(Probably) Concave Graph Matching Haggai Maron and Yaron Lipman Weizmann Institute of Science
Graph Matching πβΞ n βtr(π©ππͺπ π ) min
Graph Matching πβΞ n βtr(π©ππͺπ π ) min DS
Previous Work β’ Superiority of the indefinite relaxation β’ [Lyzinski et al. PAMI 2016] β’ Efficient graph matching via concave energies β’ [Vestner et al. CVPR 2017, Boyarski et al. 3DV 2017]
Advantages of Concave Relaxations β’ All local minima are permutation matrices πΉπππ ππ§ πΈπππππ
Many important graph matching problems are concave!
Which π΅, πΆ give rise to concave relaxations?
Concavity of Indefinite Relaxation β’ Theorem: It is sufficient that π΅ = Ξ¦ π¦ π β π¦ π , πΆ = Ξ¨(π§ π β π§ π ) where Ξ¦, Ξ¨ are positive definite functions of order one.
Concavity of Indefinite Relaxation β’ Theorem: It is sufficient that π΅ = Ξ¦ π¦ π β π¦ π , πΆ = Ξ¨(π§ π β π§ π ) where Ξ¦, Ξ¨ are positive definite functions of order one.
Concave Energies Euclidean distance in any dimension π΅ ππ = ||π¦ π β π¦ π || 2 β’ Mahalanobis distances β’ Spectral graph distances β’ Matching objects with deep descriptors Spherical distance in any dimension π΅ ππ = π π π (π¦ π , π¦ π ) [bogomolny, 2007]
Do we really need a concave relaxation?
Do we really need a concave relaxation? Image taken from Crane et al. 2017
Probably Concave Energies β’ Theorem (upper bound on the probability of convex restriction) Let π β β πΓπ and πΈ β€ β π a uniformly sampled π -dimensional subspace, then: π 1 β 2π’π π βπ/2 ππ π πΈ β» 0 β€ min π’ π=1
Probably Concave Energies β’ Theorem (upper bound on the probability of convex restriction) Let π β β πΓπ and πΈ β€ β π a uniformly sampled π -dimensional subspace, then: π 1 β 2π’π π βπ/2 ππ π πΈ β» 0 β€ min π’ π=1
Applications
Conclusion β’ A large family of concave or probably concave relaxations β’ Checking probable concavity with eigenvalue bound β’ Extension of [Lyzinsky et al. 2016] to practical matching problems
The End β’ Support β’ ERC Grant (LiftMatch) β’ Israel Science Foundation β’ Thanks for listening!
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