discrete versus continuous sparse regularizationthe deconvolution problem assumption:the signal m 0...
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![Page 1: Discrete versus Continuous sparse regularizationThe Deconvolution Problem Assumption:the signal m 0 is sparse Original Signal m 0 t 0 1 x 123 = Low-pass lter ’ t x0:5 0:5 + Noise](https://reader033.vdocuments.us/reader033/viewer/2022053022/604e8ecde897ef40610cdd36/html5/thumbnails/1.jpg)
Discrete versus Continuous sparse regularization
Vincent Duval Gabriel Peyre
CNRS - Universite Paris-Dauphine
August 29, 2014
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 1 / 22
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Outline
1 The deconvolution problem
2 Discrete regularization
3 Continuous regularization
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 2 / 22
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Outline
1 The deconvolution problem
2 Discrete regularization
3 Continuous regularization
![Page 4: Discrete versus Continuous sparse regularizationThe Deconvolution Problem Assumption:the signal m 0 is sparse Original Signal m 0 t 0 1 x 123 = Low-pass lter ’ t x0:5 0:5 + Noise](https://reader033.vdocuments.us/reader033/viewer/2022053022/604e8ecde897ef40610cdd36/html5/thumbnails/4.jpg)
Deconvolution
Measuring devices have a non sharp impulse response: our observationsare blurred of a ”true ideal scene”.
Geophysics,
Astronomy,
Microscopy,
Spectroscopy,
. . . Image courtesy of S. Ladjal
Goal: Obtain as much detail as we can from given measurements.
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 4 / 22
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The Deconvolution Problem
Consider a signal m0 defined on T = R/Z (i.e. [0, 1) with periodicboundary condition).
Perturbation model:
Original Signal
m0
t0 1
∗
Low-pass filter
ϕ
t0.5−0.5
+
Noise
w
t0 1
=
Observation
y0 + w
t0 1
Goal: recover m0 from the observation y0 + w = ϕ ∗m0 + w (orsimply y0 = ϕ ∗m0)
Ill-posed problem:I the low pass filter might not be invertible (ϕn = 0 for some
frequency n)I even though, the problem is ill-conditioned (|ϕn| � |ϕ0| for high
frequencies n)
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 5 / 22
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The Deconvolution Problem
Consider a signal m0 defined on T = R/Z (i.e. [0, 1) with periodicboundary condition).
Perturbation model:
Original Signal
m0
t0 1
∗
Low-pass filter
ϕ
t0.5−0.5
+
Noise
w
t0 1
=
Observation
y0 + w
t0 1
Goal: recover m0 from the observation y0 + w = ϕ ∗m0 + w (orsimply y0 = ϕ ∗m0)Ill-posed problem:
I the low pass filter might not be invertible (ϕn = 0 for somefrequency n)
I even though, the problem is ill-conditioned (|ϕn| � |ϕ0| for highfrequencies n)
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 5 / 22
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The Deconvolution Problem
Assumption: the signal m0 is sparse
Original Signal
m0
t0 1x1
x2
x3
∗
Low-pass filter
ϕ
t0.5−0.5
+
Noise
w
t0 1
=
Observation
y + w
t0 1
m0 =N∑i=1
aiδxi , where
ai ∈ R,xi ∈ T,N ∈ N is small.
so that we observe y + w =∑N
i=1 aiϕ(· − xi) + w.
Idea: Look for a sparse signal m such that ϕ ∗m ≈ y0 + w (or y0).
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 6 / 22
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Question
How can we use sparsity to solve the inverse problem?
Can we guarantee that the reconstructed signal is close to the originalone?
What can we say about the structure/support of the solution?
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 7 / 22
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Outline
1 The deconvolution problem
2 Discrete regularization
3 Continuous regularization
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Discretization
Define a finite grid G = { i2M
; 0 6 i 6 2M − 1} ⊂ T,
and consider signals of the form m =∑2M−1
i=0 aiδ i
2M.
Candidate Signal
m
t0 1
12M
32M
2M−12M
Write
ϕ ∗m =
2M−1∑i=0
aiϕ(· − i
2M)
=
ϕ ϕ(·− 12M
) . . . ϕ(· − 2M−12M
)
︸ ︷︷ ︸
Φ
a0
a1...
aM−1
︸ ︷︷ ︸
a
.
