Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals

1. Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals Nicolas Papadakis with contributions from J.-F. Aujol, A. I. Aviles-Rivero, A. Chambolle, T. Feld, G. Gilboa and C.-B. Schönlieb N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 0 / 18
2. Introduction - Total Variation in imaging For u of Bounded Variation on Ω Z TV (u) = sup udiv(z) z∈Cc∞ ; ||z||∞ ≤1 Ω Discrete setting in imaging X TV (u) = ||(∇u)i ||2 Denoising with TV : piecewise constant prior i • Intensively used for image regularization [Rudin, Osher, Fatemi ’92] N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 1 / 18
3. Introduction - Total Variation in imaging For u of Bounded Variation on Ω Z TV (u) = sup udiv(z) z∈Cc∞ ; ||z||∞ ≤1 Ω Discrete setting in imaging X TV (u) = ||(∇u)i ||2 Denoising with TV : piecewise constant prior i • Intensively used for image regularization [Rudin, Osher, Fatemi ’92] • Theory [Andreu, Ballester, Caselles, Chambolle, Mázon, Novaga... ’00-] N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 1 / 18
4. Introduction - Total Variation in imaging For u of Bounded Variation on Ω Z TV (u) = sup udiv(z) z∈Cc∞ ; ||z||∞ ≤1 Ω Discrete setting in imaging X TV (u) = ||(∇u)i ||2 Denoising with TV : piecewise constant prior i • Intensively used for image regularization [Rudin, Osher, Fatemi ’92] • Theory [Andreu, Ballester, Caselles, Chambolle, Mázon, Novaga... ’00-] • Algorithms [Chambolle ’04 and Pock ’11, Condat ’13] N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 1 / 18
5. Introduction - Total Variation in imaging For u of Bounded Variation on Ω Z TV (u) = sup udiv(z) z∈Cc∞ ; ||z||∞ ≤1 Ω Discrete setting in imaging X TV (u) = ||(∇u)i ||2 Denoising with TV : piecewise constant prior i • Intensively used for image regularization [Rudin, Osher, Fatemi ’92] • Theory [Andreu, Ballester, Caselles, Chambolle, Mázon, Novaga... ’00-] • Algorithms [Chambolle ’04 and Pock ’11, Condat ’13] • Non linear spectral decomposition [Gilboa ’13-] N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 1 / 18
6. Introduction - Total Variation in imaging For u of Bounded Variation on Ω Z TV (u) = sup udiv(z) z∈Cc∞ ; ||z||∞ ≤1 Ω Discrete setting in imaging X TV (u) = ||(∇u)i ||2 Denoising with TV : piecewise constant prior i • Intensively used for image regularization [Rudin, Osher, Fatemi ’92] • Theory [Andreu, Ballester, Caselles, Chambolle, Mázon, Novaga... ’00-] • Algorithms [Chambolle ’04 and Pock ’11, Condat ’13] • Non linear spectral decomposition [Gilboa ’13-] Lot of (cool) imaging applications [Benning et al. ’17, Gilboa ’18] N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 1 / 18
7. Eigenfunctions of Total Variation • Solve the non linear eigenvalue problem w.r.t subgradient of TV λu ∈ ∂TV (u) • Atoms of TV regularization in inverse problems. Ex 1D: (u) Local minima of TV ||u||2 are eigenfunctions of TV [Andreu et al. ’01] N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 2 / 18
