410

# Exact Support Recovery for Sparse Spikes Deconvolution

Initial talk at SIAM IS14, updated for the Workshop "IFIP TC7.4 Workshop on Inverse Problems and Imaging", updated for the conference SPARS 2015. ## Gabriel Peyré

December 17, 2014

## Transcript

1. Gabriel Peyré
www.numerical-tours.com
Exact Support Recovery
for Sparse Spikes
Deconvolution
Joint work with
Vincent Duval & Quentin Denoyelle
VISI N

2. Sparse Deconvolution
Neural spikes (1D)
y m0
= ?
ginal Signal
m0
t
1
x2
x3

Low-pass ﬁlter
'
t
0.5
0.5
+
Noise
w
0 1
+
y = ' ? m0 + w
m0 is “sparse”
'
w

3. Sparse Deconvolution
Neural spikes (1D)
Seismic imaging (1.5D)
y m0
= ?
ginal Signal
m0
t
1
x2
x3

Low-pass ﬁlter
'
t
0.5
0.5
+
Noise
w
0 1
+
y = ' ? m0 + w
m0 is “sparse”
'
w

4. Sparse Deconvolution
Neural spikes (1D)
Astrophysics (2D)
Seismic imaging (1.5D)
y m0
= ?
ginal Signal
m0
t
1
x2
x3

Low-pass ﬁlter
'
t
0.5
0.5
+
Noise
w
0 1
+
y = ' ? m0 + w
m0 is “sparse”
'
w

5. Sparse Deconvolution
Neural spikes (1D)
Astrophysics (2D)
Seismic imaging (1.5D)
y m0
= ?
ginal Signal
m0
t
1
x2
x3

Low-pass ﬁlter
'
t
0.5
0.5
+
Noise
w
0 1
+
y = ' ? m0 + w
m0 is “sparse”
Presented results
n
D problems
extend to
'
w

6. Overview
• Sparse Spikes Super-resolution
• Robust Support Recovery
• Asymptotic Positive Measure Recovery

m
on
T
= (
R/Z
)
d
.
Deconvolution of Measures
m

8. Discrete measure:
ma,x
=
P
N
i
=1 ai xi , a
2 RN
, x
2 TN
m
on
T
= (
R/Z
)
d
.
Deconvolution of Measures
m
a,x
m

9. Discrete measure:
ma,x
=
P
N
i
=1 ai xi , a
2 RN
, x
2 TN
m
on
T
= (
R/Z
)
d
.
Deconvolution of Measures
m
a,x
m
' 2 C2(T ⇥ T)
y = (m) + w
(
m
) =
Z
T '
(
x,
·)d
m
(
x
)
Linear measurements:

10. y
Discrete measure:
ma,x
=
P
N
i
=1 ai xi , a
2 RN
, x
2 TN
m
on
T
= (
R/Z
)
d
.
Deconvolution of Measures
m
a,x
'
m
' 2 C2(T ⇥ T)
y = (m) + w
(
m
) =
Z
T '
(
x,
·)d
m
(
x
)
Linear measurements:
Example: 1-D (
d
= 1) convolution
'
(
x, t
) =
'
(
t x
)

11. y
Discrete measure:
ma,x
=
P
N
i
=1 ai xi , a
2 RN
, x
2 TN
Minimum separation:
= mini6=j
|
xi xj
|
!
Signal-dependent recovery criteria.
m
on
T
= (
R/Z
)
d
.
Deconvolution of Measures
m
a,x
'
= 2/fc
y
= 0.5/fc
m
' 2 C2(T ⇥ T)
y = (m) + w
(
m
) =
Z
T '
(
x,
·)d
m
(
x
)
Linear measurements:
Example: 1-D (
d
= 1) convolution
'
(
x, t
) =
'
(
t x
)

12. Discrete
`1
regularization:
Computation grid
z
= (
zk)
K
k=1.
Sparse l1 Deconvolution
y
zk

13. Discrete
`1
regularization:
Computation grid
z
= (
zk)
K
k=1.
Basis-pursuit / Lasso:
Sparse l1 Deconvolution
y
zk
¯ : a 2 RK 7! (ma,z) =
X
k
ak'(zk, ·) 2 Im( )
min
a2RN
1
2
||y ¯a||2 + ||a||1

14. Discrete
`1
regularization:
Computation grid
z
= (
zk)
K
k=1.
Basis-pursuit / Lasso:
Sparse l1 Deconvolution
y
zk
¯ : a 2 RK 7! (ma,z) =
X
k
ak'(zk, ·) 2 Im( )
min
a2RN
1
2
||y ¯a||2 + ||a||1
Why `1?
q = 0
a1
a2
“`0 ball”

15. Discrete
`1
regularization:
Computation grid
z
= (
zk)
K
k=1.
Basis-pursuit / Lasso:
Sparse l1 Deconvolution
y
zk
¯ : a 2 RK 7! (ma,z) =
X
k
ak'(zk, ·) 2 Im( )
min
a2RN
1
2
||y ¯a||2 + ||a||1
q = 1 q = 2
q = 3/2
q = 1/2
Why `1? `q ball a 2 RK ;
P
k
|ak
|q 6 1
q = 0
a1
a2
“`0 ball” !

16. Discrete
`1
regularization:
Computation grid
z
= (
zk)
K
k=1.
Basis-pursuit / Lasso:
Sparse l1 Deconvolution
y
zk
¯ : a 2 RK 7! (ma,z) =
X
k
ak'(zk, ·) 2 Im( )
min
a2RN
1
2
||y ¯a||2 + ||a||1
q = 1 q = 2
q = 3/2
q = 1/2
Why `1? `q ball a 2 RK ;
P
k
|ak
|q 6 1
q = 0
a1
a2
sparse
“`0 ball” !

17. Discrete
`1
regularization:
Computation grid
z
= (
zk)
K
k=1.
Basis-pursuit / Lasso:
Sparse l1 Deconvolution
y
zk
¯ : a 2 RK 7! (ma,z) =
X
k
ak'(zk, ·) 2 Im( )
min
a2RN
1
2
||y ¯a||2 + ||a||1
q = 1 q = 2
q = 3/2
q = 1/2
Why `1? `q ball a 2 RK ;
P
k
|ak
|q 6 1
q = 0
a1
a2
sparse
convex
“`0 ball” !

18. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1

19. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1
|m|(T) =
R
|f| = ||f||L1
d
m
(
x
) =
f
(
x
)d
x

20. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1
|m
a,x
|(T) = ||a||
`
1
m
a,x
|m|(T) =
R
|f| = ||f||L1
d
m
(
x
) =
f
(
x
)d
x

21. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1
|m
a,x
|(T) = ||a||
`
1
m
a,x
|m|(T) =
R
|f| = ||f||L1
d
m
(
x
) =
f
(
x
)d
x

22. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1
min
m
1
2
|| (m) y||2 + |m|(T) (P (y))
Sparse recovery:
|m
a,x
|(T) = ||a||
`
1
m
a,x
|m|(T) =
R
|f| = ||f||L1
d
m
(
x
) =
f
(
x
)d
x

23. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1
min
m
1
2
|| (m) y||2 + |m|(T) (P (y))
Sparse recovery:
(P0(y))
min
m
{|m|(T) ; m = y}
! 0+
|m
a,x
|(T) = ||a||
`
1
m
a,x
|m|(T) =
R
|f| = ||f||L1
d
m
(
x
) =
f
(
x
)d
x

24. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1
min
m
1
2
|| (m) y||2 + |m|(T) (P (y))
Sparse recovery:
(P0(y))
min
m
{|m|(T) ; m = y}
! 0+
|m
a,x
|(T) = ||a||
`
1
m
a,x
|m|(T) =
R
|f| = ||f||L1
d
m
(
x
) =
f
(
x
)d
x
Proposition:
If dim(Im( ))
<
+1, 9(
a, x
) 2 RN ⇥ TN with
N
6 dim(Im( ))
such that
m
a,x is a solution to
P
(
y
).

25. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1
!
Algorithms: [Bredies, Pikkarainen, 2010] (proximal-based)
[Cand`
es, Fernandez-G. 2012] (root ﬁnding)
min
m
1
2
|| (m) y||2 + |m|(T) (P (y))
Sparse recovery:
(P0(y))
min
m
{|m|(T) ; m = y}
! 0+
|m
a,x
|(T) = ||a||
`
1
m
a,x
|m|(T) =
R
|f| = ||f||L1
d
m
(
x
) =
f
(
x
)d
x
Proposition:
If dim(Im( ))
<
+1, 9(
a, x
) 2 RN ⇥ TN with
N
6 dim(Im( ))
such that
m
a,x is a solution to
P
(
y
).

26. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1
!
Algorithms: [Bredies, Pikkarainen, 2010] (proximal-based)
[Cand`
es, Fernandez-G. 2012] (root ﬁnding)
min
m
1
2
|| (m) y||2 + |m|(T) (P (y))
Sparse recovery:
(P0(y))
min
m
{|m|(T) ; m = y}
! 0+
Competitors: Prony’s methods (MUSIC, ESPRIT, FRI).
|m
a,x
|(T) = ||a||
`
1
m
a,x
|m|(T) =
R
|f| = ||f||L1
d
m
(
x
) =
f
(
x
)d
x
Proposition:
If dim(Im( ))
<
+1, 9(
a, x
) 2 RN ⇥ TN with
N
6 dim(Im( ))
such that
m
a,x is a solution to
P
(
y
).

27. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1
!
Algorithms: [Bredies, Pikkarainen, 2010] (proximal-based)
[Cand`
es, Fernandez-G. 2012] (root ﬁnding)
min
m
1
2
|| (m) y||2 + |m|(T) (P (y))
Sparse recovery:
(P0(y))
min
m
{|m|(T) ; m = y}
! 0+
Competitors: Prony’s methods (MUSIC, ESPRIT, FRI).
“+”: always works when
w
= 0, less sensitive to sign.
|m
a,x
|(T) = ||a||
`
1
m
a,x
|m|(T) =
R
|f| = ||f||L1
d
m
(
x
) =
f
(
x
)d
x
Proposition:
If dim(Im( ))
<
+1, 9(
a, x
) 2 RN ⇥ TN with
N
6 dim(Im( ))
such that
m
a,x is a solution to
P
(
y
).

28. Grid-free Sparse Recovery
Grid-free regularization: total variation of measures:
|m|(T) = sup
R
⌘dm : ⌘ 2 C(T), ||⌘||1
6 1
!
Algorithms: [Bredies, Pikkarainen, 2010] (proximal-based)
[Cand`
es, Fernandez-G. 2012] (root ﬁnding)
min
m
1
2
|| (m) y||2 + |m|(T) (P (y))
Sparse recovery:
(P0(y))
min
m
{|m|(T) ; m = y}
! 0+
Competitors: Prony’s methods (MUSIC, ESPRIT, FRI).
“+”: always works when
w
= 0, less sensitive to sign.
“-”: only for convolution operator, '(x, t) = '(x t)
|m
a,x
|(T) = ||a||
`
1
m
a,x
|m|(T) =
R
|f| = ||f||L1
d
m
(
x
) =
f
(
x
)d
x
Proposition:
If dim(Im( ))
<
+1, 9(
a, x
) 2 RN ⇥ TN with
N
6 dim(Im( ))
such that
m
a,x is a solution to
P
(
y
).

29. Overview
• Sparse Spikes Super-resolution
• Robust Support Recovery
• Asymptotic Positive Measure Recovery

30. (P0(y))
Robustness and Support-stability
= 0.55/fc
= 0.45/fc
= 0.1/fc
= 0.3/fc
min
m
{|m|(T) ; m = y}
Low-pass ﬁlter supp( ˆ
'
) = [
fc, fc].
When is
m0 solution of
P0(
m0) ?

31. (P0(y))
Robustness and Support-stability
= 0.55/fc
= 0.45/fc
= 0.1/fc
= 0.3/fc
min
m
{|m|(T) ; m = y}
Low-pass ﬁlter supp( ˆ
'
) = [
fc, fc].
When is
m0 solution of
P0(
m0) ?
Theorem:
[Cand`
es, Fernandez G.]
> 1.26
fc
) m0 solves
P0(
m0).

32. (P0(y))
Robustness and Support-stability
= 0.55/fc
= 0.45/fc
= 0.1/fc
= 0.3/fc
min
m
{|m|(T) ; m = y}
Low-pass ﬁlter supp( ˆ
'
) = [
fc, fc].
are solutions of
P
(
m0 +
w
)?
How close to m0
When is
m0 solution of
P0(
m0) ?
Theorem:
[Cand`
es, Fernandez G.]
> 1.26
fc
) m0 solves
P0(
m0).

