Discreteness-Aware AMP for Reconstruction of Symmetrically Distributed Discrete...
Transcript of Discreteness-Aware AMP for Reconstruction of Symmetrically Distributed Discrete...
ˆ
b = arg min
x2RN
(q0kxk1 +
LX
`=1
q`(kx� r`1k1 + kx+ r`1k1))
subject to y = Ax
0 10 20 30 40 50
# of iterations
10−4
10−3
10−2
10−1
100
σ2
empirical (N = 100)empirical (N = 500)empirical (N = 1000)empirical (N = 2000)empirical (N = 3000)
�2t�2
t+1�2t+2
reconstruct a discrete-valued vector from its linear measurements y = Ab 2 RM
4. Simulation Results
2. Proposed DAMP Algorithm
3. State Evolution for DAMP1. Introduction
Abstract
Discreteness-Aware AMP for Reconstruction of Symmetrically Distributed Discrete VariablesRyo Hayakawa (Graduate School of Informatics, Kyoto University) Kazunori Hayashi (Graduate School of Engineering, Osaka City University)
[1] D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” in Proc. Nat. Acad. Sci., vol. 106, no. 45, pp. 18914-18919, Nov. 2009. [2] M. Nagahara, “Discrete signal reconstruction by sum of absolute values,” IEEE Signal Process. Lett., vol. 22, no. 10, pp.1575-1579, Oct. 2015.
is composed of i.i.d. variables with zero mean and variance
Success
: concave
Failure
For , calculate
: mean : Mean-Square-Error (MSE)*
estimate of parameter
SOAV (Sum-of-Absolute-Value) optimization [2]
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Proposed algorithm (DAMP: Discreteness-aware AMP)
apply the idea of AMP algorithm [1]
Initialization:x
�1 = x
0 = 0, z
�1 = 0
2
Potential applications
✦ multiuser detection in M2M communication(Machine-to-Machine)
✦ MIMO signal detection(Multiple-Input Multiple-Output)
✦ FTN signaling(Faster-than-Nyquist)
*In practice, we use the estimate of �2t
State evolution [1] predict the behavior of the MSEin the large system limit
∆0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p 0
0
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0.9
1
DAMP (bj ∈ {0,±1})AMP
Successd
d(�2)
�����#0
< 1
Failured
d(�2)
�����#0
> 1
required observation ratiofor DAMP
∆0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
recovery
rate
0
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1p0 = 0p0 = 0.4
SuccessFailure
x
t+1 = ⌘(ATz
t + x
t,��t),
z
t = y �Ax
t
+1
�z
t�1h⌘0(ATz
t�1 + x
t�1,��t�1)i
We propose a message passing-based algorithm to reconstruct a discrete-valued vector whose elements have a symmetric probability distribution. The proposed algorithm, referred to as discreteness-aware approximate message passing (DAMP), borrows the idea of the AMP algorithm for compressed sensing. We analytically evaluate the performance of DAMP via state evolution framework to derive a required number of linear measurements for the exact reconstruction with DAMP.
Discrete-valued vector reconstruction
b
soft thresholding function
⌘(u,��)b 2 {0,±1}Nfor
Successd
d(�2)
�����#0
< 1 )
Phase transition of DAMP
b 2 {0,±1}NPr(bj = 0) = p0
Pr(bj = +1) = (1� p0)/2
Pr(bj = �1) = (1� p0)/2
: observation ratio
Z ⇠ N (0, 1)X ⇠ distribution of bj
b 2 {0,±1}Np0 = 0.2� = 0.7
N = 1000b 2 {0,±1}N
rapidly increase aroundthe theoretical boundary
Purpose of this work
A by NM
reconstructb̂
❖ propose a low-complexity algorithm for the discrete-valued vector reconstruction
❖ theoretically analyze the performance of the proposed algorithm via state evolution [1]
b 2 {0,±r1, . . . ,±rL}NAssumption
(Pr(bj = r`) = Pr(bj = �r`))
parameter
use the fact that some elements of are b± r`1 0
(N � M)
approach the theoretical performance as increasesN
AcknowledgmentThis work was supported in part by the Grants-in-Aid for Scientific Research no. 15K06064 and 15H2252 from MEXT.
A
complexity: per iteration O(MN)