Uniformly Reweighted Belief Propagation: A Factor Graph Approach

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1 Uniformy Reweighted Beief Propagation: A Factor Graph Approach Henk Wymeersch Chamers University of Technoogy Gothenburg, Sweden henkw@chamers.se Federico Penna Poitecnico di Torino Torino, Itay federico.penna@poito.it Vadimir Savić Universidad Poitecnica de Madrid Madrid, Spain vadimir@gaps.ssr.upm.es Abstract Tree-reweighted beief propagation is a message passing method that has certain advantages compared to traditiona beief propagation (BP). However, it fais to outperform BP in a consistent manner, does not end itsef we to distributed impementation, and has not been appied to distributions with higher-order interactions. We propose a method caed uniformyreweighted beief propagation that mitigates these drawbacks. After having shown in previous works that this method can substantiay outperform BP in distributed inference with pairwise interaction modes, in this paper we extend it to higher-order interactions and appy it to LDPC decoding, eading performance gains over BP. I. INTRODUCTION Beief propagation (BP) [1] is a powerfu method to perform approximate inference, and has found appications in a wide variety of fieds [2]. Furthermore, it provides new interpretations of existing agorithms, such as the Viterbi agorithm [3]. BP can be seen as a message passing agorithm on a graphica mode. When this graphica mode, which represents the factorization of an un-normaized distribution, is cyce-free (i.e., a tree), BP is guaranteed to converge and can provide margina distributions as we as the normaization constant of the distribution (aso known as the partition function). Moreover, BP can be interpreted as an iterative scheme to find stationary points of a convex variationa optimization probem [4]. Unfortunatey, when the graphica mode contains cyces, the variationa probem is no onger guaranteed to be convex, so that BP may end up in a oca optimum, or fai to converge atogether. To address the probem of non-convexity, variations of BP have been proposed that are provaby convex, or aow to compute a ower or upper bound of the origina optimization probem. One such variation is tree-reweighted BP (TRW-BP), which corresponds to a convex upper approximation of the origina objective function (to be introduced ater) with reaxed constraints [5]. In its origina formuation, the authors ony provided expicit message passing rues for TRW-BP in the case of a particuar graphica mode (a Markov random fied) with a particuar factorization (pairwise interactions). The extension to higher-order interaction was described through This research is supported, in part, by the FPU feowship from Spanish Ministry of Science and Innovation, ICT project FP7-ICT WHERE-2, program CONSOLIDER-INGENIO 2010 under grant CSD COMONSENS, and the European Research Counci, under grant COOPNET No hypergraphs, though no expicit message passing rues were derived. Expicit rues for higher-order interactions were derived in [6] for fractiona BP, of which TRW-BP was shown to be a specia case. TRW-BP has been observed to outperform BP in some, though not a cases [5]. Moreover, TRW-BP requires an optimization of a distribution over spanning trees, making it unsuitabe for distributed inference probems (e.g., over wireess networks). In our prior work on distributed positioning [7] and distributed detection [8], we have proposed a nove message passing scheme, uniformy-reweighted beief propagation (URW-BP), which encompasses both BP as we TRW-BP over uniform graphs, and does not invove an optimization of a distribution over spanning trees. This method was imited to distributions with pairwise interactions. In this paper, we extend URW-BP to higher-order interaction. Rather than reying on hypergraphs as in [5], we empoy factor graphs. We formuate a variationa probem and determine the stationary points of the Lagragian, eading to expicit message passing rues for higher-order interactions. We aso appy these new message passing rues to the exampe of decoding an LDPC code, and make connections