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
ICIAM 2019
Nonlinear Spectral Decompositions with Applications in Imaging
and Data Science
July 18th 2019
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 0 / 18

Introduction - Total Variation in imaging
For u of Bounded Variation on Ω
TV(u) = sup
z∈C∞
c
; ||z||∞≤1 Ω
udiv(z)
Discrete setting in imaging
TV(u) =
i
||(∇u)i ||2
Denoising with TV:
piecewise constant prior
• Intensively used for image regularization [Rudin, Osher, Fatemi ‘92]
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 1 / 18

Introduction - Total Variation in imaging
For u of Bounded Variation on Ω
TV(u) = sup
z∈C∞
c
; ||z||∞≤1 Ω
udiv(z)
Discrete setting in imaging
TV(u) =
i
||(∇u)i ||2
Denoising with TV:
piecewise constant prior
• 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

Introduction - Total Variation in imaging
For u of Bounded Variation on Ω
TV(u) = sup
z∈C∞
c
; ||z||∞≤1 Ω
udiv(z)
Discrete setting in imaging
TV(u) =
i
||(∇u)i ||2
Denoising with TV:
piecewise constant prior
• 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

Introduction - Total Variation in imaging
For u of Bounded Variation on Ω
TV(u) = sup
z∈C∞
c
; ||z||∞≤1 Ω
udiv(z)
Discrete setting in imaging
TV(u) =
i
||(∇u)i ||2
Denoising with TV:
piecewise constant prior
• 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

Introduction - Total Variation in imaging
For u of Bounded Variation on Ω
TV(u) = sup
z∈C∞
c
; ||z||∞≤1 Ω
udiv(z)
Discrete setting in imaging
TV(u) =
i
||(∇u)i ||2
Denoising with TV:
piecewise constant prior
• 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

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:
Local minima of TV(u)
||u||2
are eigenfunctions of TV [Andreu et al 2001]
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 2 / 18

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:
Local minima of TV(u)
||u||2
are eigenfunctions of TV [Andreu et al 2001]
Calibrable sets
• C ⊂ Ω such that χC
is an eigenfunction of TV
• A calibrable set is convex with bounded curvature [Belettini et al. 2002]:
ess sup
q∈∂C
κ(q) ≤
Per(C)
|C|
[Gilboa et al. 2016]
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 2 / 18

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:
Local minima of TV(u)
||u||2
are eigenfunctions of TV [Andreu et al 2001]
Calibrable sets
• C ⊂ Ω such that χC
is an eigenfunction of TV
• A calibrable set is convex with bounded curvature [Belettini et al. 2002]:
ess sup
q∈∂C
κ(q) ≤
Per(C)
|C|
[Gilboa et al. 2016]
Cheeger set of Ω : min
C⊂Ω
TV(χC
)
|C|
Cheeger cut of Ω : min
C⊂Ω
TV(χC
)
max(|C| |Ω\C|)
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 2 / 18

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:
Local minima of TV(u)
||u||2
are eigenfunctions of TV [Andreu et al 2001]
Calibrable sets
• C ⊂ Ω such that χC
is an eigenfunction of TV
• A calibrable set is convex with bounded curvature [Belettini et al. 2002]:
ess sup
q∈∂C
κ(q) ≤
Per(C)
|C|
[Gilboa et al. 2016]
Cheeger set of Ω : min
C⊂Ω
TV(χC
)
|C|
Cheeger cut of Ω : min
C⊂Ω
TV(χC
)
max(|C| |Ω\C|)
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 2 / 18

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
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 3 / 18

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. 2008]
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 3 / 18

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. 2008]
• 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

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. 2008]
• 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 2019]
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 3 / 18

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. 2008]
• 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 2019]
- Implicit scheme for discrete TV ﬂow:
ut+1 − ut
∆t
= −pt+1 ⇔ ut+1 = argmin
u
1
2∆t
||u − ut ||2 + TV(u)
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 3 / 18

Objectives
• Stable scheme computing eigenfunctions and calibrable sets of J
• Minimization of generalized non linear Rayleigh quotient
min
u
R(u) :=
J(u)
H(u)
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 4 / 18

Objectives
• Stable scheme computing eigenfunctions and calibrable sets of J
• Minimization of generalized non linear Rayleigh quotient
min
u
R(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

Objectives
• Stable scheme computing eigenfunctions and calibrable sets of J
• Minimization of generalized non linear Rayleigh quotient
min
u
R(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 2008]
J(u) =
ij
wij |ui − uj | wij≥0
⇒ Regularization in imaging / Clustering with graph laplacian
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 4 / 18

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

Proposed ﬂow
• Local minima of Rayleigh quotient:
R(u) =
J(u)
H(u)
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 6 / 18

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
H2(u)
=
1
H(u)
(p − R(u)q)
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 6 / 18

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
H2(u)
=
1
H(u)
(p − R(u)q)
• 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

Analysis of the ﬂow
Considered problem
∂t
u =
J(u)
H(u)
q − p
• J(u) is the (non local) total variation
• H(u) = ||u||p
, p ≥ 1 is the p
norm
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 7 / 18

