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Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality

Angela Yingjun Zhang

Joint work with Xiaojun Yuan and Congmin Fan

Department of Information Engineering The Chinese University of Hong Kong

May 2017

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 1 / 45

Ultra Dense Wireless Networks

Limited bandwidth resource

◮ Radio resource management ◮ Interference mitigation ◮ Innovative frequency reuse ◮ Multi-RAT

Ultra-dense cells and devices

◮ Mobility management ◮ Small cell discovery ◮ User association

Limited backhaul/fronthaul ca- pacity Energy efficiency and green net- works

◮ BS idling and selection ◮ Massive MIMO and CoMP ◮ Energy-efficient wireless and

wired backhaul

Macrocell Cloud-RAN Fog-RAN Macrocell Picocell Femtocell Backhaul Fronthaul Processing Unit Storage Unit RRH

Cloud Data Center Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 2 / 45

Ultra Dense Wireless Networks: Goals and Challenges

Network-wide optimization Horizontal and vertical coordina- tion

Macrocell Cloud-RAN Fog-RAN Macrocell Picocell Femtocell Backhaul Fronthaul Processing Unit Storage Unit RRH

Coordination Localization Distribution Cloud Data Center

• Y. J. Zhang, L. Qian, and J. Huang, “Monotonic Optimization in Communication

and Networking Systems,” Foundations and Trends in Networking, vol. 7, no. 1,

• Oct. 2013.

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 3 / 45

Ultra Dense Wireless Networks: Goals and Challenges

Network-wide optimization Horizontal and vertical coordina- tion Low complexity and cost Distribution and parallelization Local vs. global information scalable system capacity scalable system complexity

Macrocell Cloud-RAN Fog-RAN Macrocell Picocell Femtocell Backhaul Fronthaul Processing Unit Storage Unit RRH

Coordination Localization Distribution Cloud Data Center

Scalability is what makes a large system worth investing in!

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 4 / 45

Dense Wireless Networks Modelled as Graphs

Observed node (RRH) Unobserved node (User)

U4 R7 U6 R6 R4 U3 R1 U1 R3 R2 R5 U5 U2

Graphs can model

◮ Coverage ◮ Interference and conflicts ◮ Physical and logical topology ◮ ...

Learning on graphs for

◮ Statistical inference ◮ Estimation and detection ◮ Resource allocation ◮ Optimization ◮ ... Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 5 / 45

Factor Graph

A factor graph is a bipartite graph that expresses the structure of factorization p(x) = Πsf(xs) where xs’s are subsets of x. Example: p(x) = fA(x1)fB(x2)fC(x1, x2, x3)fD(x3, x4)fE(x3, x5) Marginalization: ¯ pk(xk) =

x\xk p(x).

Maximization: ˆ pk(xk) = maxx\xk p(x).

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 6 / 45

Message Passing, Belief Propagation

Messages are exchanged between variable nodes and factor nodes ωk→a(xk)

fa xk

µa→k(xk)

fa xk

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 7 / 45

Message Passing, Belief Propagation

Messages are exchanged between variable nodes and factor nodes ωk→a(xk)

fa xk

µa→k(xk)

fa xk

Sum-Product rule ωk→a(xk) =

• c∈N(k)\a

µc→k(xk) ωk→a(xk) µa→k(xk) =

• xa\xk

fa(xa)

• j∈N(a)\k

ωj→a(xj) µa→k(xk)

fa xk fa xk

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 7 / 45

Message Passing

Compute the marginal of xk ¯ pk(xk) =

• a∈N(k)

µa→k(xk)

xk

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 8 / 45

Outline

Full-scale collaborative signal detection in C-RANs

◮ Randomized Gaussian message passing for scalable signal detection

Blind signal detection in sparse massive MIMO channels

◮ Achievable degree of freedom (DoF) ◮ Belief propagation algorithm for blind detection Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 9 / 45

Collaborative Signal Detection

Received signal y = Hx + n MMSE detection V = (HHH + N0I)−1H ˜ x = VHy = VHHx + VHn Challenges

◮ High computational complexity (e.g., O(N 3) for MMSE detection).