with another recenty proposed decoder [9]. II. THE INFERENCE PROBLEM We consider an a posteriori distribution of the foowing form p(x y) = p(y x)p(x), (1) p(y) where y is a (fixed) observation and x =[x 1,x 2,...,x N ] is the unobserved variabe of interest. We wi assume that x n is a discrete random variabe defined over a finite set. Both the ikeihood function p(y x) as we as the prior p(x) are assumed to be known. The vaue p(y) is unknown, and in sometimes referred to as the partition function. Moreover, we assume a factorization exists of the form N L p(y x)p(x) = φ n (x n ) ψ (x C ), (2) where x C x contains at east two variabes, indexed by the set C {1,..., N}. Typica goas in inference incude (i) determining the margina a posteriori distributions, p(x n y), for every component x n ; (ii) computing p(y). =1

2 For a given y and a given x, evauating p(y x)p(x) is straightforward. However, brute-force computation of p(x n y) or p(y) is intractabe, due to the high-dimensiona nature of x. A factorization of the form (2) can be convenienty expressed by a graphica mode, such as a factor graph or a Markov random fied. By executing a message passing agorithm (e.g., BP or TRW-BP) on this graphica mode, one can approximate p(x n y) as we as p(y) [4], [5]. III. PAIRWISE INTERACTIONS: TRW-BP AND BP When there are ony pairwise interactions, every C contains exacty 2 eements. For notationa convenience, we denote ψ (x m,x n ) by ψ mn (x m,x n ). As an exampe, consider a factorization p(x y) φ 1 (x 1 )φ 2 (x 2 )φ 3 (x 3 )ψ 12 (x 1,x 2 )ψ 23 (x 2,x 3 ). The corresponding Markov random fied is depicted in Fig. 1. We define neighbor set N m of variabe X m as foows: X n N m if and ony if there exists a factor ψ mn (x m,x n ). Then, the message from X m to a neighboring variabe X n is given by [5, eq. (39)] M mn (x n ) (3) k N φ m ψmn 1/ρmn (x n,x m ) M ρ km m\{n} km (x m), Mnm 1 ρmn x m where ρ nm ρ mn is the so-caed edge appearance probabiity (EAP) of factor ψ nm. The terminoogy of EAP is due to the representation of ψ nm as an edge in the Markov random fied representation of ψ nm in (2). The so-caed beiefs, which are an approximation of the a posteriori marginas p(x n y), are given by b n (x n ) φ n (x n ) mn (x n ). (4) m N n M ρnm Equations (3) (4) are iterated unti the beiefs converge. The set of vaid EAPs is non-trivia: given a Markov random fied graph G and the set T(G) of a possibe trees, we can introduce a distribution over the trees: 0 ρ(t ) 1, for T T(G), with T T(G) ρ(t ) = 1. For a given distribution ρ(t ), the EAP of edge (n, m) is then given by ρ nm = ρ(t ) I {(n, m) T }, T T(G) where I { } is the indicator function. Note that when G is a tree, ρ nm =1, for a edges (n, m). When the graph G contains cyces, ρ nm < 1 for at east one edge. BP can now be interpreted as a variation of TRW-BP, where ρ nm =1, for a edges (n, m), irrespective of the structure of G. Note that this is not a vaid choice of EAPs for a graph with cyces. IV. EXTENSION TO HIGHER-ORDER INTERACTIONS A. Factor Graphs When some of the ciques C contain 3 or more eements, a factor graph can more eeganty capture those higher-order interactions than a Markov random fied. A factor graph is a X 1 X 1 X 2 M 23 (x 3 ) M 32 (x 2 ) ψ 12 X 2 ψ 23 X 3 φ 1 φ 2 φ 3 Figure 1. Markov random fied (top) and factor graph (bottom) of the factorization φ 1 (x 1 )φ 2 (x 2 )φ 3 (x 3 )ψ 12 (x 1,x 2 )ψ 23 (x 2,x 3 ). bi-partite graph where we distinguish between variabe vertices and factor vertices. A factor vertex and a variabe vertex are connected by an edge when the the corresponding variabe appears in the corresponding factor (see Fig. 1). When ψ has x n as a variabe, we wi write N n. We now provide a derivation of TRW-BP message passing rues by foowing a variationa approach, resuting in expicit message expressions. Then, we speciaize this to our proposed method, URW-BP. B. Variationa Interpretation Given any distribution b(x), the Kuback-Leiber divergence between b(x) and p(x y) is given by KL (b p) = x b(x) og X 3 b(x) 0. (5) p(x y) If we insert (2) into (5), and perform some straightforward