE 1

Maximizing a Tree Series in the Representation Space

Guillaume Rabusseau, Fran¸ cois Denis

ICGI 2014

September 19, 2014

SLIDE 2

Overview

1

Starting Point: Metropolis Procedural Modeling

2

Problem Formulation

3

Working in the Representation Space

4

Experiments

5

Conclusion

Guillaume Rabusseau, Fran¸ cois Denis (Qarma) Maximization in the Representation Space September 19, 2014 2 / 23

SLIDE 3

Overview

1

Starting Point: Metropolis Procedural Modeling

2

Problem Formulation Preliminaries: Tree Series and the Representation Space Motivations and Problematic

3

Working in the Representation Space Complexity Study Metropolis-Hastings in the Representation Space

4

Experiments

5

Conclusion

Guillaume Rabusseau, Fran¸ cois Denis (Qarma) Maximization in the Representation Space September 19, 2014 3 / 23

SLIDE 4

Metropolis Procedural Modeling [Talton et al., 2011]

PCFG G

t1 ∈ TG t2 ∈ TG

· · · · · · PCFG G, I =

ˆ t ∈ TG

2 steps: Define a posterior distribution p(t|I) ∝ π(t)L(I|t) on TG Find ˆ t ∈ TG maximizing p(·|I) ⇒ Metropolis-Hastings

Guillaume Rabusseau, Fran¸ cois Denis (Qarma) Maximization in the Representation Space September 19, 2014 4 / 23

SLIDE 5

Metropolis-Hastings Algorithm

p : X → R+ such that Z =

X p(x) dx < ∞

⇒ 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

SLIDE 6

Metropolis-Hastings Algorithm

p : X → R+ such that Z =

X p(x) dx < ∞

⇒ ˆ p : x → p(x) /Z is a probability distribution on X. To sample from ˆ p : (i) Choose a jump distribution qx(·) (distribution on X for each x ∈ X). (ii) Build a Markov chain in X : Input : xn ∈ X Returns : xn+1 ∈ X

1: Draw a candidate x∗ ∼ qxn(·) 2: Accept x∗ (i.e. xn+1 ← x∗, otherwise xn+1 ← xn) with probability

α(xn, x∗) = min

1, p(x∗)qx∗(xn)

p(xn)qxn(x∗)

Guillaume Rabusseau, Fran¸

cois Denis (Qarma) Maximization in the Representation Space September 19, 2014 5 / 23