Optimality of Large MIMO Detection via Approximate Message Passing

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1 ptimality of Large MIM Detection via Approximate Message Passing Charles Jeon, Ramina Ghods, Arian Maleki, and Christoph Studer arxiv:5.695v [cs.it] ct 5 Abstract ptimal data detection in multiple-input multipleoutput (MIM communication systems with a large number of antennas at both ends of the wireless link entails prohibitive computational complexity. In order to reduce the computational complexity, a variety of sub-optimal detection algorithms have been proposed in the literature. In this paper, we analyze the optimality of a novel data-detection method for large MIM systems that relies on approximate message passing (AMP. We show that our algorithm, referred to as individually-optimal (I large-mim AMP (short I-LAMA, is able to perform I data detection given certain conditions on the MIM system and the constellation set (e.g., QAM or PSK are met. I. INTRDUCTIN We consider the problem of recovering the M T -dimensional data vector s MT from the noisy multiple-input multipleoutput (MIM input-output relation y = Hs + n, by performing individually-optimal (I data detection [], [3] (I s I l = arg max s l p( s l y, H. Here, s I l denotes the l-th I estimate, is a finite constellation (e.g., QAM or PSK, p( s l y, H is a probability density function assuming i.i.d. zero-mean complex Gaussian noise for the vector n C MR with variance N per complex dimension, M T and M R denotes the number of transmit and receive antennas, respectively, y C MR is the receive vector, and H C MR MT is the (known MIM system matrix. In what follows, we assume that the entries of the MIM system matrix H are i.i.d. zero-mean complex Gaussian with variance /M R, and we define the so-called system ratio as β = M T /M R. Although I detection achieves the minimum symbol errorrate [4], the combinatorial nature of the (I problem [], [3] requires prohibitive computational complexity, especially in large (or massive MIM systems [4], [5]. In order to enable data detection in such high-dimensional systems, a large number of low-complexity but sub-optimal algorithms have been proposed in the literature (see, e.g., [6] [8]. A. Contributions In this paper, we propose and analyze a novel, computationally efficient data-detection algorithm, referred to as I-LAMA (short for I large MIM approximate message An extended version of this paper including all proofs is in preparation []. C. Jeon, R. Ghods, and C. Studer are with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY; jeon@csl.cornell.edu, rghods@csl.cornell.edu, studer@cornell.edu. A. Maleki is with Department of Statistics at Columbia University, New York City, NY; arian@stat.columbia.edu. This work was supported in part by Xilinx Inc. and by the US National Science Foundation (NSF under grants ECCS-486 and CCF-438. (a MIM system with I-LAMA as the data detector. (b Equivalent decoupled system with effective noise variance σ t. Fig.. I-LAMA decouples large MIM systems (a into a set of parallel and independent AWGN channels with equal noise variance; (b equivalent system in the large-system limit, i.e., for β = M T /M R with M T. passing. We show that I-LAMA decouples the noisy MIM system into a set of independent additive white Gaussian noise (AWGN channels with equal signal-to-noise ratio (SNR; see Fig. for an illustration of this decoupling property. The state-evolution (SE recursion of AMP enables us to track the effective noise variance σt of each decoupled AWGN channel at every algorithm iteration t. Using these results, we provide precise conditions on the MIM system matrix, the system ratio β, the noise variance N, and the modulation scheme for which I-LAMA exactly solves the (I problem. B. Relevant Prior Art Initial results for I data detection in large MIM systems reach back to [9] where Verdú and Shamai analyzed the achievable rates under optimal data detection in randomlyspread CDMA systems. Tanaka [] derived expressions for the error-rate performance and the multi-user efficiency for I detection using the replica method. While Tanaka s results were limited to BPSK constellations, Guo and Verdú extended his results to arbitrary discrete input distributions [3], []. All these results study the fundamental performance of I data detection in the large-system limit, i.e., for β = M T /M R with M T. Corresponding practical detection algorithms have been proposed for BPSK constellations [], [3] to the best of our knowledge, no computationally efficient algorithms for general constellation sets and complex-valued systems have been proposed in the open literature.