Equivalent paradigm: Look for a sparse vector a ∈ R2M such thatΦa ≈ y0 (or Φa ≈ y0 + w).
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 9 / 22
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Discrete `1 regularizationDefine
‖m‖`1(G) =
{ ∑2M−1i=0 |ai| if m =
∑2M−1i=0 aiδi/2M ,
+∞ otherwise.
m0 =∑2M−1
i=0 a0,iδi/2M
t0 11
2M
32M
2M−12M
Basis Pursuit [Chen & Donoho (94)]
infm∈M(T)
‖m‖`1(G) such that Φm = y0 (PM0 (y0))
LASSO [Tibshirani (96)] or Basis Pursuit Denoising [Chen et al. (99)]
infm∈M(T)
λ‖m‖`1(G) +1
2||Φm− (y0 + w)||22 (PMλ (y0 + w))
`2-robustness ([Grasmair et al.(11)])
If m0 =∑i a0,iδi/2M is the unique solution to PM0 (y0), and mλ =
∑i aλ,i is a
solution to PMλ (y0 + w), then ‖aλ − a0‖2 = O(‖w‖2) for λ = C‖w‖2.
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 10 / 22
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Discrete `1 regularizationDefine
‖m‖`1(G) =
{ ∑2M−1i=0 |ai| if m =
∑2M−1i=0 aiδi/2M ,
+∞ otherwise.
Basis Pursuit [Chen & Donoho (94)]
infm∈M(T)
‖m‖`1(G) such that Φm = y0 (PM0 (y0))
LASSO [Tibshirani (96)] or Basis Pursuit Denoising [Chen et al. (99)]
infm∈M(T)
λ‖m‖`1(G) +1
2||Φm− (y0 + w)||22 (PMλ (y0 + w))
`2-robustness ([Grasmair et al.(11)])
If m0 =∑i a0,iδi/2M is the unique solution to PM0 (y0), and mλ =
∑i aλ,i is a
solution to PMλ (y0 + w), then ‖aλ − a0‖2 = O(‖w‖2) for λ = C‖w‖2.
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 10 / 22
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Robustness of the support (discrete problem)
Can one guarantee that Suppmλ = Suppm0?
Sufficient conditions for Suppmλ ⊆ Suppm0:I Exact Recovery Principle (ERC) [Tropp (06)]
I Weak Exact Recovery Principle (W-ERC) [Dossal & Mallat (05)]
Almost necessary and sufficient Suppmλ = Suppm0
I Fuchs criterion [Fuchs (04)]
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 11 / 22
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Robustness of the support (discrete problem)
Can one guarantee that Suppmλ = Suppm0?
Sufficient conditions for Suppmλ ⊆ Suppm0:I Exact Recovery Principle (ERC) [Tropp (06)]
I Weak Exact Recovery Principle (W-ERC) [Dossal & Mallat (05)]
Almost necessary and sufficient Suppmλ = Suppm0
I Fuchs criterion [Fuchs (04)]
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 11 / 22
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Optimality conditionsFrom convex analysis:
a0 is a solution to PM0 (y0) if and only if there exists p ∈ L2(T) such that thefunction η = Φ∗p satisfies:
∀i s.t. a0,i 6= 0, η
(i
2M
)= sign(a0,i) and ∀k,
∣∣∣∣η( k
2M
)∣∣∣∣ 6 1;
aλ is a solution to PMλ (y0 + w) if and only if the functionηλ = 1
λΦ∗(y0 + w − Φ∗aλ) satisfies:
∀i s.t. aλ,i 6= 0, ηλ
(i
2M
)= sign(aλ,i) and ∀k,
∣∣∣∣ηλ( k
2M
)∣∣∣∣ 6 1;
−1
0
1
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 12 / 22
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Fuchs theorem
For m0 =∑Ni=1 a0,iδx0,i , define
ηF = Φ∗pF , where
pF = argmin{‖p‖L2(T); (Φ∗p)(x0,i) = sign(a0,i)}
= Φ+,∗s.−1
0
1
Theorem ([Fuchs (04)])
Assume that {ϕ(· − x0,1), . . . ϕ(· − x0,N )} has full rank.If |ηF ( k
2M)| < 1 for all k such that k
2M/∈ {x0,1, . . . , x0,N}, then m0 is the unique
solution to PM0 (y0), and there exists α > 0, λ0 > 0 such that for 0 6 λ 6 λ0 and‖w‖2 6 αλ,
The solution mMλ to PMλ (y0 + w) is unique.