manipuations, we can rewrite this inequaity as a variationa probem: { N og p(y) = max H(b)+ b n (x n ) og φ n (x n ) b M(G) x n L } + b C (x C ) og ψ (x C ), (6) x C =1 where H(b) denotes the entropy of the distribution b(x), H(b) = x b(x) og b(x), and M(G) is the so-caed margina poytope, which is the set of margina distributions b k (x k ) and b C (x C ) that can be reated to a vaid distribution, which factorizes according to the same factor graph G, induced by (2). Note that the soution to (6) is b(x) =p(x y) with corresponding maximum equa to og p(y). The optimization probem (6) turns out to be convex, but, uness G is a tree, intractabe: (i) the number of constraints to describe M(G) is intractabe (ii) computing the entropy for any b(x) is intractabe. C. Standard Beief Propagation In standard BP, a tractabe soution is achieved by (i) reaxing M(G) to the oca poytope L(G), the set of margina distributions b k (x k ) and b C (x C ) that are normaized, nonnegative, and mutuay consistent, but not necessariy correspond to a vaid goba distribution; (ii) approximating the

3 entropy H(b) by the so-caed Bethe entropy N L H Bethe (b) = H(b n ) I C (b C ), where H(b n ) is the entropy of b n (x n ) and I C (b C ) is a mutua information term defined as I C (b C )= b C (x C ) b C (x C ) og x C m C b m. Substitution into (6), expressing the stationary conditions by setting the derivative of the Lagrangian to zero eads to the we-known BP message passing rues [4]: Message from variabe vertex X n to factor vertex ψ, n C : µ Xn ψ (x n )=φ n (x n ) µ ψk X n (x n ). (7) k N n\{} =1 Message from factor vertex 1 ψ to variabe vertex X n, n C : µ ψ X n (x n )= ψ (x C ) µ Xm ψ, m C \{n} (8) where refers to the summation over a variabes in x C, except x n. Beief of variabe x n : b n (x n ) φ n (x n ) µ ψ X n (x n ). (9) N n Equations (7) (9) are iterated unti the beiefs converge. When there are ony pairwise interactions, it is easy to show that (7) (9) is equivaent with (3) (4), when ρ nm =1, for a (n, m). D. Tree-reweighted Beief Propagation 1) Formuation: Starting from a factor graph of a factorization of (2), we can now generate a tree by removing a suitabe subset of factor vertices ψ, aong with a connected edges. For any such tree T, we can express the corresponding entropy as N H(b; T )= I C (b C ), H(b n ) C T where C T denotes a summation over a factor vertices that remain in the tree. For any tree and a fixed b( ), we know that H(b; T ) H(b) [10]. We can now introduce a distribution p(t ) over such trees, as we as factor appearance probabiities: ρ = p(t )I {C T }. T T(G) Given a distribution over trees, a convex upper bound on H(b) is given by N L H(b; ρ) = H(b n ) ρ I C (b C ). 1 For mathematica convenience, we ony compute messages from factor vertices corresponding to factors with at east 2 variabes. =1 We finay arrive to the foowing variationa probem, by reaxing M(G) to L(G), and repacing H(b) by H(b; ρ): { og p(y) arg max H(b; ρ)+ b L(G) N b n (x n ) og φ n (x n ) x n L } + b C (x C ) og ψ (x C ). (10) x C =1 2) Message Passing Rues: Message update rues are then obtained by expressing the Lagrangian and setting the derivative to zero. We state here the fina resuts; the compete derivation is presented in the Appendix. Message from variabe vertex X n to factor vertex ψ, n C : µ Xn ψ (x n )= (11) φ n (x n )µ ρ 1 ψ X n (x n ) µ ρ k ψ k X n (x n ). k N n\{} Message from factor vertex ψ to variabe vertex X n, n C : µ ψ X n (x n )= (12) ψ 1/ρ (x C ) µ Xm ψ. Beief of variabe x n : m C \{n} b n (x n ) φ n (x n ) µ ρ ψ X n (x n ). (13) N n As we woud expect, when ρ =1,, we find equations (7) (9) again. Moreover, when there are ony pairwise interactions, we find (3) (4). It shoud be noted that equivaent message passing rues were presented in [6] in the context of fractiona beief propagation. E. Uniformy Reweighted Beief Propagation It is now straightforward to introduce URW-BP. In principe, TRW-BP aows us optimize over a possibe distributions ρ. Nevertheess, we are not guaranteed that this wi ead to better performance than BP. Here we restrict our attention to fixed ρ = ρ,. This scheme is motivated by the fact that in graphs that are roughy uniform in structure with roughy equa properties of the factor vertices, we expect that a uniform distribution over trees woud be optima, eading to ρ = ρ1, for some 0 <ρ 1 [10], with ρ =1when the factor graph is a tree. Hence, for such uniform graphs, we expect URW-BP to achieve the best performance among BP and TRW-BP. A. Mode V. NUMERICAL EXAMPLE We consider a practica exampe for which, to the best of our knowedge, TRW-BP has not been appied before: decoding of an LDPC code. Consider an LDPC code with an L N sparse parity check matrix H. Let x denote the transmitted codeword

4 10 1 min sum sum product 10 0 min sum ρ = 1 sum product ρ = min sum ρ = 0.55 sum product ρ = BER 10 2 BER ρ E b /N 0 [db] Figure 2. Performance of URW-BP after 20 decoding iterations as a function of the scaar parameter ρ for LDPC code, at a fixed SNR (E b /N 0 =3dB). BP corresponds to ρ =1. and y the observation over a memoryess channe, with known p(y n x n ). The a posteriori distribution of interest is now p(x y) p(y x)p(x) N = p(y n x n )I{Hx = 0} = { } N L p(y n x n ) I x m =0, m C where the summation is in the binary fied, and C is the index set corresponding to the non-zero eements of the th row in H. Ceary, we can make the association φ n (x n ) p(y n x n ) and ψ (x C ) I { m C x m =0 }. B. Message Passing Rues The message from variabe node X n to check node ψ, expressed as a og-ikeihood ratio (LLR), is given by λ Xn ψ = λ ch,n + ρ k λ ψk X n λ ψ X n, k N n where =1 λ ch,n = og p(y n x n = 1) p(y n x n = 0). The messages from check nodes to variabe nodes are unchanged with respect to standard BP, since here ψ 1/ρ (x C )= ψ (x C ), irrespective of ρ. We wi consider two types of messages from check nodes to variabe nodes: sum-product messages (in the og-domain, see [2, eq. (22)]) and simpified min-sum messages (see [9, eq. (14)]). Finay, the beiefs in LLR format are given by λ b,n = λ ch,n + ρ k λ ψk X n. k N n Interestingy, in its min-sum version, URW-BP is equivaent to a recenty proposed LDPC decoder from [9], based on the divide and concur agorithm, provided that ρ is set to 2Z, where Z was introduced in [9, eq. (15)]. Figure 3. Performance of BP and URW-BP as a function of the SNR for LDPC code after 20 decoding iterations. C. Performance Exampe In Fig. 2, we pot the bit error rate (BER) performance 2 of a rate 1/2 LDPC code with N = 256 and L = 128 at a fixed SNR as a function of ρ. We observe that ρ =1does not yied the best performance and that the goba minimum in the BER is achieved by ρ 0.85 and ρ 0.55, for sumproduct and min-sum, respectivey. The optima vaue of ρ for min-sum is ower because the min-sum rue tends to overshoot the LLRs more than sum-product. In Fig. 3 we compare the performance of BP and URW-BP (with ρ = 0.85 and ρ =0.55 for sum-product and min-sum, respectivey) in terms of BER vs. SNR for the same LDPC codes. We see that URW- BP outperforms BP especiay at high SNR. The difference in BER between standard BP and URW-BP is not arge, but sti significant considering that we have used a rea LDPC code, designed such that oops are ong and have imited impact on BP decoding. The performance gap can be much greater in case of non-optimized graph configurations, i.e., with many short oops. VI. CONCLUSIONS Motivated by promising resuts in distributed processing, we have derived uniformy reweighted beief propagation (URW-BP) for distributions with higher-order interactions. We have adopted a factor graph mode, as this aows an eegant representation of higher-order interactions, as we as the message passing interpretation. The message passing rues (11), (12), and (13) easiy ead to URW-BP. URW-BP combines the potentia improved performance of TRW-BP with the distributed nature of BP. We have appied these new message passing rues to LDPC decoding and have shown that URW-BP can outperform BP, even for an LDPC code without short cyces. 2 For a decoding agorithms, we appied the foowing stopping rue: when a vaid codeword is found at a certain iteration, no further iterations are performed.