2 ur data-detection method, I-LAMA, builds upon approximate message passing (AMP [4] [6], which was initially developed for the recovery of sparse signals. AMP has been generalized to arbitrary signal priors in [7] [9] and enables a precise performance analysis via the SE recursion [4], [5]. Recently, AMP-related algorithms have been proposed for data detection [] []; these algorithms, however, lack of a theoretical performance analysis. C. Notation Lowercase and uppercase boldface letters designate vectors and matrices, respectively. For a matrix H, we define its conjugate transpose to be H H. The l-th column of H is denoted by h c l. We use to write x = N N k= x k. A multivariate complex-valued Gaussian probability density function (pdf is denoted by CN (m, K, where m is the mean vector and K the covariance matrix. E X [ ] and Var X [ ] denotes the expectation and variance operator with respect to the pdf of the random variable X, respectively. II. I-LAMA: LARGE-MIM DETECTIN USING AMP We now present I-LAMA and the SE recursion, which is used in Section III for our optimality analysis. A. The I-LAMA Algorithm We assume that the transmit symbols s l, l =,, M T, of the transmit data vector s are taken from a finite set = {a j : j =,, with constellation points a j chosen, e.g., from a QAM or PSK alphabet. We assume an i.i.d. prior p(s = M T l= p(s l, with the following distribution for each transmit symbol s l : p(s l = a p aδ(s l a. ( Here, p a designates the (known prior probability of each constellation point a and δ( is the Dirac delta function; for uniform priors, we have p a =. The I-LAMA algorithm summarized below is obtained by using the prior distribution in ( within complex Bayesian AMP. A detailed derivation of the algorithm is given in []. Algorithm. Initialize ŝ l = E S[S] for l =,, M T, r = y, and τ = β Var S [S]/N. Then, for every I-LAMA iteration t =,,, compute the following steps: z t = ŝ t + H H r t ŝ t+ = F(z t, N ( + τ t τ t+ = β N G(z t, N ( + τ t r t+ = y Hŝ t+ + τ t+ +τ t r t. The functions F(s l, τ and G(s l, τ correspond to the message mean and variance, and are computed as follows: F(ŝ l, τ = s l s l f(s l ŝ l, τds l ( G(ŝ l, τ = s l s l f(s l ŝ l, τds l F(ŝ l, τ. Here, f(s l ŝ l, τ is the posterior pdf defined by f(s l ŝ l, τ = Z p(ŝ l s l, τp(s l with p(ŝ l s l, τ CN (s l, τ and a normalization constant Z. Both functions F(ŝ l, τ and G(ŝ l, τ operate element-wise on vectors. In order to analyze the performance of I-LAMA in the large-system limit, we next summarize the SE recursion. The SE recursion in the following theorem enables us to track the effective noise variance σt for the decoupled MIM system for every iteration t (cf. Fig., which is key for the optimality analysis in Section III. A detailed derivation is given in []. Theorem. Fix the system ratio β = M T /M R and the constellation set, and let M T. Initialize σ = N +β Var S [S]. Then, the effective noise variance σ t of I-LAMA at iteration t is given by the following recursion: σ t = N + βψ(σ t. (3 The so-called mean-squared error (MSE function is defined by Ψ(σ t = E S,Z [ F ( S + σt Z, σ t S ], where F is given in ( and Z CN (,. B. I-LAMA Decouples Large MIM Systems In the large-system limit and for every iteration t, I-LAMA computes the marginal distribution of s l, l =,, M T, which corresponds to a Gaussian distribution centered around the original signal s l with variance σ t. These properties follow from [6, Sec. 6], which shows that z t = ŝ t + H H r t is distributed according to CN (s, σ t I MT. Hence, the inputoutput relation for each transmit stream z t l = ŝt l + (hc l H r t l is equivalent to the following single-stream AWGN channel: z t l = s l + n t l. Here, s l is the l-th original transmitted signal and n t l is AWGN with variance σt per complex entry. As a consequence, I-LAMA decouples the MIM system into M T parallel and independent AWGN channels with equal noise variance σt in the large-mim limit; see Fig. (b for an illustration. III. PTIMALITY F I-LAMA We now provide conditions for which I-LAMA exactly solves the (I problem. A. Fixed points of I-LAMA s State Evolution For t, the SE recursion in Theorem converges to the following fixed-point equation [], [5]: σ I = N + βψ(σ I, (4 which coincides with the fixed-point equation developed for I detection by Guo and Verdú using the replica method in [3, Eq. (34]. We note that (4 may have multiple fixed-point solutions. In the case of such non-unique fixed points, Guo and Verdú choose the solution that minimizes the free energy [3, Sec. -D], whereas I-LAMA converges, in general, to the fixed-point solution with the largest effective noise variance σ. We note that if the fixed-point solution to (4 is unique, then I-LAMA recovers the solution with minimal effective noise variance σ and thus, performs I detection. However, if there are multiple fixed-points solutions to (4, I-LAMA is, in general, sub-optimal and does not necessarily converge to the fixed-point solution with the minimal free energy. We next Convergence to another fixed-point solution is possible if I-LAMA is initialized sufficiently close to such a fixed point; see [], [3] for the details.

