This talk was presented during the Operator Splitting Workshop

1. Composite Infimal Convolutions
Zev Woodstock
Department of Mathematics
North Carolina State University
Advisor: Patrick Combettes
March 22, 2018
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

2. What do these operations have in common?
[Image processing, Stetzer et al. (2011)] Given a regularizing
seminorm | · |R, solve an optimization problem involving
φ(x) = infy∈L2
1
2 x − y 2
L2
+ |y|R.
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

3. What do these operations have in common?
[Image processing, Stetzer et al. (2011)] Given a regularizing
seminorm | · |R, solve an optimization problem involving
φ(x) = infy∈L2
1
2 x − y 2
L2
+ |y|R.
[Moreau & Pasch-Hausdorff Envelopes in Inverse Problems]
Given f : H →] − ∞, +∞],
fp(x) = infy∈H f (y) + 1
p x − y p.
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

4. What do these operations have in common?
[Image processing, Stetzer et al. (2011)] Given a regularizing
seminorm | · |R, solve an optimization problem involving
φ(x) = infy∈L2
1
2 x − y 2
L2
+ |y|R.
[Moreau & Pasch-Hausdorff Envelopes in Inverse Problems]
Given f : H →] − ∞, +∞],
fp(x) = infy∈H f (y) + 1
p x − y p.
[Statistics/machine learning]
[Square root LASSO] For A ∈ RM×N
and x ∈ RM
:
minimize Ay − x 2 + y 1 with y ∈ RN
.
[Combettes et al. (2017)] L : H → G is bounded and linear,
solve an optimization problem involving functions of the form
φ(x) = infLy=x |||y|||.
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

5. What do these operations have in common?
[Image processing, Stetzer et al. (2011)] Given a regularizing
seminorm | · |R, solve an optimization problem involving
φ(x) = infy∈L2
1
2 x − y 2
L2
+ |y|R.
[Moreau & Pasch-Hausdorff Envelopes in Inverse Problems]
Given f : H →] − ∞, +∞],
fp(x) = infy∈H f (y) + 1
p x − y p.
[Statistics/machine learning]
[Square root LASSO] For A ∈ RM×N
and x ∈ RM
:
minimize Ay − x 2 + y 1 with y ∈ RN
.
[Combettes et al. (2017)] L : H → G is bounded and linear,
solve an optimization problem involving functions of the form
φ(x) = infLy=x |||y|||.
[Interpolation spaces, Peetre (1970)] If X1 and X2 are contin-
uously embedded in a topological vector space H:
x = inf
y1∈X1,y2∈X2
y1+y2=x
y1
p
X1
+ y2
p
X2
1/p
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

6. What do these operations have in common?
[Inverse problem, Youla (1978)] V1 and V2 are closed vector
subspaces of a real Hilbert space H. Given x ∈ V2,
find y ∈ V1 such that projV2
y = x.
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

7. What do these operations have in common?
[Inverse problem, Youla (1978)] V1 and V2 are closed vector
subspaces of a real Hilbert space H. Given x ∈ V2,
find y ∈ V1 such that projV2
y = x.
[Approximation theory, Combettes-Reyes (2010)] Given closed
vector subspaces (Vi )i∈I and (∀i ∈ I) xi ∈ Vi ,
find y ∈ H such that (∀i ∈ I) projVi
y = xi .
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

8. What do these operations have in common?
[Inverse problem, Youla (1978)] V1 and V2 are closed vector
subspaces of a real Hilbert space H. Given x ∈ V2,
find y ∈ V1 such that projV2
y = x.
[Approximation theory, Combettes-Reyes (2010)] Given closed
vector subspaces (Vi )i∈I and (∀i ∈ I) xi ∈ Vi ,
find y ∈ H such that (∀i ∈ I) projVi
y = xi .
[Minimal-norm interpolation] Given values of x : R → R at
points (ti )i∈I in R:
[Favard (1940)]
minimize y(k)
∞ subject to (∀i ∈ I) y(ti ) = x(ti )
[Kunkle (2002)]
minimize y W k,2 subject to (∀i ∈ I) y(ti ) = x(ti )
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

9. Unifying two convolutions
Let H and G be real Hilbert spaces, suppose f and h are proper
functions from H to ] − ∞, +∞], and let L : H → G.
Infimal convolution: (f h)(x) = infy∈H f (y) + h(x − y).
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