Network Size Complexity/Cost

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 10 / 45

Channel Sparsity = Scalability ?

Antenna Selection [Mehanna’2013], [Hoy- dis’2013], [Wang’2015], [Liu’2014] Clustering [Papadogiannis’2008], [Zhang’ 2009], [Lee’2014] By converting HHH + N0I into a doubly bordered block diagonal matrix, the optimal computationally time is O(N 2) by parallel computing.

• C. Fan, Y. J. Zhang, and X. Yuan, “Dynamic nested clustering for parallel PHY-layer pro-

cessing in cloud-RANs,” IEEE Transactions on Wireless Commununications, vol. 15, no. 3, pp. 1881-1894, Mar. 2016.

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 11 / 45

Bipartite Graph Representation

d0 RRH User d0 Distance threshold

Observed node (RRH) Unobserved node (User)

U4 R7 U6 R6 R4 U3 R1 U1 R3 R2 R5 U5 U2

The graph is random due to random locations of users. The graph is sparse but locally dense.

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 12 / 45

Inference Over a Bipartite Graph

MMSE is equivalent to MAP in a Gaussian channel with Gaussian signals

• x = arg max

x

p(x|y, H) = arg max

x

p(y1|xI1) · · · p(yn|xIn) · · · p(yN|xIN ) × p(x1) · · · p(xk) · · · p(xK),

x1 x2 x3 x4 x5 x6

1

( ) p x Factor node Variable node Factor node

1

1

( | )

I

p y x

6

( ) p x

7

7

( | )

I

p y x

. . .

4

4

( | )

I

p y x

. . .

2

( ) p x

3

( ) p x

4

( ) p x

5

( ) p x

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 13 / 45

Gaussian Message Passing

Assumption: n ∼ CN(0, N0I), x ∼ CN(0, I) Iteration: For all n, k such that Hn,k = 0, compute

v(t)

yn→xk =

1 P|Hn,k|2  N0 + P

• j=k

|Hn,j|2v(t−1)

xj→yn

  m(t)

yn→xk =

1 P

1 2 Hn,k

 yn − P

1 2

j=k

Hn,jm(t−1)

xj→yn

  v(t)

xk→yn =

 

• Hj,k=0,j=n

1 v(t)

yj→xk

+ 1  

−1

m(t)

xk→yn = v(t) xk→yn

 

• Hj,k=0,j=n

m(t)

yj→xk

v(t)

yj→xk

 

m(t+1) = Ωm(t) + z, where Ω is a function of the limit point of variances.

• Y. Weiss and W. T. Freeman, ”Correctness of belief propagation in Gaussian graphical models

for arbitrary topology,” Neural Computation, vol. 13, no. 10, pp. 2173-2200, 2001.

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 14 / 45

Message Passing for C-RAN: Scalability

Complexity per iteration: linear with the network size ✓ Convergence ? Convergence speed ?

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 15 / 45

Convergence

Convergence is guaranteed

• nly

when the factor graph is a tree Likely to converge when the factor graph is

◮ locally sparse ◮ with i.i.d. edge weights

Gaussian Message Passing converges when

◮ A = HHH + N0I is strictly diag-

• nally dominant

◮ ρ(Ω) < 1

The factor graph of C-RAN is loopy The factor graph of C-RAN is

◮ locally dense ◮ globally

sparse with distance- dependent edge weights

In C-RAN

◮ A is not always diagonally domi-

nant

◮ ρ(Ω) > 1 sometimes

100 200 300 400 500 600 700 800 Number of RRHs 0.97 0.975 0.98 0.985 0.99 0.995 1 Empirical probability of convergence

m(t+1) = Ωm(t) + z

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 16 / 45

Asynchronous Gaussian Message Passing

Synchronous vs. Asynchronous ⇒ Jacobi vs. Gauss-Seidel Example: update order {1, 3, 2, 4}

1

( ) p x

Factor node Variable node Factor node

1

1

( | )