SuppmM = Suppm0, that is mMλ =
∑Ni=1 a
Mλ,iδx0,i , and sign(aλ,i) = sign(a0,i),
aMλ,I = a0,I + Φ+xIw − λ(Φ∗xIΦxI )−1 sign(a0,I).
If |ηF ( k2M
)| > 1 for some k, the support is not stable.
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 13 / 22
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Fuchs theorem
For m0 =∑Ni=1 a0,iδx0,i , define
ηF = Φ∗pF , where
pF = argmin{‖p‖L2(T); (Φ∗p)(x0,i) = sign(a0,i)}
= Φ+,∗s.
Theorem ([Fuchs (04)])
Assume that {ϕ(· − x0,1), . . . ϕ(· − x0,N )} has full rank.If |ηF ( k
2M)| < 1 for all k such that k
2M/∈ {x0,1, . . . , x0,N}, then m0 is the unique
solution to PM0 (y0), and there exists α > 0, λ0 > 0 such that for 0 6 λ 6 λ0 and‖w‖2 6 αλ,
The solution mMλ to PMλ (y0 + w) is unique.
SuppmM = Suppm0, that is mMλ =
∑Ni=1 a
Mλ,iδx0,i , and sign(aλ,i) = sign(a0,i),
aMλ,I = a0,I + Φ+xIw − λ(Φ∗xIΦxI )−1 sign(a0,I).
If |ηF ( k2M
)| > 1 for some k, the support is not stable.
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 13 / 22
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But. . .
1 1
When the grid is too thin, the Fuchs criterion cannot hold⇒ the support is not stable.
Theorem (D.-Peyre 2013)
If the Non Degenerate Source Condition holds, and M is large enough,the support at low noise contained in
⋃i∈I(i ∪ i+ εi) where εi = ±1 .
In general, the support at low noise is made at mostof pairs of consecutive spikes: the original one andone of its immediate neighbours.
t0 1
12M
32M
2M−12M
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 14 / 22
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Outline
1 The deconvolution problem
2 Discrete regularization
3 Continuous regularization
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Continuous setting (total variation)
As proposed in [de Castro & Gamboa (12), Candes & Fernandez-Granda (13), Bredies &
Pikkarainen (13), Tang et al. (12)], consider the
Total variation of measures:
|m|(T) = sup
{∫Tψdm;ψ ∈ C(T), ‖ψ‖∞ 6 1
}m0 =
∑Ni=1 a0,iδx0,i
t0 1
x0,1
x0,2
x0,3
Example : If m =∑Ni=0 aiδxi , then |m|(T) =
∑Ni=0 |ai|.
If m = fdL, then |m|(T) =∫T |f(t)|dt.
Rationale: the extreme points of {m ∈M(T), |m|(T) 6 1} are the Diracmasses: δx for x ∈ T.
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Total variation regularization
Total variation of measures:|m|(T) = sup
{∫T ψdm;ψ ∈ C(T), ‖ψ‖∞ 6 1
}Basis Pursuit for measures [de Castro & Gamboa (12), Candes &
Fernandez-Granda (13)],
infm∈M(T)
|m|(T) such that Φm = y0 (P0(y0))
LASSO for measures [Bredies & Pikkarainen (13), Azais et al. (13)]
infm∈M(T)
λ|m|(T) +1
2||Φm− (y0 + w)||22 (Pλ(y0 + w))
Numerical methods for solving P0(y0) and Pλ(y0 + w) are proposed in [Bredies &
Pikkarainen (13), Candes & Fernandez-Granda (13)]
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Identifiability for discrete measuresMinimum separation distance of a measure m:
∆(m) = minx,x′∈Suppm,x 6=x′
|x− x′|
Ideal Low Pass filter: ϕ(t) = sin(2fc+1)πt)sinπt
i.e ϕn = 1 for |n| 6 fc, 0 otherwise.
m =∑N
i=1 aiδxi
t0 1
x1
x2
x3
∆(m)
Theorem ((Candes and Fernandez-Granda, 2013))
Let ϕ be the ideal low-pass filter. There exists a constant C > 0 suchthat, for any (discrete) measure m0 with ∆(m0) > C
fc, m0 is the unique
solution of
infm∈M(T)
|m|(T) such that Φm = y0 (P0(y0))
where y0 = Φm0.