33. ! [Cand`
es, Fernandez-G. 2012]
(P0(y))
Robustness and Support-stability
= 0.55/fc
= 0.45/fc
= 0.1/fc
= 0.3/fc
min
m
{|m|(T) ; m = y}
Low-pass ﬁlter supp( ˆ
'
) = [
fc, fc].
are solutions of
P
(
m0 +
w
)?
!
[Fernandez-G.][de Castro 2012]
Weighted
L2
error:
Support localization:
How close to m0
When is
m0 solution of
P0(
m0) ?
Theorem:
[Cand`
es, Fernandez G.]
> 1.26
fc
) m0 solves
P0(
m0).

34. ! [Cand`
es, Fernandez-G. 2012]
(P0(y))
Robustness and Support-stability
= 0.55/fc
= 0.45/fc
= 0.1/fc
= 0.3/fc
min
m
{|m|(T) ; m = y}
Low-pass ﬁlter supp( ˆ
'
) = [
fc, fc].
are solutions of
P
(
m0 +
w
)?
!
[Fernandez-G.][de Castro 2012]
General kernels?
Exact support recovery?
Open problems:
Weighted
L2
error:
Support localization:
How close to m0
When is
m0 solution of
P0(
m0) ?
Theorem:
[Cand`
es, Fernandez G.]
> 1.26
fc
) m0 solves
P0(
m0).

35. From Primal to Dual
min
m
|m|(T) +
1
2
|| m y||2
P (y)
9

36. From Primal to Dual
min
m
|m|(T) +
1
2
|| m y||2
P (y)
9
= min
m
h
sup
||⌘||1
61
h⌘, mi +
1
2
|| m y||2
i

37. From Primal to Dual
min
m
|m|(T) +
1
2
|| m y||2
P (y)
9
m 2 @◆||·||1
61
(⌘)
= min
m
h
sup
||⌘||1
61
h⌘, mi +
1
2
|| m y||2
i

38. From Primal to Dual
min
m
|m|(T) +
1
2
|| m y||2
P (y)
9
m 2 @◆||·||1
61
(⌘)
= min
m
h
sup
||⌘||1
61
h⌘, mi +
1
2
|| m y||2
i
= sup
||⌘||1
61
h
min
m
h⌘, mi +
1
2
|| m y||2
i

39. From Primal to Dual
min
m
|m|(T) +
1
2
|| m y||2
P (y)
9
Ideal low-pass ﬁlter:
! ⌘
=
⇤p
trigonometric polynomial.
m 2 @◆||·||1
61
(⌘)
⌘ = ⇤
m y
= ⇤p
= min
m
h
sup
||⌘||1
61
h⌘, mi +
1
2
|| m y||2
i
= sup
||⌘||1
61
h
min
m
h⌘, mi +
1
2
|| m y||2
i

40. From Primal to Dual
min
m
|m|(T) +
1
2
|| m y||2
P (y)
9
Ideal low-pass ﬁlter:
! ⌘
=
⇤p
trigonometric polynomial.
m 2 @◆||·||1
61
(⌘)
⌘ = ⇤
m y
= ⇤p
= min
m
h
sup
||⌘||1
61
h⌘, mi +
1
2
|| m y||2
i
= sup
||⌘||1
61
h
min
m
h⌘, mi +
1
2
|| m y||2
i
D (y)
= sup
|| ⇤p||1
61
hp, yi
2
||p||2

41. From Primal to Dual
min
m
|m|(T) +
1
2
|| m y||2
P (y)
9
Ideal low-pass ﬁlter:
! ⌘
=
⇤p
trigonometric polynomial.
m 2 @◆||·||1
61
(⌘)
, ⌘ 2 @|m|(T)
⌘ = ⇤
m y
= ⇤p
= min
m
h
sup
||⌘||1
61
h⌘, mi +
1
2
|| m y||2
i
= sup
||⌘||1
61
h
min
m
h⌘, mi +
1
2
|| m y||2
i
D (y)
= sup
|| ⇤p||1
61
hp, yi
2
||p||2

42. From Primal to Dual
min
m
|m|(T) +
1
2
|| m y||2
P (y)

;
8
t,
|

(
t
)| 6 1
8
i, ⌘
(
xi) = sign(
ai)
@|m
a,x
|(T) =
9
Ideal low-pass ﬁlter:
! ⌘
=
⇤p
trigonometric polynomial.
m 2 @◆||·||1
61
(⌘)
, ⌘ 2 @|m|(T)
⌘ = ⇤
m y
= ⇤p
= min
m
h
sup
||⌘||1
61
h⌘, mi +
1
2
|| m y||2
i
= sup
||⌘||1
61
h
min
m
h⌘, mi +
1
2
|| m y||2
i
D (y)
= sup
|| ⇤p||1
61
hp, yi
2
||p||2

43. From Primal to Dual
min
m
|m|(T) +
1
2
|| m y||2
P (y)

;
8
t,
|

(
t
)| 6 1
8
i, ⌘
(
xi) = sign(
ai)
@|m
a,x
|(T) =
9
!
Interpolates spikes location and sign.
Ideal low-pass ﬁlter:
! ⌘
=
⇤p
trigonometric polynomial.
m 2 @◆||·||1
61
(⌘)
, ⌘ 2 @|m|(T)
⌘ = ⇤
m y
= ⇤p
= min
m
h
sup
||⌘||1
61
h⌘, mi +
1
2
|| m y||2
i
= sup
||⌘||1
61
h
min
m
h⌘, mi +
1
2
|| m y||2
i
D (y)
= sup
|| ⇤p||1
61
hp, yi
2
||p||2

44. From Primal to Dual
min
m
|m|(T) +
1
2
|| m y||2
P (y)

;
8
t,
|

(
t
)| 6 1
8
i, ⌘
(
xi) = sign(
ai)
@|m
a,x
|(T) =
9
!
Interpolates spikes location and sign.
! |⌘
(
t
)
|2
= 1: polynomial equation of supp(
m
).
Ideal low-pass ﬁlter:
! ⌘
=
⇤p
trigonometric polynomial.
m 2 @◆||·||1
61
(⌘)
, ⌘ 2 @|m|(T)
⌘ = ⇤
m y
= ⇤p
= min
m
h
sup
||⌘||1
61
h⌘, mi +
1
2
|| m y||2
i
= sup
||⌘||1
61
h
min
m
h⌘, mi +
1
2
|| m y||2
i
D (y)
= sup
|| ⇤p||1
61
hp, yi
2
||p||2

