Deep Learning-Based AMP for Massive MIMO Detection

Yang Yang,Shaoping Chen,*,Xiqi Gao

1 Hubei Key Laboratory of Intelligent Wireless Communications,South-Central Minzu University,Wuhan 430074,China

2 National Mobile Communications Research Laboratory,Southeast University,Nanjing 210096,China

*The corresponding author,email:spchen@scuec.edu.cn

Abstract:Low-complexity detectors play an essential role in massive multiple-input multiple-output(MIMO)transmissions.In this work,we discuss the perspectives of utilizing approximate message passing(AMP)algorithm to the detection of massive MIMO transmission.To this end,we need to efficiently reduce the divergence occurrence in AMP iterations and bridge the performance gap that AMP has from the optimum detector while making use of its advantage of low computational load.Our solution is to build a neural network to learn and optimize AMP detection with four groups of specifically designed learnable coefficients such that divergence rate and detection mean squared error(MSE)can be significantly reduced.Moreover,the proposed deep learning-based AMP has a much faster converging rate,and thus a much lower computational complexity than conventional AMP,providing an alternative solution for the massive MIMO detection.Extensive simulation experiments are provided to validate the advantages of the proposed deep learning-based AMP.

Keywords:approximate message passing;convergence;machine learning

I.INTRODUCTION

Massive multiple-input multiple-output(MIMO)system with hundreds of antennas at the base-station(BS)that serves multiple mobile user-equipment(UE)at the same spectrum band had spurred much research interest as a key element of the 5th generation mobile communication system,due to its high spectral and power efficiency.However,the complexity of traditional MIMO detectors,e.g.,the maximum likelihood(ML)detector and the iterative detection and decoding scheme,whose BER performance approaches ML detector,is too high to be applied in the massive MIMO systems.Hence,it is worthwhile to develop simple yet effective detectors for massive MIMO systems[1–3].As an efficient iterative algorithm,approximate message passing(AMP)has gained significant attentions for its light computational load without much compromise in performance.

AMP was first proposed in[4]to solve the sparse linear inverse problem for compressed sensing.Afterwards,some significant progresses have been made to enable AMP converge faster,or converge into a more accurate result[5–7].Recently,an approach to improve AMP algorithm following the line of machine learning was presented in[8],whereby AMP-unfolded deep network was first proposed which can achieve a much better performance than the standard AMP.Although AMP along with its improved schemes has many advantages,there still exist some challenges,e.g.,it may suffer from divergence if not properly controlled[4].Moreover,to ease the derivation,an assumption of Gaussian distribution about the detected signals is made and the computational complexity is significantly reduced under this assumption.However,different from sparse linear inverse problem where AMP was first applied,the signals to be detected in massive MIMO detection are not sparse but of finite alphabets and the Gaussian assumption is not met in practice.Thus,the AMP and its learning-based version cannot be directly applied in massive MIMO detection[9–11].

Inspired by the work in[8],we consider employing machine learning to optimize AMP algorithm for massive MIMO detection.Different from conventional AMP where the parameters are obtained by theoretical analysis and approximation,our proposed method introduces specifically designed coefficients whose values are learned from a large number of data.Via machine learning,the divergence rate of AMP can be controlled to a very low level such that detection errors caused by it can be corrected by interleaving and channel coding in practice.What is more,via machine learning,a faster converging rate along with a lower detection bit error rate(BER)can be achieved.

The rest of the paper is organized as follows.In Section II,we briefly review the system model for massive MIMO detection.The AMP detector is introduced in Section III.Our proposed method is then given in Section IV.Simulation results are given in Section V.The conclusions are drawn in Section VI.

II.SYSTEM MODEL

We consider a massive MIMO communication,consisting of one BS equipped withMantennas servingNUE’s each equipped with one antenna.In this work,we focus on the uplink detection problem,where the frequency domain signals from UEs are denoted by the vector∈CN×1which are drawn from the constellation.The received signal vector∈CM×1is given by

where y and x are real vectors consisting of the real and imaginary part of~y and~x respectively.H consists of the real and imaginary part of~H as follows,

In the massive MIMO detection,the block fading channel is usually assumed where channel matrix H stays constant during a coherence block and changes from block to block according to some ergodic property.We estimate channel H and recover the information transmitted from the UE’s,which is referred to the uplink detection given as

where‖·‖2returns the 2-norm of a vectors.Ω is the real equivalent constellation of the complex.For more detail of the system model see[2,12].