I

p y x

2

2

( | )

I

p y x

4

4

( | )

I

p y x

2

( ) p x

3

( ) p x

4

( ) p x

3

3

( | )

I

p y x

x2 x4 x3 x1

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 17 / 45

Asynchronous Gaussian Message Passing

Example: update order {1, 3, 2, 4}

1

( ) p x

Factor node Variable node Factor node

1

1

( | )

I

p y x

2

2

( | )

I

p y x

4

4

( | )

I

p y x

2

( ) p x

3

( ) p x

4

( ) p x

3

3

( | )

I

p y x

x2 x4 x3 x1

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 18 / 45

Asynchronous Gaussian Message Passing

Example: update order {1, 3, 2, 4}

1

( ) p x

Factor node Variable node Factor node

1

1

( | )

I

p y x

2

2

( | )

I

p y x

4

4

( | )

I

p y x

2

( ) p x

3

( ) p x

4

( ) p x

3

3

( | )

I

p y x

x2 x4 x3 x1

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 19 / 45

Asynchronous Gaussian Message Passing

Example: update order {1, 3, 2, 4}

1

( ) p x

Factor node Variable node Factor node

1

1

( | )

I

p y x

2

2

( | )

I

p y x

4

4

( | )

I

p y x

2

( ) p x

3

( ) p x

4

( ) p x

3

3

( | )

I

p y x

x2 x4 x3 x1

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 20 / 45

Asynchronous Gaussian Message Passing

Example: update order {1, 3, 2, 4}

1

( ) p x

Factor node Variable node Factor node

1

1

( | )

I

p y x

2

2

( | )

I

p y x

4

4

( | )

I

p y x

2

( ) p x

3

( ) p x

4

( ) p x

3

3

( | )

I

p y x

x2 x4 x3 x1

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 21 / 45

Asynchronous Gaussian Message Passing

Messages are updated in a sequential order Example: update order {1, 3, 2, 4}

10 20 30 40 50 60

Number of iterations

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Relative Error

Converges!

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 22 / 45

Asynchronous Gaussian Message Passing

Convergence heavily depends on the schedule Lack of systematic way to find a convergent schedule Stricter convergence condition required for general scheduling

10 20 30 40 50 60 0.5 1 1.5 2 2.5 3 3.5

Number of iterations Relative Error

1

( ) p x

Factor node Variable node Factor node

1

1

( | )

I

p y x

2

2

( | )

I

p y x

4

4

( | )

I

p y x

2

( ) p x

3

( ) p x

4

( ) p x

3

3

( | )

I

p y x

x2 x4 x3 x1

Update schedule {1, 2, 3, 4}. Diverges!

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 23 / 45

Randomized Gaussian Message Passing (RGMP)

Define Σ as Σ {σ|σ is a permutation of {1, · · · , K}}. At each iteration, draw a permutation σ uniformly from Σ. Update the messages at the variable node in the order of σ.

10 20 30 40 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Number of iterations Relative Error

• C. Fan, X. Yuan, Y. J. Zhang ,“Scalable Uplink Signal Detection in C-RANs via Randomized

Gaussian Message Passing,” to appear, IEEE Transactions on Wireless Communications.

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 24 / 45

RGMP Converges

100 200 300 400 500 600 700

Number of RRHs

0.975 0.98 0.985 0.99 0.995 1

Empirical probability of convergence RGMP Synchronous GMP

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 25 / 45

Convergence Analysis

Evolution of the mean:

◮ Synchronous GMP: m(t+1) = Ωm(t) + z. ◮ Randomized GMP:

m(t+1)

σt(i) =

• j<i

Ωσt(i),σt(j)m(t+1)

σt(j) +

• j>i

Ωσt(i),σt(j)m(t)

σt(j) + zσt(i),

◮ or

m(t+1) = L−1

σt Rσtm(t) + L−1 σt z,

where Rσt(σt(i), σt(j)) =

• Ωσt(i),σt(j),

i < j 0,

• therwise. ,

Lσt = −Ω + I + Rσt

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 26 / 45

Convergence Analysis

Synchronous GMP: ρ(Ω) < 1. Randomized GMP:

Theorem 1

The sequence of means in RGMP converges in expectation to the true conditional mean iff the spectral radius ρ(Λ) < 1, where Λ Eσ[L−1

σ Rσ] = 1

K!