45. Asymptotic Dual and Certificate
min
m
|m|(T) +
1
2
|| m y||2
P (y)
D (y)
p def.
= argmax
|| ⇤p||1
61
hp, yi
2
||p||2

46. Asymptotic Dual and Certificate
min
m
|m|(T) +
1
2
|| m y||2
P (y)
D (y)
P0(y)
! 0+
m0
2 argmin
m=y
|m|(T)
p def.
= argmax
|| ⇤p||1
61
hp, yi
2
||p||2

47. Asymptotic Dual and Certificate
min
m
|m|(T) +
1
2
|| m y||2
P (y)
D (y)
P0(y)
! 0+
m0
2 argmin
m=y
|m|(T)
D0(y)
D0(
y
) = argmax
|| ⇤p||1
61
hp, yi
p def.
= argmax
|| ⇤p||1
61
hp, yi
2
||p||2

48. Asymptotic Dual and Certificate
min
m
|m|(T) +
1
2
|| m y||2
P (y)
D (y)
P0(y)
! 0+
m0
2 argmin
m=y
|m|(T)
p0
def.
= argmax
p2D0(y)
1
2
||p||2
! 0+
D0(y)
D0(
y
) = argmax
|| ⇤p||1
61
hp, yi
p def.
= argmax
|| ⇤p||1
61
hp, yi
2
||p||2

49. Asymptotic Dual and Certificate
min
m
|m|(T) +
1
2
|| m y||2
P (y)
D (y)
P0(y)
! 0+
m0
2 argmin
m=y
|m|(T)
p0
def.
= argmax
p2D0(y)
1
2
||p||2
! 0+
D0(y)
D0(
y
) = argmax
|| ⇤p||1
61
hp, yi
p def.
= argmax
|| ⇤p||1
61
hp, yi
2
||p||2
D0(y) = {p ; ⇤p 2 @|m0
|(T)}
Lemma:

50. Asymptotic Dual and Certificate
min
m
|m|(T) +
1
2
|| m y||2
P (y)
D (y)
P0(y)
! 0+
m0
2 argmin
m=y
|m|(T)
p0
def.
= argmax
p2D0(y)
1
2
||p||2
! 0+
D0(y)
D0(
y
) = argmax
|| ⇤p||1
61
hp, yi
= 1/fc
= 0.6/fc
⌘0 ⌘0
p def.
= argmax
|| ⇤p||1
61
hp, yi
2
||p||2
D0(y) = {p ; ⇤p 2 @|m0
|(T)}
Lemma:
Deﬁnition: for any
m0 solution of
P0(
y
),
⌘0 = ⇤p0 = argmin
⌘= ⇤[email protected]|m0
|(T)
||p||

51. −1
1 η
0
η
V
Vanishing Derivative Pre-certificate
9⌘0
() m0 solves
P0(
m0)
Input measure: m0 = m
a,x
.
⌘0
def.
= argmin
⌘= ⇤p
||
p
|| s.t.

8
i, ⌘
(
xi) = sign(
ai)
,
||

||1
6 1
.

52. −1
1 η
0
η
V
⌘0 = ⌘V
Vanishing Derivative Pre-certificate
9⌘0
() m0 solves
P0(
m0)
Input measure: m0 = m
a,x
.
⌘0
def.
= argmin
⌘= ⇤p
||
p
|| s.t.

8
i, ⌘
(
xi) = sign(
ai)
,
||

||1
6 1
.
⌘V
def.
= argmin
⌘= ⇤p
||
p
|| s.t.

8
i, ⌘
(
xi) = sign(
ai)
,
8
i, ⌘
0(
xi) = 0
.

53. −1
1 η
0
η
V
⌘0 = ⌘V
Vanishing Derivative Pre-certificate
9⌘0
() m0 solves
P0(
m0)
Input measure: m0 = m
a,x
.
⌘0
def.
= argmin
⌘= ⇤p
||
p
|| s.t.

8
i, ⌘
(
xi) = sign(
ai)
,
||

||1
6 1
.
⌘V
def.
= argmin
⌘= ⇤p
||
p
|| s.t.

8
i, ⌘
(
xi) = sign(
ai)
,
8
i, ⌘
0(
xi) = 0
.
where
Ax
(
b
) =
P
i b
1
i '
(
xi,
·) +
b
2
i '
0(
xi,
·)
Proposition:

V
= ⇤A+
x
(sign(a); 0)

54. −1
1 η
0
η
V
⌘0 = ⌘V
Vanishing Derivative Pre-certificate
9⌘0
() m0 solves
P0(
m0)
Input measure: m0 = m
a,x
.
⌘0
def.
= argmin
⌘= ⇤p
||
p
|| s.t.

8
i, ⌘
(
xi) = sign(
ai)
,
||

||1
6 1
.
⌘V
def.
= argmin
⌘= ⇤p
||
p
|| s.t.

8
i, ⌘
(
xi) = sign(
ai)
,
8
i, ⌘
0(
xi) = 0
.
()
Non-degenerate certiﬁcate:
⌘ 2 ND(m
a,x
) :
8
t /
2 {
x1, . . . , xN
}
,
|

(
t
)|
<
1 and 8
i, ⌘
00(
xi) 6= 0
where
Ax
(
b
) =
P
i b
1
i '
(
xi,
·) +
b
2
i '
0(
xi,
·)
Proposition:

V
= ⇤A+
x
(sign(a); 0)

55. −1
1 η
0
η
V
⌘0
6= ⌘V
−1
1 η
0
η
V
⌘0 = ⌘V
Vanishing Derivative Pre-certificate
9⌘0
() m0 solves
P0(
m0)
Theorem:
⌘V
2 ND(m0) =) ⌘V = ⌘0
Input measure: m0 = m
a,x
.
⌘0
def.
= argmin
⌘= ⇤p
||
p
|| s.t.

8
i, ⌘
(
xi) = sign(
ai)
,
||

||1
6 1
.
⌘V
def.
= argmin
⌘= ⇤p
||
p
|| s.t.