III.AMP DETECTION

The complexity of exhaustive search for Eq.(4)scales exponentially withNand it is impractical for a largeNin massive MIMO’s.Hence,how to design a detector with a low complexity has been an active research topic.AMP is an iterative algorithm with a low complexity,which is designed based on the expectation of the posterior probability distribution of the received signals as[9]

wheref(x|y)is the conditional distribution function of x given y.Rewritef(x|y)using Bayes formula as follows,

Suppose each element of x and y are independent from each other.Then,Eq.(6)can be changed into

Figure 1.Factor graph of posteriori distribution f(x|y).

A factor graph is employed to illustrate the posteriori distribution in Eq.(7)as given in Figure 1.As a matter of fact,a numerically efficient method can be employed to approximate each marginal pdff(xn|y)by a set of message passing operations that go from factor nodes to variable nodes(i.e.,rm→n)and from variable nodes to factor nodes(i.e.,xm→n),as illustrated in Figure 1.The message passing equations are constructed as follows,[9]

where a superscriptrtdenotes the number of iterations and the symbol～=denotes identity between probability distributions up to a normalization constant.

Suppose all random variables ofxtn→mfollow Gaussian distributions,satisfying

Whereμtn→m,νtn→mdenote its mean and variance respectively.Thus,rtm→ncan be worked out through Eq.(8)according to Gaussian approximation method as

where

wherehmnis the(m,n)-th entry of H.Base on Eq.(8)and Eq.(10),we come across a method to updatextn→mas

Henceforth,we have the method to update the mean and variance ofxtn→mas

Nevertheless,the computation load in equations Eq.(8)is still heavy as it requires to calculate 2MNmessages.With the goal of further simplifying the updating,AMP is reorganized via soft thresholding function as[4,9]

whereη(z;λ0)is the soft thresholding function defined by

The functionη′(;)is the derivative ofη(;).The operation〈b〉denotes the average of vector b,and[b]jdenotes thej-th component of the vector.Vector x is initialized as a zero vector,and r is initialized as a vector of one’s generally.For detailed derivation of the AMP,please see[4,9].

Based on Eq.(13),a multi-layer network is obtained via unfolding the message passing as shown in Figure 2,where the number of layers is determined by the number of AMP iterations.It well known thatη(z;λ0)plays a vital role to rebuild a sparse x[9].However,it does not mean thatη(z;λ0)can be absent where x is not sparse,as the message will diverge to infinity without the restriction fromη(z;λ0).

IV.DEEP LEARNING-BASED AMP DETECTOR

AMP is a low complexity yet effective algorithm and has a great application potential in many scenarios.However,AMP may suffer from divergence if not properly controlled.It has been extensively investigated in the academic society,and a theoretical conclusion of what causes the divergence is still absent.In addition,AMP is derived under the assumption that the signals are Gaussian,which is not met in massive MIMO transmissions.Our motivation in the paper is to modify conventional AMP algorithm and optimize it via machine learning such that it can be applied for massive MIMO detection.

Although many investigations are proposed to optimize AMP,most of them focus on how to adjust key coefficients in a heuristic manner and thus a performance gain is limited.In addition,repeated manual adjustment is troublesome and hard to achieve an optimized solution,e.g.,divergence may occur if not properly controlled.

Fortunately,deep learning provides a promising solution,which helps us to find the optimized coefficients for AMP.This is achieved via learning from a tremendous amount of massive MIMO data.Thus,a learnable AMP with an obviously better performance over traditional AMP is allowable,even though we cannot explain exactly how it works[13–15].

Our investigation shows that whether AMP converges or not greatly depends on the channel matrix H along with the received signals y.To mitigate the occurrence of divergence,we introduce two learnable coefficients(θ,ψ)for H and y,respectively.The modified version of Eq.(13)is given by

Although it is hard to work out the optimum solution of(θ,ψ),a suboptimal solution is available via machine learning.Extensive investigations via machine learning show that(θ,ψ)does have a dominant effect on the convergence of AMP.To explain this effect visually,some intermediate learning results are shown in Figure 4 and Figure 5,Section V,which show how the normalized mean square error(NMSE)of AMP will vary with the increase of layers(iterations).The results show that the proposed parameters do have an impact on the convergence performance of AMP,and there is a great potential of optimization learning.