3 provide conditions for which there is exactly one (unique fixed point with minimum effective noise variance σ and as a consequence I-LAMA solves the (I problem. B. Exact Recovery Thresholds (ERTs We start by analyzing I-LAMA in the noiseless setting. We provide conditions on the system ratio β and the constellation set, which guarantee exact recovery of an unknown transmit signal s MT in the large-system limit, i.e., β is fixed and M T. In particular, we show that if β < β max, where βmax is the so-called exact recovery threshold (ERT, then I-LAMA perfectly recovers s ; for β β max, perfect recovery is not guaranteed, in general. To make this behavior explicit, we need the following technical result; the proof is given in Appendix A. Lemma. Fix the constellation set. If Var S [S] is finite, then there exists a non-negative gap σ Ψ(σ with equality if and only if σ =. As σ, the MSE Ψ(σ and as σ, MSE Ψ(σ Var S [S]. For all σ >, Lemma guarantees that Ψ(σ < σ. Suppose that for some β >, βψ(σ < σ also holds for all σ >. Then, as long as β > is not too large to also ensure βψ(σ < σ for all σ >, there will only be a single fixed point at σ =. Therefore, LAMA can still perfectly recover the original signal s by Theorem since Ψ(σ =. Leveraging the gap between Ψ(σ and σ will allow us to find the exact recovery threshold (ERT of LAMA for values of β >. For the fixed (discrete constellation set, the largest β that ensures βψ(σ < σ is precisely the ERT defined next. Definition. Fix and let N =. Then, the exact recovery threshold (ERT that enables perfect recovery of the original signal s using I-LAMA is given by = min { (Ψ(σ σ. (5 With Definition, we state Theorem 3, which establishes optimality in the noiseless case; the proof is given in Appendix B. Theorem 3. Let N = and fix a discrete set. If β <, then I-LAMA perfectly recovers the original signal s from y = Hs + n in the large system limit. Note that for a given constellation set, the ERT β max can be computed numerically using (5. Furthermore, the signal variance, Var S [S], has no impact on the ERT as the MSE function Ψ(σ and σ scale linearly with Var S [S]. Table I summarizes ERTs β max for common QAM and PSK constellation sets. C. ptimality Conditions for I-LAMA We now study the optimality of I-LAMA in the presence of noise, where exact recovery is no longer guaranteed. In particular, we provide conditions for which I-LAMA converges to the fixed point with minimal effective noise variance σ, which corresponds to solving the (I problem. We assume the initialization in Algorithm. I-LAMA may recover the original signal for β if initialized appropriately; see, e.g., [3]. TABLE I ERTS β MAX, MRTS βmin, AND CRITICAL NISE LEVELS N min (βmin AND N max (β max F I-LAMA FR CMMN CNSTELLATIN SETS Constellation β min N min (βmin βmax N max ( BPSK QPSK QAM QAM PSK PSK TABLE II SUMMARY F (SUB-PTIMALITY REGIMES F I-LAMA β β min β min <β <βmax β N < N min (β optimal optimal suboptimal N min(β N N max (β optimal (sub-optimal 3 suboptimal N max (β < N optimal optimal optimal Note that such a minimum free-energy solution is also the fixed point for the I detector in [3, Eq. (34]. We call the fixed point with minimum effective noise variance optimal fixed point; other fixed points are called suboptimal fixed points. We identify three different operation regimes for I-LAMA depending on the system ratio β (see Table II. To make these three regimes explicit, we need the following definition. Definition. Fix the constellation set. Then, the minimum recovery threshold (MRT β min is defined by { (dψ(σ β min = min. (6 The definition of MRT shows that for all system ratios β β min, the fixed point of (4 is unique. The following lemma establishes