8
i, ⌘
(
xi) = sign(
ai)
,
8
i, ⌘
0(
xi) = 0
.
()
Non-degenerate certiﬁcate:
⌘ 2 ND(m
a,x
) :
8
t /
2 {
x1, . . . , xN
}
,
|

(
t
)|
<
1 and 8
i, ⌘
00(
xi) 6= 0
where
Ax
(
b
) =
P
i b
1
i '
(
xi,
·) +
b
2
i '
0(
xi,
·)
Proposition:

V
= ⇤A+
x
(sign(a); 0)

56. Support Stability Theorem
⌘ = ⇤p !0
! ⌘0 = ⇤p0
supp(m ) ⇢ {|⌘ | = 1}

57. Support Stability Theorem
⌘ = ⇤p !0
! ⌘0 = ⇤p0
supp(m ) ⇢ {|⌘ | = 1}
If ⌘0
2 ND(m0) then supp(m ) ! supp(m0)
⌘0

x1 x2
x
?
2
x
?
1

58. Support Stability Theorem
x
?
i
max
Noiseless
w
= 0. min
⇠ ||w||
x
?
i
max
||w||
⌘ = ⇤p !0
! ⌘0 = ⇤p0
supp(m ) ⇢ {|⌘ | = 1}
If ⌘0
2 ND(m0) then supp(m ) ! supp(m0)
⌘0

x1 x2
x
?
2
x
?
1
Theorem:
the solution of
P
(
y
) for
y
= (
m0) +
w
is
for (
||w||/ ,
) =
O
(1),
[Duval, Peyr´
e 2014]
If

V
2
ND(
m0) for
m0 =
m
a,x, then
m =
P
N
i
=1
a?
i x
?
i
where ||(x, a) (x?
, a?)|| = O(||w||)

59. When is Non-degenerate ?
⌘V
' '
⌘V
⌘V
⌘V
⌘V
⌘V
⌘V
Input measure: m0 = m
a, x
, ! 0

60. When is Non-degenerate ?
⌘V
' '
⌘V
⌘V
⌘V
⌘V
⌘V
⌘V
Input measure:
Theorem:
[Tang, Recht, 2013]
Valid for:
'
(
x
) =
e x
2
/
2
'
(
x
) = (1 + (
x/
)2) 1
. . .
9C,
(
> C
) =
)
(
⌘V is non degenerate)
m0 = m
a, x
, ! 0

61. Overview
• Sparse Spikes Super-resolution
• Robust Support Recovery
• Asymptotic Positive Measure Recovery

62. Recovery of Positive Measures
m = (
R
e 2i⇡ktdm(t))fc
k= fc
Theorem:
let and
[de Castro et al. 2011]
! m0 is recovered when there is no noise.
⌘S(
t
) = 1

QN
i=1
sin(

(
t xi))2
for
N 6 fc and

small enough,
⌘S
2 ¯
D
(
m0).
-1
1
-1
1
-1
1
-1
1
⌘S ⌘S
⌘S
⌘S
Input measure: m0 = m
a,x
where a 2 RN
+
.

63. Recovery of Positive Measures
m = (
R
e 2i⇡ktdm(t))fc
k= fc
Theorem:
let and
[de Castro et al. 2011]
! m0 is recovered when there is no noise.
⌘S(
t
) = 1

QN
i=1
sin(

(
t xi))2
for
N 6 fc and

small enough,
⌘S
2 ¯
D
(
m0).
-1
1
-1
1
-1
1
-1
1
⌘S ⌘S
⌘S
⌘S
[Morgenshtern, Cand`
es, 2015] discrete
`1
robustness.
[Demanet, Nguyen, 2015] discrete
`0
robustness.
Input measure: m0 = m
a,x
where a 2 RN
+
.
!
behavior as
8
i, xi
!
0 ?

64. Recovery of Positive Measures
m = (
R
e 2i⇡ktdm(t))fc
k= fc
Theorem:
let and
[de Castro et al. 2011]
! m0 is recovered when there is no noise.
!
noise robustness of support recovery ?
⌘S(
t
) = 1

QN
i=1
sin(

(
t xi))2
for
N 6 fc and

small enough,
⌘S
2 ¯
D
(
m0).
-1
1
-1
1
-1
1
-1
1
⌘S ⌘S
⌘S
⌘S
[Morgenshtern, Cand`
es, 2015] discrete
`1
robustness.
[Demanet, Nguyen, 2015] discrete
`0
robustness.
Input measure: m0 = m
a,x
where a 2 RN
+
.
!
behavior as
8
i, xi
!
0 ?

65. Comparison of Certificates
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
⌘S ⌘V

66. Asymptotic of Vanishing Certificate
1
⌘V
Vanishing Derivative pre-certiﬁcate:
⌘V
def.
= argmin
⌘= ⇤p
||p||
m0 = m
a, x
where ! 0
s.t. 8
i,

(
xi) = 1
,

0(
xi) = 0
.

67. Asymptotic of Vanishing Certificate
1
⌘V
1
1
1
⌘V
⌘V
⌘W
Vanishing Derivative pre-certiﬁcate:
⌘V
def.
= argmin
⌘= ⇤p
||p||
m0 = m
a, x
where ! 0
s.t. 8
i,

(
xi) = 1
,

0(
xi) = 0
.

68. Asymptotic of Vanishing Certificate
1
⌘V
1
1
1
⌘V
⌘V
⌘W
s.t.

⌘(0) = 1,
⌘0(0) = . . . = ⌘(2N 1)(0) = 0.
Asymptotic pre-certiﬁcate:
⌘W
def.
= argmin
⌘= ⇤p
||p||
Vanishing Derivative pre-certiﬁcate:
⌘V
def.
= argmin
⌘= ⇤p
||p||
! 0
m0 = m
a, x
where ! 0
s.t. 8
i,

(
xi) = 1
,

0(
xi) = 0
.