The authors in[7,8]showed that the thresholdτtplays an important role in the convergence speed of AMP iterations and accordingly presented a multilayer network by introducing a coefficientαtfor the threshold that can be learned.By optimizing the threshold via machine learning,an obvious performance gain is allowed for AMP iterations.In addition,we introduce another coefficientβtfor more performance gain.A learnable message passing detector with more efficient learnable coefficients is obtained by transforming Eq.(15)into

Figure 3 illustrates the proposed methold via unfolding the equations in Eq.(16)into a multilayer network.The difference between our proposed learning-based AMP and that presented in[8]includes two folds.First,we explicitly employ threshold update operation that was first presented in[4].The threshold update(the third Eq.(16))with optimized coefficientsαtwill further speed up the convergence.Second,additional learnable coefficients are introduced to mitigate the occurrence of divergence in AMP iterations.

Figure 2.Multilayer network from unfolding the AMP iterations.

Figure 3.Multilayer network from unfolding the proposed message passing methods.

whereEandRare the convergence and divergence loss,respectively.The divergence lossRis evaluated by the divergence rate and the convergence lossEis evaluated by the MSE of the converged outputs.To minimize the sum ofEandRcan help to reduce the divergence loss and optimize the converged outputs at the same time.

There should be enough training data for a network to achieve a good performance.Generally,thousands or tens of thousands of training data are recommended with the goal of predicting the unknown x with a high accuracy from a newly received y[15].Suppose there areNdtraining data available denoted byThe detailed process to compute them is given in Algorithm 1,where 0 and 1 denote vectors with all 0’s and 1’s,respectively.

Different from traditional AMP where the parameters are obtained by theoretical analysis and approximation,the proposed method includes 4 learnable coefficients that can be optimized via learning.The massive MIMO channel varies in different scenarios,so the traditional method with constant parameters may suffer from performance degradation in a changing scenario.In contrast,the proposed learnable method is more flexible and adaptive to channel variations,and thus may adapt to massive MIMO detections.Different from the learnable AMP scheme presented in[8],where learnable coefficients for threshold is introduced to achieve a faster convergence speed and a lower detection error,our proposed learnable AMP employs threshold update and introduce more learnable coefficients such that a significant lower divergence rate is achieved than the learnable AMP in[8]while maintaining a faster convergence rate and a lower detection error.

Algorithm 1.Proposed quadratic loss for learning.Require:Ξt,H,{xd,yd}Nd d=1 Ensure:E,R 1:Et=0,Rt=0,Nc=0 2:for d=1:Nd do 3:xt=0,rt=1,τt=0.5 4:for t=1:T do 5: (xt,rt,τt)6: =AMP（yd,Ξt,H,xt-1,rt-1,τt-1）7:end for 8:if images/BZ_80_371_908_399_954.png(xt-xd)2images/BZ_80_599_908_627_954.png 〉then 9: Et=Et+(xt-xd)2images/BZ_80_819_994_847_1040.png 10: Nc=Nc+1 11:else 12: Rt=Rt+1 13:end if 14:end for 15:E=Et/Nc,R=Rt/Nd images/BZ_80_591_994_619_1040.png <〈x2d

There are various solutions to deal with the MIMO detection problem following different technical routes,and some already achieve very low BER recently[10,11].However,AMP is still worthwhile to be further investigated,although it does not have a beautiful BER curve in comparison with some other ideas.As a distributed algorithm,AMP can be conducted into a group of simple computing units,so it is easy to be realized via the programmable chips and does not require high cost powerful centralized processors.Hence,AMP has been a popular method for massive MIMO detection problem for its potential application in various mobile,small,and low energy consumption devices,which will become the main stream of the future networks like internet of things.