10. Unifying two convolutions
Let H and G be real Hilbert spaces, suppose f and h are proper
functions from H to ] − ∞, +∞], and let L : H → G.
Infimal convolution: (f h)(x) = infy∈H f (y) + h(x − y).
Infimal postcomposition: (L f )(x) = infLy=x f (y).
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

11. Unifying two convolutions
Let H and G be real Hilbert spaces, suppose f and h are proper
functions from H to ] − ∞, +∞], and let L : H → G.
Infimal convolution: (f h)(x) = infy∈H f (y) + h(x − y).
Infimal postcomposition: (L f )(x) = infLy=x f (y).
For g : G →] − ∞, +∞], we define the composite infimal
convolution (provided L f is proper):
((L f ) g) (x) = inf
z∈G



 inf
y∈H
Ly=z
f (y)

 + g(y − z)


= inf
y∈H
f (y) + g(x − Ly).
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

12. What do we already know?
((L f ) g) (x) = inf
y∈H
(f (y) + g(x − Ly)) (1)
Special cases: L f = (L f ) ι{0} and f g = (Id f ) g.
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

13. What do we already know?
((L f ) g) (x) = inf
y∈H
(f (y) + g(x − Ly)) (1)
Special cases: L f = (L f ) ι{0} and f g = (Id f ) g.
We have investigated several topological, algebraic, and preser-
vative properties of this operation (e.g. lower semicontinuity,
coercivity, norms, ...).
Convex analytical properties: the conjugate and subdiferential
are known.
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

14. What do we already know?
((L f ) g) (x) = inf
y∈H
(f (y) + g(x − Ly)) (1)
Special cases: L f = (L f ) ι{0} and f g = (Id f ) g.
We have investigated several topological, algebraic, and preser-
vative properties of this operation (e.g. lower semicontinuity,
coercivity, norms, ...).
Convex analytical properties: the conjugate and subdiferential
are known.
Amenable to proximal splitting methods for block-iterative al-
gorithms.
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

15. What do we already know?
((L f ) g) (x) = inf
y∈H
(f (y) + g(x − Ly)) (1)
Special cases: L f = (L f ) ι{0} and f g = (Id f ) g.
We have investigated several topological, algebraic, and preser-
vative properties of this operation (e.g. lower semicontinuity,
coercivity, norms, ...).
Convex analytical properties: the conjugate and subdiferential
are known.
Amenable to proximal splitting methods for block-iterative al-
gorithms.
Observation: This model allows us to impose more structure in
penalty functions.
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

16. Open Questions
[Norm interpolation] Let |||·|||H and |||·|||G be norms. If
L : H → G is linear and bounded, (L |||·|||H) |||·|||G is a
norm.
If |||·|||H and |||·|||G are Hilbertian, is (L |||·|||H) |||·|||G also
Hilbertian?
What if |||·|||H and |||·|||G are polyhedral?
What if |||·|||H and |||·|||G are atomic?
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

17. Open Questions
Under qualifying conditions, [Becker-Combettes (2014)] showed
∂ ((L f ) g) = (L◦(∂f )−1
◦L∗
+(∂g)−1
)−1
= (L ∂f ) ∂g.
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

18. Open Questions
Under qualifying conditions, [Becker-Combettes (2014)] showed
∂ ((L f ) g) = (L◦(∂f )−1
◦L∗
+(∂g)−1
)−1
= (L ∂f ) ∂g.
Let A : H → 2H, let B : G → 2G, and let L : H → G be linear
and bounded.
(L A) B = L ◦ A−1
◦ L∗
+ B−1 −1
Appears in [Penot-Z˘alinescu (2005)],[Bot¸-L´aszl´o (2012)], where
monotonicity properties are characterized.
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution

19. Open Questions
Under qualifying conditions, [Becker-Combettes (2014)] showed
∂ ((L f ) g) = (L◦(∂f )−1
◦L∗
+(∂g)−1
)−1
= (L ∂f ) ∂g.
Let A : H → 2H, let B : G → 2G, and let L : H → G be linear
and bounded.
(L A) B = L ◦ A−1
◦ L∗
+ B−1 −1
Appears in [Penot-Z˘alinescu (2005)],[Bot¸-L´aszl´o (2012)], where
monotonicity properties are characterized.
What non-variational equilibrium problems are of the form
0 ∈ ( i∈I(Li Ai) Bi)x?
Can we refine the splitting algorithms discussed in [Becker-
Combettes (2014)]?
Zev Woodstock (zwoodst@ncsu.edu) Composite Infimal Convolution