8. Eigenfunctions of Total Variation • Solve the non linear eigenvalue problem w.r.t subgradient of TV λu ∈ ∂TV (u) • Atoms of TV regularization in inverse problems. Ex 1D: (u) Local minima of TV ||u||2 are eigenfunctions of TV [Andreu et al. ’01] Calibrable sets • C ⊂ Ω such that χC is an eigenfunction of TV • A calibrable set is convex with bounded curvature [Belettini et al. ’02]: ess sup κ(q) ≤ q∈∂C Per (C) |C| [Gilboa et al. ’16] N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 2 / 18
9. Eigenfunctions of Total Variation • Solve the non linear eigenvalue problem w.r.t subgradient of TV λu ∈ ∂TV (u) • Atoms of TV regularization in inverse problems. Ex 1D: (u) Local minima of TV ||u||2 are eigenfunctions of TV [Andreu et al. ’01] Calibrable sets • C ⊂ Ω such that χC is an eigenfunction of TV • A calibrable set is convex with bounded curvature [Belettini et al. ’02]: ess sup κ(q) ≤ q∈∂C Per (C) |C| [Gilboa et al. ’16] Cheeger set of Ω : min C⊂Ω N. Papadakis TV (χC ) |C| Cheeger cut of Ω : min TV (χC ) C⊂Ω max(|C| |Ω\C|) Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 2 / 18
10. Eigenfunctions of Total Variation • Solve the non linear eigenvalue problem w.r.t subgradient of TV λu ∈ ∂TV (u) • Atoms of TV regularization in inverse problems. Ex 1D: (u) Local minima of TV ||u||2 are eigenfunctions of TV [Andreu et al. ’01] Calibrable sets • C ⊂ Ω such that χC is an eigenfunction of TV • A calibrable set is convex with bounded curvature [Belettini et al. ’02]: ess sup κ(q) ≤ q∈∂C Per (C) |C| [Gilboa et al. ’16] Cheeger set of Ω : min C⊂Ω N. Papadakis TV (χC ) |C| Cheeger cut of Ω : min TV (χC ) C⊂Ω max(|C| |Ω\C|) Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 2 / 18
11. Minimization of a non linear quotient min u N. Papadakis TV (u) H(u) • H(u) = ||u||2 : eigenfunctions of TV • H(u) = ||u||1 : calibrable sets of TV Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 3 / 18
12. Minimization of a non linear quotient min u TV (u) H(u) • H(u) = ||u||2 : eigenfunctions of TV • H(u) = ||u||1 : calibrable sets of TV Numerical approaches • Maximum Cheeger set with a projection algorithm [Carlier et al. ’08] N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 3 / 18
13. Minimization of a non linear quotient min u TV (u) H(u) • H(u) = ||u||2 : eigenfunctions of TV • H(u) = ||u||1 : calibrable sets of TV Numerical approaches • Maximum Cheeger set with a projection algorithm [Carlier et al. ’08] • PDE associated to the TV flow  u(0) = u0 ∂t u = −p p ∈ ∂TV (u) - u0 is an eigenfunction: u(t) = (1 − tλ)+ u0 N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 3 / 18
14. Minimization of a non linear quotient min u TV (u) H(u) • H(u) = ||u||2 : eigenfunctions of TV • H(u) = ||u||1 : calibrable sets of TV Numerical approaches • Maximum Cheeger set with a projection algorithm [Carlier et al. ’08] • PDE associated to the TV flow  u(0) = u0 ∂t u = −p p ∈ ∂TV (u) - u0 is an eigenfunction: u(t) = (1 − tλ)+ u0 - At instinction time, u(t) is an eigenfunction [Bungert et al. ’19] N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 3 / 18
