Geometry of Diversity and Determinantal Point Processes ...

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<strong>Determinantal</strong> point process • Given an N × N symmetric p.s.d. matrix L P(Y = Y ) det(L Y ) P({2, 4}) ∝ ∣ L 22 L 24 L 42 L 44 ∣ [Macchi, 1975]
DPP inference • Normalization: P(Y ) ∝ det(L Y ) • Marginals, conditioning (N 3 or faster) • Exact sampling (N 3 or faster) • MAP / mode is NP-hard, but log-submodular
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Geometry of Diversity and Determina
Page 3 and 4:
Eichner & Ferrari BMVC09
Page 5 and 6: Human pose estimation: what’s so
Page 7 and 8: Human pose estimation: what’s so
Page 9 and 10: Local signal is weak = true right e
Page 11 and 12: Graphical Models vs. Tyranny of Sma
Page 17 and 18: Accuracy of Wrist Localization 90 8
Page 19 and 20: The Catch • Problem: inference ex
Page 23 and 24: A distribution over sets of poses?
Page 25 and 26: Image search: “jaguar” Relevanc
Page 27 and 28: Summarization
Page 29 and 30: Graphical models?
Page 31 and 32: Graphical models? 0/1 sentence i
Page 33 and 34: Graphical models?
Page 35 and 36: Determinantal point processes (DPPs
Page 37 and 38: Determinantal point processes Quali
Page 39 and 40: Discrete point process • N items
Page 41 and 42: Independent point process • Each
Page 43 and 44: Feature function g on items in Y
Page 45 and 46: Feature function g on items in Y
Page 49 and 50: L = L ij = g(i) ⊤ g(j)
Page 51 and 52: Determinantal point process L = P(Y
Page 53 and 54: Determinantal point process • Giv
Page 55: Determinantal point process • Giv
Page 59 and 60: DPP inference • Marginals: P(A Y
Page 61 and 62: P(A Y ) = det(K A )
Page 63 and 64: P(A Y ) = det(K A ) P(i Y ) = det(K
Page 65 and 66: Point process samples Independent D
Page 67 and 68: L =
Page 69 and 70: L = L ij = q(i) (i) (j)q(j)
Page 71 and 72: L = L ij = q(i) (i) (j)q(j) q(i)
Page 73 and 74: q(i) (i) Vol 2 P({i, j}) q(j) (j)
Page 75 and 76: q(i) (i) Vol 2 P({i, j}) q(j) (j) I
Page 77 and 78: Quality vs. diversity • Intuitive
Page 79 and 80: Quality vs. diversity • Intuitive
Page 81 and 82: Eigendecomposition N∑ L = λ n v
Page 83 and 84: 1 2 3 4 5 6 v 4 v 6 v 1 v 2 v 3 v 5
Page 85 and 86: Elementary DPP v 1 v 2 v 3 v 5 P {2
Page 87 and 88: Key insight Every DPP is a “facto
Page 89 and 90: ∑ P ∝ 1 2 3 4 5 6 J⊆{1,...,N}
Page 91 and 92: Choose elementary DPP P J PHASE ONE
Page 93 and 94: Sampling algorithm Choose elementar
Page 95 and 96: Consequences • Phase one determin
Page 97 and 98: What if we need exactly k diverse i
Page 99 and 100: k-DPPs • Simple idea: condition D
Page 101 and 102: k-DPP ∑ P ∝ λ n J⊆{1,...,N}
Page 103 and 104: ∑ P ∝ λ n J⊆{1,...,N} P J
Page 105 and 106: k-DPP sampling • Solution: recurs
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Quality features • Dubbed the “
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Quality features 2. Pizza Hut has r
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Quality features Mean similarity 2.
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Diversity features • φ fixed to
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Diversity features • φ fixed to
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System ROUGE-1F ROUGE-1R R-SU4F Beg
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System ROUGE-1F ROUGE-1R R-SU4F Beg
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? ? ? ?
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Y Y Y I""""""""really""""""""love""
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Structured DPPs • Exponentially m
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1. Factorization • Quality scores
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1. Factorization • Quality scores
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Quality
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Quality x x
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Diversity
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Diversity Low diversity
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Diversity Low diversity High divers
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2. Dual representation Φ Q Q Φ
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2. Dual representation L = C = 2 N
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2. Dual representation L = C = 2 N
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3. Second-order message passing
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3. Second-order message passing •
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Multiple-pose estimation
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Pose accuracy Overall F Arms F 1 1
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News happens Apr 3: Instagram reach
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News happens Apr 9: Facebook buys I
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News happens
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iraq iraqi killed baghdad arab mari
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hotel kitchen casa inches post shad
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Scale • ~35,000 articles per six
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Scale • ~35,000 articles per six
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Results: Human summaries & Turk rat
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Results: Human summaries & Turk rat
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k-SDPPs vs. Tyranny of Small Decisi
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• DPPs capture global, negative c
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