• σ∈Σ

(L−1

σ Rσ).

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 27 / 45

RGMP vs Synchronous Message Passing

ρ(Λ) < 1 is less stringent than ρ(Ω) < 1 in C-RANs.

0.2 0.4 0.6 0.8 1 1.2 1.4

Spectral radius

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Empirical CDF ;($), N=12, K=9, r=691m ;(+), N=12, K=9, r=691m ;($), N=8, K=6, r=564m ;(+), N=8, K=6, r=564m

0.9 1 1.1 0.99 0.995 1 Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 28 / 45

RGMP Enables Fully Distributed Operation

Traditional message passing algorithms require centralized coordination and scheduling The RGMP algorithm converges when each node randomly and unco-

• rdinatedly chooses when to update

its message when some nodes accidentally choose to update messages at the same time when message passing incurs delay

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Spectral radius Empirical CDF () with M=2 () with M=10 () (0.9+0.1I) (0.5+0.5I) 0.9 1 1.1 0.96 0.98 1

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 29 / 45

Convergence Rate

50 100 150 200 250 300 350 400 450 500

Number of iterations

10-8 10-6 10-4 10-2 100

Relative Error RGMP GAMP PCG ADMM

user density 8 per square kilometre, RRH density 10 per square kilometre, network size 4 square kilometers

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 30 / 45

Convergence Rate

10 20 30 40 50 60 70 20 40 60 80 100 120 140 160 180 200 Network area (km2) Number of iterations RGMP B−RGMP with M=10 GMP with damping,=0.5 GAMP CG

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 31 / 45

RGMP Achieves Scalability

Complexity per iteration: linear in the number of edges/the network size. ✓ Convergence ✓ Convergence speed: constant regardless of network size ✓

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 32 / 45

Outline

Full-scale collaborative signal detection in C-RANs

◮ Randomized Gaussian message passing for scalable signal detection

Blind signal detection in sparse massive MIMO channels

◮ Achievable degree of freedom (DoF) ◮ Belief propagation algorithm for blind detection Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 33 / 45

Massive MIMO

Transmit terminals: K single-antenna users Receive terminal: BS with N antennas N ≫ K ≫ 1 Ideally, C ∝ K log(SNR) In practice, C ∝ K

• 1 − K

T

• log(SNR)

◮ coherent detection:

due to overhead of training-based channel estimation

◮ blind detection: due to reduction of detec-

tion accuracy

• X. Yuan, C. Fan. and Y. J. Zhang, “Fundamental limits of training-based multiuser MIMO

systems,” arXiv: 1511.08977

• L. Zheng and D. N. C. Tse, “Communication on the Grassmann manifold: A geometric

approach to the noncoherent multiple-antenna channel,” IEEE Trans. Information Theory, Feb. 2002.

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 34 / 45

Sparsity of Massive MIMO Channel

Physical channel is sparse in the angular do- main at the receive antenna array Sparsity level: ρ = |S| NK ≪ 1

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 35 / 45

Existing Work: Training by Compressed Sensing

Use compressed sensing to reduce the number of pilots needed

=

✕ Received signal Yp Channel H Pilot Xp

Degree of freedom: K

• 1 − cK

T

• , where 0 < c < 1.
• W. U. Bajwa, J. Haupt, A. M. Sayeed, and R. Nowak, “Compressed channel sensing: A new

approach to estimating sparse multipath channels,” IEEE Proceedings, Feb. 2013.

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 36 / 45

Blind Signal Detection with Channel Sparsity

=

✕ Received signal Y Channel H Transmit signal X

Y H is sparse Factorizing Y to obtain

Y is bilinear in H and X. What is the performance limit of the blind signal detection? How to design an efficient algorithm to obtain the estimates?