Analysis of the ﬂow
Considered problem
∂t
u =
J(u)
H(u)
q − p
• J(u) is the (non local) total variation
• H(u) = ||u||p
, p ≥ 1 is the p
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

Analysis of the ﬂow
Considered problem
∂t
u =
J(u)
H(u)
q − p
• J(u) is the (non local) total variation
• H(u) = ||u||p
, p ≥ 1 is the p
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

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
q − p (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

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
q − p (1)
J, H: Absolutely one homogeneous functional
p ∈ ∂J(u) ⇒ J(u) = p, u
• Norm conservation d
dt
1
2
||u||2 = u, ut = J(u)
H(u)
u, q − u, p = 0
• u does not vanish
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
q − p (1)
J, H: Absolutely one homogeneous functional
p ∈ ∂J(u) ⇒ J(u) = p, u
• Norm conservation d
dt
1
2
||u||2 = u, ut = J(u)
H(u)
u, q − u, p = 0
• u does not vanish
H: 2
norm
q =
u
||u||2
∈ ∂||u||2 ⇒ ∂t
u =
J(u)
||u||2
2
u − p
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
q − p (1)
J, H: Absolutely one homogeneous functional
p ∈ ∂J(u) ⇒ J(u) = p, u
• Norm conservation d
dt
1
2
||u||2 = u, ut = J(u)
H(u)
u, q − u, p = 0
• u does not vanish
H: 2
norm
q =
u
||u||2
∈ ∂||u||2 ⇒ ∂t
u =
J(u)
||u||2
2
u − p
• Convergence of the PDE ⇔ eigenfunction of J
∂t
u = 0 ⇔ p =
J(u)
||u||2
2
u ∈ ∂J(u)
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
q − p (1)
J: (non-local) Total Variation
J(u) =
i
||(∇u)i ||2
J(u) =
ij
wij |ui − uj |, wij ≥ 0
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
q − p (1)
J: (non-local) Total Variation
J(u) =
i
||(∇u)i ||2
J(u) =
ij
wij |ui − uj |, wij ≥ 0
Any p ∈ ∂J(u) checks
1 - Zero mean
¯
p = p, 1 = 0
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
(q − ¯
q1) − p (1)
J: (non-local) Total Variation
J(u) =
i
||(∇u)i ||2
J(u) =
ij
wij |ui − uj |, wij ≥ 0
Any p ∈ ∂J(u) checks
1 - Zero mean
¯
p = p, 1 = 0
• Adaptation of the ﬂow:
¯
u0 = 0 ⇒ u(t) = 0
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
(q − ¯
q1) − p (1)
J: (non-local) Total Variation
J(u) =
i
||(∇u)i ||2
J(u) =
ij
wij |ui − uj |, wij ≥ 0
Any p ∈ ∂J(u) checks
1 - Zero mean
¯
p = p, 1 = 0
• Adaptation of the ﬂow:
¯
u0 = 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

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
(q − ¯
q1) − p (1)
J: (non-local) Total Variation
J(u) =
i
||(∇u)i ||2
J(u) =
ij
wij |ui − uj |, wij ≥ 0
Any p ∈ ∂J(u) checks
1 - Zero mean
¯
p = p, 1 = 0
• Adaptation of the ﬂow:
¯
u0 = 0 ⇒ u(t) = 0
• Constant norm + zero mean
⇒ steady points u∗ are non trivial
2 - Co-area
p ∈ ∂J(1u>
), ∀ ∈ [umin, umax ]
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
(q − ¯
q1) − p (1)
J: (non-local) Total Variation
J(u) =
i
||(∇u)i ||2
J(u) =
ij
wij |ui − uj |, wij ≥ 0
Any p ∈ ∂J(u) checks
1 - Zero mean
¯
p = p, 1 = 0
• Adaptation of the ﬂow:
¯
u0 = 0 ⇒ u(t) = 0
• Constant norm + zero mean
⇒ steady points u∗ are non trivial
2 - Co-area
p ∈ ∂J(1u>
), ∀ ∈ [umin, umax ]
• For H(u) = ||u||1
, take
q = (1u>0 − 1u≤0) ∈ ∂||u||1
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
(q − ¯
q1) − p (1)
J: (non-local) Total Variation
J(u) =
i
||(∇u)i ||2
J(u) =
ij
wij |ui − uj |, wij ≥ 0
Any p ∈ ∂J(u) checks
1 - Zero mean
¯
p = p, 1 = 0
• Adaptation of the ﬂow:
¯
u0 = 0 ⇒ u(t) = 0
• Constant norm + zero mean
⇒ steady points u∗ are non trivial
2 - Co-area
p ∈ ∂J(1u>
), ∀ ∈ [umin, umax ]
• For H(u) = ||u||1
, take
q = (1u>0 − 1u≤0) ∈ ∂||u||1
• If u∗ a steady point of (1) then
⇒ v = 1u∗>0 − |1u∗>0| is a calibrable set
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18

