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]

1

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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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)