15. Minimization of a non linear quotient min u TV (u) H(u) • H(u) = ||u||2 : eigenfunctions of TV • H(u) = ||u||1 : calibrable sets of TV Numerical approaches • Maximum Cheeger set with a projection algorithm [Carlier et al. ’08] • PDE associated to the TV flow  u(0) = u0 ∂t u = −p p ∈ ∂TV (u) - u0 is an eigenfunction: u(t) = (1 − tλ)+ u0 - At instinction time, u(t) is an eigenfunction [Bungert et al. ’19] - Implicit scheme for discrete TV flow: ut+1 − ut = −pt+1 ∆t N. Papadakis ⇔ ut+1 = argmin u 1 ||u − ut ||2 + TV (u) 2∆t Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 3 / 18
16. Objectives • Stable scheme computing eigenfunctions and calibrable sets of J • Minimization of generalized non linear Rayleigh quotient min R(u) := u N. Papadakis J(u) H(u) Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 4 / 18
17. Objectives • Stable scheme computing eigenfunctions and calibrable sets of J • Minimization of generalized non linear Rayleigh quotient min R(u) := u J(u) H(u) • Finite dimension setting X ⊂ Rn • J : X → R of full domain, proper, convex, lower semi-continuous and absolutely one-homogeneous: J(αu) = |α|J(u) N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 4 / 18
18. Objectives • Stable scheme computing eigenfunctions and calibrable sets of J • Minimization of generalized non linear Rayleigh quotient min R(u) := u J(u) H(u) • Finite dimension setting X ⊂ Rn • J : X → R of full domain, proper, convex, lower semi-continuous and absolutely one-homogeneous: J(αu) = |α|J(u) Discrete Non-Local Total Variation [Gilboa and Osher ’08] X J(u) = wij |ui − uj | wij≥0 ij ⇒ Regularization in imaging / Clustering with graph laplacian N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 4 / 18
19. Outline 1 Proposed flow - Absolutely one homogeneous functionals - Computation of eigenfunctions and calibrable sets 2 Discretization of the flow 3 Illustration and application to graph clustering N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 5 / 18
20. Proposed flow • Local minima of Rayleigh quotient: R(u) = N. Papadakis J(u) H(u) Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 6 / 18
21. Proposed flow • Local minima of Rayleigh quotient: R(u) = J(u) H(u) • “Derivative” of R, for p ∈ ∂J(u) and q ∈ ∂H(u): H(u)p − J(u)q 1 = (p − R(u)q) H 2 (u) H(u) N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 6 / 18
22. Proposed flow • Local minima of Rayleigh quotient: R(u) = J(u) H(u) • “Derivative” of R, for p ∈ ∂J(u) and q ∈ ∂H(u): H(u)p − J(u)q 1 = (p − R(u)q) H 2 (u) H(u) • New PDE [Nossek and Gilboa ’18, Aujol et al. ’18, Feld et al. ’19] ∂t u = R(u)q − p • When R(u)q is locally Lipschitz: results from [Brezis ’73]: ∂t u = A(u) − p N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 6 / 18
23. Analysis of the flow Considered problem ∂t u = J(u) q−p H(u) • J(u) is the (non local) total variation • H(u) = ||u||p , p ≥ 1 is the ℓp norm N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 7 / 18
24. Analysis of the flow Considered problem ∂t u = J(u) q−p H(u) • J(u) is the (non local) total variation • H(u) = ||u||p , p ≥ 1 is the ℓp norm Bounded subgradients in finite dimension [Burger et al. ’16]: ∃CH < ∞ such that ∥q∥ ≤ CH , ∀q ∈ ∂H N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 7 / 18