Remark: 1 6 C 6 1.87.
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Robustness?
Weak-* robustness ([Bredies & Pikkarainen (13)])
If m0 =∑i a0,iδx0,i is the unique solution to P0(y0), mλ is a solution to
Pλ(y0 + w), then mλ∗⇀m0 as λ→ 0+, ‖w‖22/λ→ 0.
(see also [Azais et al. (13), Fernandez-Granda (13)] for robustness of local averages in the case of the ideal LPF)
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Robustness?
Weak-* robustness ([Bredies & Pikkarainen (13)])
If m0 =∑i a0,iδx0,i is the unique solution to P0(y0), mλ is a solution to
Pλ(y0 + w), then mλ∗⇀m0 as λ→ 0+, ‖w‖22/λ→ 0.
(see also [Azais et al. (13), Fernandez-Granda (13)] for robustness of local averages in the case of the ideal LPF)
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Optimality conditions
From convex analysis:
m0 =∑Ni=1 a0,iδx0,i is a solution to P0(y0) if and only if there exists p ∈ L2(T)
such that the function η = Φ∗p satisfies:
∀1 6 i 6 N, η (x0,i) = sign(a0,i) and ∀t ∈ T, |η (t)| 6 1;
mλ =∑N′
i=1 aλ,iδxλ,i is a solution to Pλ(y0 + w) if and only if the functionηλ = 1
λΦ∗(y0 + w − Φ∗aλ) satisfies:
∀1 6 i 6 N ′, ηλ (xλ,i) = sign(aλ,i) and ∀t ∈ T, |ηλ (t)| 6 1;
−1
0
1
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Robustness of the support (continuous problem)
For m0 =∑Ni=1 ai0δxi,0 , define
ηV = Φ∗pV , where
pV = argmin{‖p‖L2(T); (Φ∗p)(x0,i) = sign(ai,0),
and (Φ∗p)′(x0,i) = 0}−1
0
1
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Robustness of the support (continuous problem)
For m0 =∑Ni=1 ai0δxi,0 , define
ηV = Φ∗pV , where
pV = argmin{‖p‖L2(T); (Φ∗p)(x0,i) = sign(ai,0),
and (Φ∗p)′(x0,i) = 0}
Theorem ((D.-Peyre 2013))
Assume that Γx0= (ϕ(· − x0,1), . . . ϕ(· − x0,N , ϕ′(· − x0,1), . . . ϕ′(· − x0,1)) has
full rank, and that m0 satisfies the Non Degenerate Source Condition, i.e.ηV = Φ∗pV satisfies
|ηV (t)| < 1 for all t ∈ T \ {x0,1, . . . , x0,N},η′′V (x0,i) 6= 0 for all 1 6 i 6 N .
Then there exists, α > 0, λ0 > 0 such that for 0 6 λ 6 λ0 and ‖w‖2 6 αλ,
the solution mλ to Pλ(y + w) is unique and has exactly N spikes,
mλ =∑Ni=1 aλ,iδxλ,i ,
the mapping (λ,w) 7→ (aλ, xλ) is C1.
Vincent Duval, Gabriel Peyre Discrete vs continuous August 29, 2014 21 / 22
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Conclusion
Low noise stability analysis for the discrete and continuous frameworks
Imposing a grid leads to the apparition of parasitic spikes
Experiment yourself the Sparse Spikes Deconvolution on Numericaltours!
www.numerical-tours.com
Preprint:Exact Support Recovery for Sparse Spikes Deconvolution (2013),
V. Duval & G. Peyre,submitted to FoCM
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