69. Asymptotic Certificate
1
1
1
1
⌘V = ⌘W
⌘W
⌘W
⌘W
N = 1
N = 2
N = 3
N = 4
(2N 1)
-Non degenerate:
()
⌘W
(2N)(0) 6= 0

8 t 6= 0, |⌘W (t)| < 1
⌘W
2 NDN

70. Asymptotic Certificate
1
1
1
1
⌘V = ⌘W
⌘W
⌘W
⌘W
N = 1
N = 2
N = 3
N = 4
(2N 1)
-Non degenerate:
()
⌘W
(2N)(0) 6= 0

8 t 6= 0, |⌘W (t)| < 1
⌘W
2 NDN
Lemma:
! ⌘W govern stability as
!
0.
If ⌘W
2 NDN , 9 0 > 0,
8 < 0, ⌘
V
2 ND(m
x,a
)

71. Asymptotic Robustness
Theorem:
the solution of
P
(
y
) for
y
= (
m0) +
w
is
for
w , w
2N 1
,
2N 1 =
O
(1)
P
N
i
=1
a?
i x
?
i
||w
||
N = 2
||w
||
N = 1
If ⌘
W
2 ND
N
, letting m0 = m
a, x
, then
where ||(x, a) (x?, a?)|| = O

||w|| +
2N 1

[Denoyelle, D., P. 2015]

72. Asymptotic Robustness
Theorem:
the solution of
P
(
y
) for
y
= (
m0) +
w
is
x
?
i
0
↵ < 2N 1
Noise:
w
=
w0.
for
w , w
2N 1
,
2N 1 =
O
(1)
P
N
i
=1
a?
i x
?
i
Regularization: = 0

0
max
↵ = 2N 1
x
?
i
x0
||w
||
N = 2
||w
||
N = 1
If ⌘
W
2 ND
N
, letting m0 = m
a, x
, then
where ||(x, a) (x?, a?)|| = O

||w|| +
2N 1

y = m
a, x
+ w
[Denoyelle, D., P. 2015]

73. When is Non-degenerate ?
⌘W
Proposition: one has
⌘(2N)
W (0)
<
0.
!
“locally” non-degenerate.

74. When is Non-degenerate ?
⌘W
Proposition: one has
⌘(2N)
W (0)
<
0.
!
“locally” non-degenerate.
'
ˆ
'
⌘W
⌘W
⌘W
N = 2
N = 3
N = 4

75. Gaussian Deconvolution
Gaussian convolution: '
(
x, t
) =
e
|
x t
|2
2 2
Proposition: ⌘W (
x
) =
e
x
2
4 2
N 1
X
k=0
(
x/
2 )2k
k
!
In particular,
⌘W is non-degenerate.
(
m
) def.
=
Z
'
(
x,
·)d
m
(
x
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.2
0
0.2
0.4
0.6
0.8
1
1 1
1 1
!
Gaussian deconvolution is support-stable.
N = 1 N = 2
N = 3 N = 4
'(0, ·)
⌘W
⌘W ⌘W
⌘W

76. Laplace Transform Inversion
(
m
) def.
=
Z
'
(
x,
·)d
m
(
x
)
'
(
x, t
) =
e xt
Laplace transform:
'
(
x,
·)
x
= 2
x
= 20
t
[with E. Soubies]

77. Laplace Transform Inversion
(
m
) def.
=
Z
'
(
x,
·)d
m
(
x
)
'
(
x, t
) =
e xt
Laplace transform:
'
(
x,
·)
x
= 2
x
= 20
t
(m1)
(m2)
x
m1
t
m2
x
[with E. Soubies]

78. Laplace Transform Inversion
(
m
) def.
=
Z
'
(
x,
·)d
m
(
x
)
'
(
x, t
) =
e xt
Laplace transform:
'
(
x,
·)
x
= 2
x
= 20
Total internal reﬂection ﬂuorescence microscopy (TIRFM)
t
(m1)
(m2)
x
m1
t
m2
x
[with E. Soubies]
[Boulanger et al. 2014]
varying the azimuth φ during the exposure time and can be
modeled by the following expression:
gðθÞ =
Z2π
0
Z∞
0
Z∞
−∞
Iðz; α; φÞρ

θ − α
Ω=cos θ

f
À
z
Á
dαdzdφ;
where fðzÞ is the density of fluorophores in the medium con-
volved by the emission point spread function and ρð · Þ represents
the laser beam profile of divergence Ω. The function Iðz; α; φÞ
slope of the glass slide recovered (Fig. 2D), the latter falling within
the confidence interval deducted from the accuracy of the mea-
surement of the different characteristic dimensions of the sample.
Finally, from the dispersion of the estimated depth around the
average slope (Fig. 2D), we can conclude that the localization
precision obtained with this approach is higher than the corre-
sponding precision given by estimating the location of the beads in
the WF image stack as already mentioned (17).
Estimating the 3D density of fluorophores convolved by the
emission point spread function then would simply boil down to
inverting the linear system. Some care has to be taken when
inverting such system, as the inverse problem is at best badly con-
ditioned. Nevertheless, constraints can be imposed to the solution
such as positivity, and, in the case of time-lapse acquisitions, a
multiframe regularization can be used in addition to the spatial and
temporal regularization smoothness to solve the reconstruction
problem. Moreover, to be effective, such a positivity constraint
requires a correct knowledge of the background level. As a conse-
quence, for each multiangle image stack, a background image is
obtained by driving the beam out of the objective. Given that
several convex constraints have to be satisfied at the same time, we
propose to rely on a flavor of the PPXA algorithm (26) to estimate
the tridimensional density of fluorophores (Fig. S4). More detailed
information on how noise, object depth, and the required number
of angles can be taken into account is discussed in SI Imaging Model
and Reconstruction and Fig. S5. Finally, to take into account the
variations of the medium index, we select an effective index within
a predefined range by minimizing the reconstruction error at each
pixel under a spatial smoothness constraint (Fig. S6). It is worth
noting that the computation time for the reconstruction on 10
planes from a stack 512 × 512 images corresponding to 21 in-
cidence angles ranges from 1 to 5 min depending on the number
of iterations.
Imaging in Vitro and in Vivo Actin Assembly. The proposed multi-
angle TIRF image reconstruction approach was then tested on
complex samples such as actin network architectures for which
spatial resolution and dynamics remain an issue. We first chal-
lenged the spatial organization of actin nucleation geometry
using an in vitro assay based on micropatterning method (27).
A
B
C
D
Fig. 2. Experimental validation of the multiangle TIRF model. (A) Schema
of the system designed to create a slope of fluorescent beads. (B) Overlay
of the maximum intensity projection of image stack acquired with WF and
TIRF illumination. (Scale bar: 5 μm.) The evolution of the intensity versus the
illumination angle θ of two selected beads are plotted in C with the corre-
sponding fitting theoretical model (continuous line) for their estimated
depth (respectively 10 and 89 nm). (D) Depth of all of the beads estimated by
fitting the theoretical TIRF model (in red) and the depth of the same beads
estimated by fitting a Gaussian model in the WF image stack (in green).
BIOPHYSICS AND
COMPUTATIONAL BIOLOGY
slope of the glass slide recovered (Fig. 2D),
the confidence interval deducted from the
surement of the different characteristic dim
Finally, from the dispersion of the estima
average slope (Fig. 2D), we can conclude
precision obtained with this approach is h
sponding precision given by estimating the l
the WF image stack as already mentioned
Estimating the 3D density of fluoroph
emission point spread function then woul
inverting the linear system. Some care h
inverting such system, as the inverse proble
ditioned. Nevertheless, constraints can be i
such as positivity, and, in the case of tim
multiframe regularization can be used in add
temporal regularization smoothness to so
problem. Moreover, to be effective, such
requires a correct knowledge of the backgro
quence, for each multiangle image stack,
obtained by driving the beam out of the
several convex constraints have to be satisfie
propose to rely on a flavor of the PPXA alg
A
B
C
D
Fig. 2. Experimental validation of the multiangle TIRF model. (A) Schema
of the system designed to create a slope of fluorescent beads. (B) Overlay
of the maximum intensity projection of image stack acquired with WF and
✓(t)
y(t)
light
depth
x
cell
y(t) = m(t)
✓(t)
! multiple angles ✓(t).