25. Analysis of the flow Considered problem ∂t u = J(u) q−p H(u) • J(u) is the (non local) total variation • H(u) = ||u||p , p ≥ 1 is the ℓp norm Bounded subgradients in finite dimension [Burger et al. ’16]: ∃CH < ∞ such that ∥q∥ ≤ CH , ∀q ∈ ∂H ✓ Full time discrete analysis • Existence and uniqueness in time continuous setting [Brezis ’73] for H(u) = ||u||p with p ≥ 2 • Moreau-Yosida regularization to have a locally-Lipschitz mapping when p ∈ [1; 2) N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 7 / 18
26. Properties of the proposed flow ∂t u = J(u) q−p H(u) (1) J, H: Absolutely one homogeneous functional p ∈ ∂J(u) ⇒ J(u) = ⟨p, u⟩ N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
27. Properties of the proposed flow ∂t u = J(u) q−p H(u) (1) J, H: Absolutely one homogeneous functional p ∈ ∂J(u) ⇒ J(u) = ⟨p, u⟩
    J(u) ⟨u, q⟩ − ⟨u, p⟩ = 0 • Norm conservation dtd 12 ||u||2 = ⟨u, ut ⟩ = H(u) • u does not vanish N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
28. Properties of the proposed flow ∂t u = J(u) q−p H(u) (1) J, H: Absolutely one homogeneous functional p ∈ ∂J(u) ⇒ J(u) = ⟨p, u⟩
    J(u) ⟨u, q⟩ − ⟨u, p⟩ = 0 • Norm conservation dtd 12 ||u||2 = ⟨u, ut ⟩ = H(u) • u does not vanish H: ℓ2 norm q= N. Papadakis u ∈ ∂||u||2 ||u||2 ⇒ ∂t u = J(u) u−p ||u||22 Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
29. Properties of the proposed flow ∂t u = J(u) q−p H(u) (1) J, H: Absolutely one homogeneous functional p ∈ ∂J(u) ⇒ J(u) = ⟨p, u⟩
    J(u) ⟨u, q⟩ − ⟨u, p⟩ = 0 • Norm conservation dtd 12 ||u||2 = ⟨u, ut ⟩ = H(u) • u does not vanish H: ℓ2 norm q= u ∈ ∂||u||2 ||u||2 ⇒ ∂t u = J(u) u−p ||u||22 • Convergence of the PDE ⇔ eigenfunction of J ∂t u = 0 ⇔ p = N. Papadakis J(u) u ∈ ∂J(u) ||u||22 Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
30. Properties of the proposed flow ∂t u = J(u) q−p H(u) (1) J: (non-local) Total Variation J(u) = X i N. Papadakis ||(∇u)i ||2 J(u) = X wij |ui − uj |, wij ≥ 0 ij Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
31. Properties of the proposed flow ∂t u = J(u) q−p H(u) (1) J: (non-local) Total Variation J(u) = X ||(∇u)i ||2 i J(u) = X wij |ui − uj |, wij ≥ 0 ij Any p ∈ ∂J(u) checks 1 - Zero mean p̄ = ⟨p, 1⟩ = 0 N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
32. Properties of the proposed flow ∂t u = J(u) (q − q̄1) − p H(u) (1) J: (non-local) Total Variation J(u) = X ||(∇u)i ||2 i J(u) = X wij |ui − uj |, wij ≥ 0 ij Any p ∈ ∂J(u) checks 1 - Zero mean p̄ = ⟨p, 1⟩ = 0 • Adaptation of the flow: ū0 = 0 ⇒ u(t) = 0 N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
33. Properties of the proposed flow ∂t u = J(u) (q − q̄1) − p H(u) (1) J: (non-local) Total Variation J(u) = X ||(∇u)i ||2 i J(u) = X wij |ui − uj |, wij ≥ 0 ij Any p ∈ ∂J(u) checks 1 - Zero mean p̄ = ⟨p, 1⟩ = 0 • Adaptation of the flow: ū0 = 0 ⇒ u(t) = 0 • Constant norm + zero mean ⇒ steady points u ∗ are non trivial N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
34. Properties of the proposed flow ∂t u = J(u) (q − q̄1) − p H(u) (1) J: (non-local) Total Variation J(u) = X ||(∇u)i ||2 i J(u) = X wij |ui − uj |, wij ≥ 0 ij Any p ∈ ∂J(u) checks 1 - Zero mean p̄ = ⟨p, 1⟩ = 0 2 - Co-area p ∈ ∂J(1u>ϵ ), ∀ϵ ∈ [umin , umax ] • Adaptation of the flow: ū0 = 0 ⇒ u(t) = 0 • Constant norm + zero mean ⇒ steady points u ∗ are non trivial N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