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 37 / 45

Performance Limit: Degree of Freedom

Theorem 2

Suppose that ρ ≤

q K

• T

log K and K ≤ T 2.

Then, for any η > 0 and N ≥ TK3 + ηK2, the DoF of the massive MIMO system is lower bounded by DoFblind ≥

• 1 − e−η

K

• 1 − 1

T

• .

e−η can be viewed as the detection failure caused by sparsity pattern recogni- tion. The DoF lower bound can approach K

• 1 − 1

T

• when N is large engough.

The fractional DoF loss of 1

T compared with the ideal case is due to ambiguity.

The loss is independent of K.

• J. Zhang, X. Yuan, and Y. J. Zhang, “Blind Signal Detection in Massive MIMO: Exploiting

the Channel Sparsity,” arXiv: 1704.00949, 2017

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 38 / 45

Scalable Capacity

Traditional approaches Approaches exploiting channel sparsity Training Based Blind Detection Training Based Blind Detection K

• 1 − K

T

• K
• 1 − K

T

• K
• 1 − cK

T

• K
• 1 − 1

T

• Angela Yingjun Zhang

(IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 39 / 45

Detection Algorithm: Message Passing

Maximum a Posteriori estimation: Similar to sparse dictionary learning: ( ˆ H, ˆ X) = arg min

H,X ||Y − HX||2 2 + λ||H||1.

Existing algorithms:

◮ K-SVD [AhaElaBru’06], SPAMS [MairalBacPonSap’10], ER-SpUD [SpiWan-

Wri’12], BiG-AMP [ParSchCev,14]

◮ Dictionary must be “overcomplete”, i.e., T ≤ K. ◮ In massive MIMO systems, T > K is more relevant. Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 40 / 45

Projected BiG-AMP

pX(X) pH(H) xk,t hn,k yn,t cn,t cn,k,t zn,k,t n k t

Step 1: Project Y onto the row space of X Step 2: Perform message passing on the factor graph to obtain the esti- mates of H and X. Step 3: Eliminate the phase ambigu- ity by one-symbol pilot Step 4: Eliminate the permutation ambiguity by insert a transmitter la- bel in each codeword

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 41 / 45

Throughput Comparison

N = 500, K = 50, T = 100, and ρ = 0.3

5 10 15 20 25 30 35 40 45 50 100 200 300 400 500 600 700 800 900 1000

100 200 300 400 500 600 700 800 900 1000 5 10 15 20 25 30 35 40 45 50

(dB) Achievable rate (bit/channel use) Perfect channel knowledge Blind detection without exploiting channel sparsity Training-based MIMO coherence detection Compressed sensing based MIMO coherent detection Blind detection by exploiting channel sparsity

The proposed scheme achieves a DoF close to the ideal case The DoF loss is not affected by the number of users K

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 42 / 45

Throughput Comparison

N = 500, K = 50, T = 100, and ρ = 0.3 The proposed scheme significantly

• utperforms

the

• ther

dictionary learning algorithms

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 43 / 45

Conclusions

Machine learning based algorithms for signal processing in dense wireless net- works Randomized GMP for collaborative signal detection in C-RAN

◮ Fully cooperative signal detection with scalable overhead and complexity ◮ Ensures convergence by effectively breaking short loops through randomisation ◮ Eliminates the need of central coordination, synchronization, and scheduling ◮ Enables fully distributed operation with constant complexity per node

Blind detection for massive MIMO channels

◮ Exploits the angular domain channel sparsity ◮ Achieves a DoF close to the case with perfect CSI ◮ Has less stringent requirement on the channel coherence time T and the number

• f users K

Learning for

◮ resource allocation ◮ end-to-end system optimization ◮ data driven design

www.ie.cuhk.edu.hk/∼ yjzhang

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 44 / 45

Thank You

Angela Yingjun Zhang (IE, CUHK) Message Passing in Ultra Dense Wireless Networks: Scalability and Optimality May 2017 45 / 45