79. Laplace Transform Inversion
(
m
) def.
=
Z
'
(
x,
·)d
m
(
x
)
'
(
x, t
) =
e xt
Laplace transform:
'
(
x,
·)
x
= 2
x
= 20
Total internal reﬂection ﬂuorescence microscopy (TIRFM)
t
(m1)
(m2)
x
m1
t
m2
x
[with E. Soubies]
[Boulanger et al. 2014]
varying the azimuth φ during the exposure time and can be
modeled by the following expression:
gðθÞ =
Z2π
0
Z∞
0
Z∞
−∞
Iðz; α; φÞρ

θ − α
Ω=cos θ

f
À
z
Á
dαdzdφ;
where fðzÞ is the density of fluorophores in the medium con-
volved by the emission point spread function and ρð · Þ represents
the laser beam profile of divergence Ω. The function Iðz; α; φÞ
slope of the glass slide recovered (Fig. 2D), the latter falling within
the confidence interval deducted from the accuracy of the mea-
surement of the different characteristic dimensions of the sample.
Finally, from the dispersion of the estimated depth around the
average slope (Fig. 2D), we can conclude that the localization
precision obtained with this approach is higher than the corre-
sponding precision given by estimating the location of the beads in
the WF image stack as already mentioned (17).
Estimating the 3D density of fluorophores convolved by the
emission point spread function then would simply boil down to
inverting the linear system. Some care has to be taken when
inverting such system, as the inverse problem is at best badly con-
ditioned. Nevertheless, constraints can be imposed to the solution
such as positivity, and, in the case of time-lapse acquisitions, a
multiframe regularization can be used in addition to the spatial and
temporal regularization smoothness to solve the reconstruction
problem. Moreover, to be effective, such a positivity constraint
requires a correct knowledge of the background level. As a conse-
quence, for each multiangle image stack, a background image is
obtained by driving the beam out of the objective. Given that
several convex constraints have to be satisfied at the same time, we
propose to rely on a flavor of the PPXA algorithm (26) to estimate
the tridimensional density of fluorophores (Fig. S4). More detailed
information on how noise, object depth, and the required number
of angles can be taken into account is discussed in SI Imaging Model
and Reconstruction and Fig. S5. Finally, to take into account the
variations of the medium index, we select an effective index within
a predefined range by minimizing the reconstruction error at each
pixel under a spatial smoothness constraint (Fig. S6). It is worth
noting that the computation time for the reconstruction on 10
planes from a stack 512 × 512 images corresponding to 21 in-
cidence angles ranges from 1 to 5 min depending on the number
of iterations.
Imaging in Vitro and in Vivo Actin Assembly. The proposed multi-
angle TIRF image reconstruction approach was then tested on
complex samples such as actin network architectures for which
spatial resolution and dynamics remain an issue. We first chal-
lenged the spatial organization of actin nucleation geometry
using an in vitro assay based on micropatterning method (27).
A
B
C
D
Fig. 2. Experimental validation of the multiangle TIRF model. (A) Schema
of the system designed to create a slope of fluorescent beads. (B) Overlay
of the maximum intensity projection of image stack acquired with WF and
TIRF illumination. (Scale bar: 5 μm.) The evolution of the intensity versus the
illumination angle θ of two selected beads are plotted in C with the corre-
sponding fitting theoretical model (continuous line) for their estimated
depth (respectively 10 and 89 nm). (D) Depth of all of the beads estimated by
fitting the theoretical TIRF model (in red) and the depth of the same beads
estimated by fitting a Gaussian model in the WF image stack (in green).
BIOPHYSICS AND
COMPUTATIONAL BIOLOGY
slope of the glass slide recovered (Fig. 2D),
the confidence interval deducted from the
surement of the different characteristic dim
Finally, from the dispersion of the estima
average slope (Fig. 2D), we can conclude
precision obtained with this approach is h
sponding precision given by estimating the l
the WF image stack as already mentioned
Estimating the 3D density of fluoroph
emission point spread function then woul
inverting the linear system. Some care h
inverting such system, as the inverse proble
ditioned. Nevertheless, constraints can be i
such as positivity, and, in the case of tim
multiframe regularization can be used in add
temporal regularization smoothness to so
problem. Moreover, to be effective, such
requires a correct knowledge of the backgro
quence, for each multiangle image stack,
obtained by driving the beam out of the
several convex constraints have to be satisfie
propose to rely on a flavor of the PPXA alg
A
B
C
D
Fig. 2. Experimental validation of the multiangle TIRF model. (A) Schema
of the system designed to create a slope of fluorescent beads. (B) Overlay
of the maximum intensity projection of image stack acquired with WF and
✓(t)
y(t)
light
depth
x
cell
y(t) = m(t)
✓(t)
! multiple angles ✓(t).
N = 1
N = 2
N = 3
⌘W
⌘W
⌘W
¯
x
= 2 ¯
x
= 20
Non-translation-invariant operator
¯
x
x1 x2
!
⌘W depends on ¯
x!