35. Properties of the proposed flow ∂t u = J(u) (q − q̄1) − p H(u) (1) J: (non-local) Total Variation J(u) = X ||(∇u)i ||2 i J(u) = X wij |ui − uj |, wij ≥ 0 ij Any p ∈ ∂J(u) checks 1 - Zero mean p̄ = ⟨p, 1⟩ = 0 • Adaptation of the flow: ū0 = 0 ⇒ u(t) = 0 2 - Co-area p ∈ ∂J(1u>ϵ ), ∀ϵ ∈ [umin , umax ] • For H(u) = ||u||1 , take q = (1u>0 − 1u≤0 ) ∈ ∂||u||1 • Constant norm + zero mean ⇒ steady points u ∗ are non trivial N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
36. Properties of the proposed flow ∂t u = J(u) (q − q̄1) − p H(u) (1) J: (non-local) Total Variation J(u) = X ||(∇u)i ||2 i J(u) = X wij |ui − uj |, wij ≥ 0 ij Any p ∈ ∂J(u) checks 1 - Zero mean p̄ = ⟨p, 1⟩ = 0 • Adaptation of the flow: ū0 = 0 ⇒ u(t) = 0 2 - Co-area p ∈ ∂J(1u>ϵ ), ∀ϵ ∈ [umin , umax ] • For H(u) = ||u||1 , take q = (1u>0 − 1u≤0 ) ∈ ∂||u||1 • Constant norm + zero mean • If u ∗ a steady point of (1) then ⇒ steady points u ∗ are non trivial ⇒ v = 1u ∗ >0 − |1u ∗ >0 | is a calibrable set N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
37. Properties of the proposed flow ∂t u = J(u) (q − q̄1) − p H(u) (1) J: (non-local) Total Variation J(u) = X ||(∇u)i ||2 i J(u) = X wij |ui − uj |, wij ≥ 0 ij Any p ∈ ∂J(u) checks 1 - Zero mean p̄ = ⟨p, 1⟩ = 0 • Adaptation of the flow: ū0 = 0 ⇒ u(t) = 0 2 - Co-area p ∈ ∂J(1u>ϵ ), ∀ϵ ∈ [umin , umax ] • For H(u) = ||u||1 , take q = (1u>0 − 1u≤0 ) ∈ ∂||u||1 • Constant norm + zero mean • If u ∗ a steady point of (1) then ⇒ steady points u ∗ are non trivial ⇒ v = 1u ∗ >0 − |1u ∗ >0 | is a calibrable set N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18
38. Illustration: Eigenfunctions and Calibrable set of TV J(u) = Xq (ui+1,j − ui,j )2 + (ui,j+1 − ui,j )2 ij • H(u) = qP 2 ij ui,j u N. Papadakis p Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 9 / 18
39. Illustration: Eigenfunctions and Calibrable set of TV J(u) = Xq (ui+1,j − ui,j )2 + (ui,j+1 − ui,j )2 ij • H(u) = qP • H(u) = P 2 ij ui,j u u N. Papadakis p ij |ui,j | p v = 1u>0 Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 9 / 18
40. Illustration: Calibrable set of non-local TV For an image f : J(u) = X wij |ui − uj | ij H(u) = X |ui,j | ij with wij = exp(−||fi − fj ||2 ) u N. Papadakis p v = 1u>0 Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 10 / 18
41. Discrete flow Semi-explicit scheme ( uk +1/2 −uk ∆t uk +1 N. Papadakis = R(uk )qk − pk +1/2 , qk ∈ ∂H(uk ), pk +1/2 ∈ ∂J(uk +1/2 ) u +1/2 = ||ukk+1/2 ||2 Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 11 / 18
42. Discrete flow Semi-explicit scheme ( uk +1/2 −uk ∆t uk +1 = R(uk )qk − pk +1/2 , qk ∈ ∂H(uk ), pk +1/2 ∈ ∂J(uk +1/2 ) u +1/2 = ||ukk+1/2 ||2 Properties • Non decreasing norm ||uk ||22 ≤ ⟨uk +1/2 , uk ⟩ ≤ ||uk +1/2 ||22 • H(uk +1/2 ) is bounded by some constant C • Non increasing ratio R(uk +1 ) ≤ R(uk ) N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 11 / 18
