Blind Deconvolution and Compressed Sensing
Dominik Stöger
Technische Universität München,
Department of Mathematics,
Applied Numerical Analysis
[email protected]
Joint work with Peter Jung (TU Berlin), Felix Krahmer (TU München)
supported by DFG priority program “Compressed Sensing in Information Processing” (CoSIP)
Overview
Introduction and Problem Formulation
Recovery guarantees
Proof sketch
Discussion
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Introduction and Problem Formulation
Overview
Introduction and Problem Formulation
Recovery guarantees
Proof sketch
Discussion
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Blind Deconvolution and Compressed Sensing
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Introduction and Problem Formulation
Blind Deconvolution
=
z
*
(y,x)
bilinear inverse problem: z = B(x, y )
ambiguities, constraining x and/or y
Many applications:
imaging (blind deblurring)
radar, e.g., ground penetrating radar (GPR), radar imaging
speech recognition
wireless communication
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Introduction and Problem Formulation
Blind Deconvolution and Demixing
A problem in Wireless Communication:
r different devices
device i delivers message mi
Linear encoding:
xi = Ci mi with Ci ∈ RL×N
Channel model:
wi = Bi hi , where Bi ∈ RL×K
Received signal:
y=
r
X
i=1
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wi ∗ xi ∈ RL
Goal: recover all mi from y
Blind Deconvolution and Compressed Sensing
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Introduction and Problem Formulation
Assumptions on Bi and Ci
y=
r
X
i=1
wi ∗ xi =
r
X
Bi hi ∗ Ci mi
i=1
Assume wi is concentrated on the first few entries, i.e., Bi extends
hi by zeros
(Our analysis will include more general Bi )
Choice of Ci is arbitrary ⇒ randomize
Choose Ci to have i.i.d. standard normal entries
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Introduction and Problem Formulation
Lifting
There are unique linear maps Ai : RK ×N → RL such that for
arbitrary hi and mi
wi ∗ xi = Bi hi ∗ Ci mi = Ai (hi mi∗ ) = Ai (Yi )
y=
r
X
Ai (hi mi∗ ) = A (X0 ) ,
i=1
where
X0 = (h1 m1∗ , · · · , hr mr∗ )
Low rank matrix recovery problem
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Recovery guarantees
Overview
Introduction and Problem Formulation
Recovery guarantees
Proof sketch
Discussion
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Recovery guarantees
A convex approach for recovery
r = 1 investigated in [Ahmed, Recht, Romberg, 2012]
r ≥ 1 semidefinite program (SDP) [Ling, Strohmer, 2015]
minimize
r
X
kYi k∗
subject to
i=1
r
X
Ai (Yi ) = y .
i=1
k · k∗ : nuclear norm, i.e., the sum of the singular values
Recovery is guaranteed with high probability, if
L ≥ C r 2 K + µ2h N log3 L log r
coherence parameter: 1 ≤ µh = maxi
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Blind Deconvolution and Compressed Sensing
(SDP)
kĥi k∞
khi k2
≤
√
K
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Recovery guarantees
Linear scaling in r
Number of degrees of freedom: r (K + N)
Recovery guarantee of Ling and Strohmer (up to log-factors)
L ≥ C r 2 K + µ2h N log · · ·
Optimal in K and R, suboptimal in r
Conjecture by Strohmer: Number of required measurements scales
linear in r
This is supported by numerical experiments
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Recovery guarantees
Main result
Theorem (Jung, Krahmer, S., 2016)
Let α ≥ 1. Assume that
L ≥ Cα r K log2 K + Nµ2h log2 L log (γ0 r ) ,
where
(1)
s NL
+ α log L
γ0 = N log
2
and Cα is a universal constant only depending on α. Then with
probability 1 − O (L−α ) the recovery program is successful, i.e. there
exists X0 is the unique minimizer of (SDP).
(Near) optimal dependence on K , N , and r
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Proof sketch
Overview
Introduction and Problem Formulation
Recovery guarantees
Proof sketch
Discussion
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Proof sketch
Proof overview
Two main steps in the proof:
Establishing sufficient conditions for recovery
⇒ approximate dual certificate
Constructing the dual certificate via Golfing Scheme
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Proof sketch
Proof overview
Two main steps in the proof:
Establishing sufficient conditions for recovery
⇒ approximate dual certificate
Constructing the dual certificate via Golfing Scheme
Crucial new ingredient for both steps:
Restricted isometry property on 2r -dimensional space
n
T = (u1 m1∗ + h1 v1∗ , · · · , ur mr∗ + hr vr∗ ) :
o
u1 , · · · , ur ∈ RK , v1 , · · · , vr ∈ RN
Intuition for T :
Directions of change when slightly varying the mi ’s and hi ’s
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Proof sketch
Restricted isometry property 1
Definition
We say that A fulfills the restricted isometry property on T for some
δ > 0, if for all X = (X1 , · · · , Xr ) ∈ T
r r
r 2
X
X
X
2
2
Ai (Xi ) ≤ (1 + δ)
(1 − δ)
Xi .
Xi ≤ i=1
F
i=1
`2
i=1
F
[Ling, Strohmer, 2015]: Each operator Ai acts almost isometrically on
o
n
Ti = umi∗ + hi v ∗ : u ∈ RK , v ∈ RN .
and Ai , Aj are incoherent
⇒ r 2 -bottleneck
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Proof sketch
Restricted isometry property 2
Observe:
Restricted isometry property for some δ > 0 is equivalent to
r
r
X
X
Ai (Xi ) k2`2 −
kXi k2F δ ≥ sup k
X ∈T
i=1
i=1
r
r
X
h X
i
= sup k
Ai (Xi ) k2`2 − E k
Ai (Xi ) k2`2 .
X ∈T
i=1
i=1
Suprema of chaos processes: The last term can be bounded
using results from [Krahmer, Mendelson, Rauhut, 2014].
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Discussion
Overview
Introduction and Problem Formulation
Recovery guarantees
Proof sketch
Discussion
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Discussion
Open questions
Generalization to more general random matrices
Faster algorithms
What if only a few number of devices are active? Does one obtain
better recovery guarantees?
Generalization to sparsity assumption on h
(instead of a subspace assumption)
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