a fundamental relationship between MRT and ERT; the proof is given in Appendix C. Lemma 4. The MRT never exceeds the ERT. We next define the minimum critical and maximum guaranteed noise variance, N min (β and N max (β, that determine boundaries for the optimality regimes when β > β min. Definition 3. Fix β (β βmax. Then, the minimum critical noise N min (β that ensures convergence to the optimal fixed point is defined by { N min (β = min σ βψ(σ : β dψ(σ =. Definition 4. Fix β > β min. Then, the maximum guaranteed noise N max (β that ensures convergence to the optimal fixed point is defined by { N max (β = max σ βψ(σ : β dψ(σ =. We recall that all the zero crossings of the function g(σ, β, N = N + βψ(σ σ (7 3 For certain constellation sets (e.g., 6-PSK, there exist sub-intervals in [N min (β,n max (β] where I-LAMA is still optimal; see [] for the details.

4 (a β β min : I-LAMA always converges to the unique, optimal fixed point (FP irrespective of N. (b β (β βmax : I-LAMA converges to the optimal FP if N < N min(β or N > N max (β. (c β β max : I-LAMA converges to the optimal fixed point if N > N max (β. Fig.. Plot of the function (7 for three regimes (a β β min, (b β (βmin, βmax, and (c β βmax for QPSK modulation, uniform priors, and Var S [S] = E s =. The optimal fixed points are designated by ; suboptimal fixed points are designated by. correspond to all fixed points of the SE recursion of I-LAMA; we use this function to study the algorithm s optimality. Figure illustrates our optimality analysis for a large-mim system with QPSK constellations. We show (7 depending on the effective noise variance σ and for different system ratios β. The regimes β β β (βmin, βmax, and β βmax are shown in Fig. (a, Fig. (b, and Fig. (c, respectively. The special case for β = with N = corresponds to the solid blue line, along with the corresponding (unique fixed point at the origin. In the following three paragraphs, we discuss the three operation regimes of I-LAMA in detail. (i β β min : In this region, the SE recursion of I-LAMA always converges to the unique, optimal fixed point. For β < β the slope of (7 for all σ is strictly-negative. Hence, as (7 is always decreasing, there exists exactly one unique fixed point of the SE recursion regardless of the noise variance N. Thus, I-LAMA converges to the optimal fixed point and consequently, solves the (I problem. We emphasize that we still obtain exactly one fixed point even when β is equal to the MRT. Since β = β min, there exists at least one σ that satisfies β min d Ψ(σ σ =. By =σ definition of β (7 at σ implies that σ is a saddle-point, so (7 has exactly one zero at σ. We observe that if σ is unique, then N min (β min = N max (β min. For all other σ σ, the construction of σ implies that β min d Ψ(σ <, so the fixed point of (7 remains to be unique. The green, dash-dotted and red, dotted line in Fig. (a shows (7 for β = β min with N = and N = N min (β min = N max (β min, respectively. In both cases, we see that the SE recursion of I-LAMA converges to the unique fixed point. (ii β min < β < βmax : In this region, the SE recursion of I-LAMA converges to the unique, optimal fixed point if N < N min (β or N > N max (β. The green, dash-dotted line, cyan, dashed line, and magenta, dotted line in Fig. (b shows (7 for β = (β min+βmax / with N =, N > N max (β and N < N min, respectively. We note that for the three cases, the fixed point is unique, labeled in Fig. (b by a circle. n the other hand, the red, dotted line in Fig. (b shows (7 with β under noise N [N min (β, N max (β ]. In this case, however, we observe that SE recursion of I-LAMA converges to the rightmost suboptimal fixed point labeled by the crossed circle. Hence, I-LAMA does