80. Laplace Transform Inversion
(
m
) def.
=
Z
'
(
x,
·)d
m
(
x
)
'
(
x, t
) =
e xt
Laplace transform:
'
(
x,
·)
x
= 2
x
= 20
Total internal reﬂection ﬂuorescence microscopy (TIRFM)
t
(m1)
(m2)
x
m1
t
m2
x
[with E. Soubies]
[Boulanger et al. 2014]
varying the azimuth φ during the exposure time and can be
modeled by the following expression:
gðθÞ =
Z2π
0
Z∞
0
Z∞
−∞
Iðz; α; φÞρ

θ − α
Ω=cos θ

f
À
z
Á
dαdzdφ;
where fðzÞ is the density of fluorophores in the medium con-
volved by the emission point spread function and ρð · Þ represents
the laser beam profile of divergence Ω. The function Iðz; α; φÞ
slope of the glass slide recovered (Fig. 2D), the latter falling within
the confidence interval deducted from the accuracy of the mea-
surement of the different characteristic dimensions of the sample.
Finally, from the dispersion of the estimated depth around the
average slope (Fig. 2D), we can conclude that the localization
precision obtained with this approach is higher than the corre-
sponding precision given by estimating the location of the beads in
the WF image stack as already mentioned (17).
Estimating the 3D density of fluorophores convolved by the
emission point spread function then would simply boil down to
inverting the linear system. Some care has to be taken when
inverting such system, as the inverse problem is at best badly con-
ditioned. Nevertheless, constraints can be imposed to the solution
such as positivity, and, in the case of time-lapse acquisitions, a
multiframe regularization can be used in addition to the spatial and
temporal regularization smoothness to solve the reconstruction
problem. Moreover, to be effective, such a positivity constraint
requires a correct knowledge of the background level. As a conse-
quence, for each multiangle image stack, a background image is
obtained by driving the beam out of the objective. Given that
several convex constraints have to be satisfied at the same time, we
propose to rely on a flavor of the PPXA algorithm (26) to estimate
the tridimensional density of fluorophores (Fig. S4). More detailed
information on how noise, object depth, and the required number
of angles can be taken into account is discussed in SI Imaging Model
and Reconstruction and Fig. S5. Finally, to take into account the
variations of the medium index, we select an effective index within
a predefined range by minimizing the reconstruction error at each
pixel under a spatial smoothness constraint (Fig. S6). It is worth
noting that the computation time for the reconstruction on 10
planes from a stack 512 × 512 images corresponding to 21 in-
cidence angles ranges from 1 to 5 min depending on the number
of iterations.
Imaging in Vitro and in Vivo Actin Assembly. The proposed multi-
angle TIRF image reconstruction approach was then tested on
complex samples such as actin network architectures for which
spatial resolution and dynamics remain an issue. We first chal-
lenged the spatial organization of actin nucleation geometry
using an in vitro assay based on micropatterning method (27).
A
B
C
D
Fig. 2. Experimental validation of the multiangle TIRF model. (A) Schema
of the system designed to create a slope of fluorescent beads. (B) Overlay
of the maximum intensity projection of image stack acquired with WF and
TIRF illumination. (Scale bar: 5 μm.) The evolution of the intensity versus the
illumination angle θ of two selected beads are plotted in C with the corre-
sponding fitting theoretical model (continuous line) for their estimated
depth (respectively 10 and 89 nm). (D) Depth of all of the beads estimated by
fitting the theoretical TIRF model (in red) and the depth of the same beads
estimated by fitting a Gaussian model in the WF image stack (in green).
BIOPHYSICS AND
COMPUTATIONAL BIOLOGY
slope of the glass slide recovered (Fig. 2D),
the confidence interval deducted from the
surement of the different characteristic dim
Finally, from the dispersion of the estima
average slope (Fig. 2D), we can conclude
precision obtained with this approach is h
sponding precision given by estimating the l
the WF image stack as already mentioned
Estimating the 3D density of fluoroph
emission point spread function then woul
inverting the linear system. Some care h
inverting such system, as the inverse proble
ditioned. Nevertheless, constraints can be i
such as positivity, and, in the case of tim
multiframe regularization can be used in add
temporal regularization smoothness to so
problem. Moreover, to be effective, such
requires a correct knowledge of the backgro
quence, for each multiangle image stack,
obtained by driving the beam out of the
several convex constraints have to be satisfie
propose to rely on a flavor of the PPXA alg
A
B
C
D
Fig. 2. Experimental validation of the multiangle TIRF model. (A) Schema
of the system designed to create a slope of fluorescent beads. (B) Overlay
of the maximum intensity projection of image stack acquired with WF and
✓(t)
y(t)
light
depth
x
cell
y(t) = m(t)
✓(t)
! multiple angles ✓(t).
N = 1
N = 2
N = 3
⌘W
⌘W
⌘W
¯
x
= 2 ¯
x
= 20
Non-translation-invariant operator
¯
x
x1 x2
!
⌘W depends on ¯
x!
Proposition:
In particular,
⌘W is non-degenerate.
⌘W (
x
) = 1

x
¯
x
x
+ ¯
x
◆2N

81. Deconvolution of measures:
! L2
errors are not well-suited.
Weak-* convergence.
Optimal transport distance.
Exact support estimation.
...
Conclusion

82. Deconvolution of measures:
! L2
errors are not well-suited.
Weak-* convergence.
Optimal transport distance.
Exact support estimation.
...
Conclusion
Low-noise behavior:
! dictated by ⌘0.
! checkable via ⌘V .
!
asymptotic via
⌘W .

83. Lasso on discrete grids:
Deconvolution of measures:
! L2
errors are not well-suited.
Weak-* convergence.
Optimal transport distance.
Exact support estimation.
...
similar ⌘0-analysis applies.
!
Relate discrete and continuous recoveries.
Conclusion
Low-noise behavior:
! dictated by ⌘0.
! checkable via ⌘V .
!
asymptotic via
⌘W .

84. Lasso on discrete grids:
Deconvolution of measures:
! L2
errors are not well-suited.
Weak-* convergence.
Optimal transport distance.
Exact support estimation.
...
similar ⌘0-analysis applies.
!
Relate discrete and continuous recoveries.
Open problem:
other regularizations (e.g. piecewise constant) ?
Conclusion
Low-noise behavior:
! dictated by ⌘0.
! checkable via ⌘V .
!
asymptotic via
⌘W .
see [Chambolle, Duval, Peyr´
e, Poon 2016] for TV denoising.