In addition,complexity is no doubt an important factor in low cost device of Industrial applications.Both MMSE detector and the OAMP-Net method given in[13]require to perform complex matrix inversion operations,whose computational complexity of orderO（N3）is much heavier than the message passing based AMP method of a complexity of orderO(TN).Although the proposed scheme includes more coefficients to be learned and thus has a heavier training complexity,it still has a similar on-line detection complexity of orderO(TN)as conventional AMP and the learnable AMP,since the training operation is performed off-line.In addition,considering the proposed AMP converges faster than existing AMPs(less iterationsTis needed for the same performance to be achieved),the proposed AMP has a lower detection complexity than them.

V.SIMULATIONS

This section provides the simulation results to validate the advantages of the proposed method.As ML detector is too complicated to be realized for massive MIMO simulations,we choose to compare it with those of MMSE detector[2],conventional AMP in[4],tied learned AMP given in[8],and the OAMP-Net in[13].The simulations are performed with a multiuser MIMO system as described in Section II with SCM channel model H withM=100 andN=100.The normalized average powerof the transmitted signals is set to be 1,and the received signalto-noise-ratio(SNR)isThe NMSE,defined in the following,is calculated by Monte Carlo simulations with 106frames of channel and signal realizations.

Figure 4 and Figure 5 show the NMSE performance comparison against the number of layers for two groups of channel and signal realizations.The trace of H employed in the simulations shown in Figure 5 is ten times greater than the trace of H in Figure 4.Thus,the power gain of the channels experimented in Figure 5 is much higher than Figure 4.The result also shows that no matter AMP and tied learned AMP converge or not,the NMSE performance of the proposed deep learning-based AMP can well converge.Moreover,it is obvious that the proposed method converges faster and has a lower NMSE as shown in Figure 4,even though AMP and tied learned AMP all converge after about 20 iterations.

Figure 4.NMSE against the number of layers,where SNR=13 dB.

Figure 5.NMSE against the number of layers,where SNR=13 dB.

Figure 6 shows the divergence rate of each method in different SNR scenarios,where both the proposed method and tired learned AMP are trained at SNR=13 dB.We choose the 13 dB scenario to ensure the impact of AWGN can be learned well,on the other hand,the signal is available for detection.The result shows that the divergence rates of tired learned AMP along with the conventional AMP are higher than 10-1,that is far away from the practical requirement.In contrast,the divergence rate of the proposed method decreases nearly 2 order of magnitude around 13 dB compared with other methods,and it continues to decrease when SNR is beyond 13 dB.Hence,the proposed learning method controls the divergence rate to a very low level,and it also is robust to SNR change.

Figure 7 shows the NMSE comparison versus SNR.The results show that the proposed deep learningbased AMP performs obviously better than conventional AMP.Although the performance of the proposed AMP is very close to MMSE detector around SNR=13 dB,the performance gap is enlarged when SNR is away from 13 dB.The reason is that the learned AMPs are learned at SNR=13 dB such that they achieve the best performance at this SNR value.We may customize different learnable AMPs that are trained at several SNR values to achieve a good performance at whole reasonable SNR values.

Figure 6.Divergence rate comparison versus SNR.

Figure 7.The NMSE of converging outputs versus SNR.

Figure 8.The BER of each method to detect 4-QAM signals.

Figure 9.The BER of each method to detect 16-QAM signals.

Figure 8 and Figure 9 show the BER comparison in the detection of 4-QAM and 16-QAM signals respectively.The results show that MMSE and OAMP-Net have a lower BER than others,but this gain is accomplished by complex computation of matrix inversion in both MMSE and OAMP-Net.AMP,tied learned AMP,and the proposed AMP have a similar complexity,but the proposed method has much lower BER,which is very close to MMSE and OAMP-Net.Thus,the proposed deep learning-based AMP achieves a good tradeoff between complexity and performance.

VI.CONCLUSION

We have presented a deep learning-based AMP for massive MIMO detection.We build a deep neural network to learn and optimize AMP detection with four groups of specifically designed learnable coefficients.The simulation results showed that the proposed scheme had much lower divergence rate and detection MSE than the conventional AMP.In addition,we need to further investigate the reasons why divergence occurs in AMP iterations and analyze what impact it will has in detection MSE.Accordingly,optimized interleaving and coding scheme is investigated to correct the errors caused by divergence.We will conduct the research of these problems in our future work.

ACKNOWLEDGEMENT

This work was supported by the National Natural Science Foundation of China under Grants 61801523,61971452,and 91538203.