not, in general, solve the (I problem when N min (β N N max (β. (iii β β max : In this region, the SE recursion of I- LAMA converges to the unique, optimal fixed point when N > N max (β. As β β max, the low noise N < N min (β (or high SNR region of optimality disappears because N min (β as β from (5. The green, dash-dotted line and red, dotted line in Fig. (c shows (7 for β = β max with N = and < N N max (β, respectively. We observe that the SE recursion of I-LAMA converges to the suboptimal fixed point when β = β max even with N =. n the other hand, the cyan, dashed line refers to (7 for β = β max with N > N max (β. While the noiseless case resulted the SE recursion of I-LAMA to converge to the suboptimal fixed point, we observe that for strong noise (or equivalently low SNR, the SE recursion of I-LAMA actually recovers the I solution. Therefore, when β, I-LAMA solves the (I problem when the noise is greater than the maximum guaranteed noise N max (β. As a final remark, we note that the ERT β max and MRT β min in Table I do not depend on Var S[S]; the critical noise levels N min (β and N max (β, however, depend on Var S [S]. D. ERT, MRT, and Critical Noise Levels The ERT, MRT, as well as N min (β and N max (β for common constellations are summarized in Table I. We assume equally likely priors with the transmit signal normalized to E s = Var S [S] =. 4 We note that the calculations of ERT and MRT for the simplest case of BPSK constellations involve computations of a logistic-normal integral for which no closed-form expression is known [4]. Consequently, the following results were obtained via numerical integration for computing the MSE function Ψ(σ. As noted in Table I 4 The critical noise levels depend linearly on E s. Hence, we assume that E s = without loss of generality.

5 for a QPSK system under complex-valued noise, the ERT is βqpsk max.855, and the MRT is given as βmin QPSK.475. The MRTs for 6-QAM and 64-QAM indicate that small system ratios β < are required to always guarantee that I-LAMA solves the (I problem in the presence of noise. For instance, we require β β64-qam min.844, i.e. M T.844M R, to ensure that I-LAMA solves the I problem for 64-QAM in the large system limit. As β β64-qam max.573, I-LAMA is only optimal for N > N max (β64-qam max From Table I, we see that I-LAMA is a suitable candidate algorithm for the detection of higher-order QAM constellations in massive multiuser MIM systems as one typically assumes M R M T [5]. IV. CNCLUSINS We have presented the I-LAMA algorithm along with the state-evolution recursion. Using these results, we have established conditions on the MIM system matrix, the noise variance N, and the constellation set for which I-LAMA exactly solves the (I problem. While the presented results are exclusively for the large-system limit, our own simulations indicate that I-LAMA achieves near-optimal performance in realistic, finite-dimensional systems; see [] for more details. APPENDIX A PRF F LEMMA Since the variance of S is finite, denote Var S [S] = σ s. By [6, Prop. 5], we have the following upper bound: Ψ(σ σ s σ s + σ σ = + σ /σs σ. (8 Here, equality holds for all σ if and only if S is complex normal with variance σs [6]. Note that if σ =, then (8 is achieved for any σs. If σ >, then Ψ(σ < σ by (8. The first part follows directly from (8 as Ψ(σ is nonnegative. The second part requires one to realize that σ also implies F(, σ a ap a = E S [S], and hence, [ lim σ Ψ(σ E S S E S [S] ] = Var S [S]. APPENDIX B PRF F THEREM 3 We assume the initialization in Algorithm. Since N =, if LAMA perfectly recovers the original signal s, then the fixed point in (4 is unique at σ =. 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