(Probably) Concave Graph Matching Haggai Maron and Yaron Lipman - - PowerPoint PPT Presentation

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(Probably) Concave Graph Matching Haggai Maron and Yaron Lipman - - PowerPoint PPT Presentation

Jun 21, 2023 447 likes β€’623 views

(Probably) Concave Graph Matching Haggai Maron and Yaron Lipman Weizmann Institute of Science Graph Matching n tr( ) min Graph Matching n tr( ) min DS Previous Work

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SLIDE 1

(Probably) Concave Graph Matching

Haggai Maron and Yaron Lipman Weizmann Institute of Science

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SLIDE 2

Graph Matching

min

π‘ŒβˆˆΞ n βˆ’tr(π‘©π‘Œπ‘ͺπ‘Œπ‘ˆ)

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SLIDE 3

Graph Matching

min

π‘ŒβˆˆΞ n βˆ’tr(π‘©π‘Œπ‘ͺπ‘Œπ‘ˆ) DS

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SLIDE 4

Previous Work

  • Superiority of the indefinite relaxation
  • [Lyzinski et al. PAMI 2016]
  • Efficient graph matching via concave energies
  • [Vestner et al. CVPR 2017, Boyarski et al. 3DV 2017]
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SLIDE 5

Advantages of Concave Relaxations

  • All local minima are permutation matrices

πΉπ‘œπ‘“π‘ π‘•π‘§ πΈπ‘π‘›π‘π‘—π‘œ

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SLIDE 6

Many important graph matching problems are concave!

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SLIDE 7

Which 𝐡, 𝐢 give rise to concave relaxations?

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SLIDE 8

Concavity of Indefinite Relaxation

  • Theorem: It is sufficient that

𝐡 = Ξ¦ 𝑦𝑗 βˆ’ π‘¦π‘˜ , 𝐢 = Ξ¨(𝑧𝑗 βˆ’ π‘§π‘˜) where Ξ¦, Ξ¨ are positive definite functions of order one.

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SLIDE 9

Concavity of Indefinite Relaxation

  • Theorem: It is sufficient that

𝐡 = Ξ¦ 𝑦𝑗 βˆ’ π‘¦π‘˜ , 𝐢 = Ξ¨(𝑧𝑗 βˆ’ π‘§π‘˜) where Ξ¦, Ξ¨ are positive definite functions of order one.

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SLIDE 10

Concave Energies

Euclidean distance in any dimension

  • Mahalanobis distances
  • Spectral graph distances
  • Matching objects with deep descriptors

Spherical distance in any dimension [bogomolny, 2007]

π΅π‘—π‘˜ = ||𝑦𝑗 βˆ’ π‘¦π‘˜||2 π΅π‘—π‘˜ = π‘’π‘‡π‘œ (𝑦𝑗, π‘¦π‘˜)

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SLIDE 11

Do we really need a concave relaxation?

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SLIDE 12

Do we really need a concave relaxation?

Image taken from Crane et al. 2017

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SLIDE 13

Probably Concave Energies

  • Theorem (upper bound on the probability of convex restriction)

Let 𝑁 ∈ ℝ𝑛×𝑛 and 𝐸 ≀ ℝ𝑛 a uniformly sampled 𝑒-dimensional subspace, then: 𝑄𝑠 𝑁

𝐸 ≻ 0 ≀ min 𝑒 𝑗=1 π‘œ

1 βˆ’ 2π‘’πœ‡π‘— βˆ’π‘’/2

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SLIDE 14

Probably Concave Energies

  • Theorem (upper bound on the probability of convex restriction)

Let 𝑁 ∈ ℝ𝑛×𝑛 and 𝐸 ≀ ℝ𝑛 a uniformly sampled 𝑒-dimensional subspace, then: 𝑄𝑠 𝑁

𝐸 ≻ 0 ≀ min 𝑒 𝑗=1 π‘œ

1 βˆ’ 2π‘’πœ‡π‘— βˆ’π‘’/2

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SLIDE 15

Applications

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SLIDE 16

Conclusion

  • A large family of concave or probably concave relaxations
  • Checking probable concavity with eigenvalue bound
  • Extension of [Lyzinsky et al. 2016] to practical matching problems
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SLIDE 17

The End

  • Support
  • ERC Grant (LiftMatch)
  • Israel Science Foundation
  • Thanks for listening!