Properties of the proposed ﬂow
∂t
u =
J(u)
H(u)
(q − ¯
q1) − p (1)
J: (non-local) Total Variation
J(u) =
i
||(∇u)i ||2
J(u) =
ij
wij |ui − uj |, wij ≥ 0
Any p ∈ ∂J(u) checks
1 - Zero mean
¯
p = p, 1 = 0
• Adaptation of the ﬂow:
¯
u0 = 0 ⇒ u(t) = 0
• Constant norm + zero mean
⇒ steady points u∗ are non trivial
2 - Co-area
p ∈ ∂J(1u>
), ∀ ∈ [umin, umax ]
• For H(u) = ||u||1
, take
q = (1u>0 − 1u≤0) ∈ ∂||u||1
• If u∗ a steady point of (1) then
⇒ v = 1u∗>0 − |1u∗>0| is a calibrable set
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 8 / 18

Illustration: Eigenfunctions and Calibrable set of TV
J(u) =
ij
(ui+1,j − ui,j )2 + (ui,j+1 − ui,j )2
• H(u) = ij
u2
i,j
u p
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 9 / 18

Illustration: Eigenfunctions and Calibrable set of TV
J(u) =
ij
(ui+1,j − ui,j )2 + (ui,j+1 − ui,j )2
• H(u) = ij
u2
i,j
u p
• H(u) = ij
|ui,j |
u p v = 1u>0
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 9 / 18

Illustration: Calibrable set of non-local TV
For an image f:
J(u) =
ij
wij |ui − uj | H(u) =
ij
|ui,j |
with wij = exp(−||fi − fj ||2)
u p v = 1u>0
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 10 / 18

Discrete ﬂow
Semi-explicit scheme
uk+1/2
−uk
∆t
= R(uk )qk − pk+1/2
, qk ∈ ∂H(uk ), pk+1/2
∈ ∂J(uk+1/2
)
uk+1 = uk+1/2
||uk+1/2
||2
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 11 / 18

Discrete ﬂow
Semi-explicit scheme
uk+1/2
−uk
∆t
= R(uk )qk − pk+1/2
, qk ∈ ∂H(uk ), pk+1/2
∈ ∂J(uk+1/2
)
uk+1 = uk+1/2
||uk+1/2
||2
Properties
• Non decreasing norm ||uk ||2
2
≤ uk+1/2
, uk ≤ ||uk+1/2
||2
2
• 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

Discrete ﬂow
Semi-explicit scheme
uk+1/2
−uk
∆t
= R(uk )qk − pk+1/2
, qk ∈ ∂H(uk ), pk+1/2
∈ ∂J(uk+1/2
)
uk+1 = uk+1/2
||uk+1/2
||2
Properties
• Non decreasing norm ||uk ||2
2
≤ uk+1/2
, uk ≤ ||uk+1/2
||2
2
• 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

Discrete ﬂow
uk+1/2
−uk
∆t
= R(uk )qk − pk+1/2
, qk ∈ ∂H(uk ), pk+1/2
∈ ∂J(uk+1/2
)
uk+1 = uk+1/2
||uk+1/2
||2
Proof
Observing that
uk+1/2
= argmin
u
1
2∆t
||u − uk ||2
2
− R(uk ) qk , u + J(u)
we get
1
2∆t
||uk+1
− uk
||2
2
− R(uk
) qk
, uk+1
+ J(uk+1
) ≤ R(uk
)H(uk
) − J(uk
)
1
2∆t
||uk+1
− uk
||2
2
+ J(uk+1
) ≤ R(uk
)H(uk+1
)
1
2∆tH(uk+1
)
||uk+1
− uk
||2
2
+
J(uk+1
)
H(uk+1
)
≤ R(uk
)
1
2C∆t
||uk+1
− uk
||2
2
+ R(uk+1
) ≤ R(uk
)
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 12 / 18

Illustration: Eigenfunction of TV
J(u) =
ij
(ui+1,j − ui,j )2 + (ui,j+1 − ui,j )2 H(u) =
ij
u2
i,j
100
iterations
10000
iterations
u p Pointwise ratio p/u
J(uk ) ||uk+1 − uk ||2
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 13 / 18

Application to image segmentation with 1
norm
Image u Segmentation 1u>0
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 14 / 18

Application to graph binary clustering with 1
norm
Initialisation Converged state
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 15 / 18

Extension to multi class clustering
• Finding L eigenfunctions ul
• Linear coupling L
l=1
ul (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

Extension to multi class clustering
• Finding L eigenfunctions ul
• Linear coupling L
l=1
ul (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
• Label a few nodes
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 16 / 18

Extension to multi class clustering
• Finding L eigenfunctions ul
• Linear coupling L
l=1
ul (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
• Label a few nodes and diffuse the labels
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 16 / 18

Illustration: Chest Xray data set
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 17 / 18

Conclusion
• New ﬂow for absolutely one homogeneous functionals
• Numerical estimation of eigenfunctions
• Data clustering
Perspectives
• Existence in the inﬁnite dimension setting [Bungert et al. 2019]
• Convergence of the discrete scheme for multi-class clustering
N. Papadakis Eigenfunctions and calibrable sets of absolutely one-homogeneous functionals 18 / 18