43. Discrete flow Semi-explicit scheme ( uk +1/2 −uk ∆t uk +1 = R(uk )qk − pk +1/2 , qk ∈ ∂H(uk ), pk +1/2 ∈ ∂J(uk +1/2 ) u +1/2 = ||ukk+1/2 ||2 Properties • Non decreasing norm ||uk ||22 ≤ ⟨uk +1/2 , uk ⟩ ≤ ||uk +1/2 ||22 • H(uk +1/2 ) is bounded by some constant C • Non increasing ratio R(uk +1 ) ≤ R(uk ) Theorem The sequence uk converges to u ∗ and ∃q ∗ ∈ ∂H(u ∗ ), p∗ ∈ ∂J(u ∗ ) such that: R(u ∗ )q ∗ = p∗ N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 11 / 18
44. Discrete flow ( uk +1/2 −uk ∆t = R(uk )qk − pk +1/2 , qk ∈ ∂H(uk ), pk +1/2 ∈ ∂J(uk +1/2 ) u +1/2 = ||ukk+1/2 ||2 uk +1 Proof Observing that uk +1/2 = argmin u 1 ||u − uk ||22 − R(uk )⟨qk , u⟩ + J(u) 2∆t we get 1 ||uk +1 − uk ||22 − R(uk )⟨qk , uk +1 ⟩ + J(uk +1 ) ≤ R(uk )H(uk ) − J(uk ) 2∆t 1 ||uk +1 − uk ||22 + J(uk +1 ) ≤ R(uk )H(uk +1 ) 2∆t J(uk +1 ) 1 ||uk +1 − uk ||22 + ≤ R(uk ) 2∆tH(uk +1 ) H(uk +1 ) 1 ||uk +1 − uk ||22 + R(uk +1 ) ≤ R(uk ) 2C∆t N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 12 / 18
45. Illustration: Eigenfunction of TV J(u) = Xq (ui+1,j − ui,j )2 + (ui,j+1 − ui,j )2 H(u) = sX 2 ui,j ij 10000 iterations 100 iterations ij u p J(u k ) N. Papadakis Pointwise ratio p/u ||u k +1 − u k ||2 Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 13 / 18
46. Application to image segmentation with ℓ1 norm Image N. Papadakis u Segmentation 1u>0 Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 14 / 18
47. Application to graph binary clustering with ℓ1 norm Initialisation N. Papadakis Converged state Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 15 / 18
48. Extension to multi class clustering • Finding L eigenfunctions u l PL • Linear coupling l=1 u l (x) = 0, for all node x: - One-homogeneous constraint - Compatible with zero mean property of eigenfunction - Fast projection w.r.t simplex [Bresson et al. ’13, Hein et al. ’14] N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 16 / 18
49. Extension to multi class clustering • Finding L eigenfunctions u l PL • Linear coupling l=1 u l (x) = 0, for all node x: - One-homogeneous constraint - Compatible with zero mean property of eigenfunction - Fast projection w.r.t simplex [Bresson et al. ’13, Hein et al. ’14] Transductive learning [Aviles-Rivero et al. ’19] • Label a few nodes N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 16 / 18
50. Extension to multi class clustering • Finding L eigenfunctions u l PL • Linear coupling l=1 u l (x) = 0, for all node x: - One-homogeneous constraint - Compatible with zero mean property of eigenfunction - Fast projection w.r.t simplex [Bresson et al. ’13, Hein et al. ’14] Transductive learning [Aviles-Rivero et al. ’19] • Label a few nodes and diffuse the labels N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 16 / 18
51. Illustration: Chest Xray data set N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 17 / 18
52. Conclusion • New flow for absolutely one homogeneous functionals • Numerical estimation of eigenfunctions • Data clustering Perspectives • Existence in the infinite dimension setting [Bungert et al. ’19] • Convergence of the discrete scheme for multi-class clustering N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 18 / 18