5 APPENDIX Here we provide the derivation eading to the message passing rues for TRW-BP with higher-order interactions, foowing the reasoning from [10, Section 4.1.3]. The Loca Poytope: We first describe the oca poytope L(G). Any eement b L(G) is described by singe-variabe beiefs b n (x n ), n =1,...,N and higher-order beiefs b (x C ), = 1,..., L and must satisfy the foowing conditions: non-negativity, normaization, and consistency: b n (x n ) 0, b C (x C ) 0, x n b n (x n ) = 1, x C b C (x C ) = 1, b C (x C ) = b n (x n ),n C. We wi enforce the consistency constraints expicity with Lagrange mutipiers λ C (x n ), and the non-negativity and normaization constraints impicity. The Lagrangian: The Lagrangian is given by L(b, λ) =H(b; ρ)+ + + L =1 N x n b n (x n ) og φ n (x n ) b C (x C ) og ψ (x C ) x C { [ L λ C b m ]} b C (x C ), x m x m =1 m C subject to non-negativity and normaization constraints. Stationary Points: We now determine the stationary points of the Lagrangian. The derivatives with respect to the beiefs are given by L(b, λ) b n (x n ) = κ n og b n(x n ) φ n (x n ) + λ C (x n ) (14) N n and L(b, λ) b C (x C ) = (15) b C (x C ) κ C ρ og m C b m + og ψ (x C ) λ C, m C where κ n and κ C are constants. Setting the derivatives to zero and taking exponentias eads to b n (x n ) φ n (x n ) N i exp (λ C (x n )) (16) b C (x C ) ψ 1/ρ (x C ) b m ( λc ). (17) (x m C exp m) ρ We introduce ( ) λc µ ψ m = exp, (18) so that (16) and (17) can be written as b n (x n ) φ n (x n ) N n µ ρ ψ X n (x n ) (19) b C (x C ) ψ 1/ρ (x C ) b m µ ψ X m C m. (20) ρ Observe that (19) is exacty (13). Note that normaization and non-negativity are inherenty satisfied. Message Passing Rues: Finay, we express the consistency constraints to revea the message passing rues. Essentiay, our objective is to write a vaid expression that does not contain beiefs. Summing out a variabes except x n in (20) yieds b n (x n )= b n (x n ) µ ψ X n (x n ) ψ 1/ρ (x C ) m C \{n} b m µ ψ X m, (21) so that, after canceing out b n (x n ) and moving µ ψ X n (x n ) to the eft hand side, we find that µ ψ X n (x n )= ψ 1/ρ (x C ) (22) m C \{n} φ m k N m µ ρ k ψ k X m. µ ψ X m Introducing the message from variabe vertex X m to factor vertex ψ, µ Xm ψ = φ m k N m µ ρ k ψ k X m µ ψ X m we immediatey find that µ ψ X n (x n )= ψ 1/ρ (x C ) m C \{n} µ Xm ψ. (23) (24) Here, (24) is exacty (12) and (23) is by construction the same as (11). REFERENCES [1] J. Pear, Probabiistic Reasoning in Inteigent Systems. Networks of pausibe inference, Morgan Kaufmann, [2] F. R. Kschischang, B. J. Frey, and H.-A. Loeiger, Factor graphs and the sum-product agorithm, IEEE Transactions on Information Theory, vo. 47, no. 2, pp , Feb [3] H. A. Loeiger, An introduction to factor graphs, Signa Processing Magazine, IEEE, vo. 21, no. 1, pp. pp , Jan [4] J. S. Yedidia, W. T. Freeman, and Y. Weiss, Constructing free-energy approximations and generaized beief propagation agorithms, IEEE Transactions on Information Theory, vo. 51, no. 7, pp , Juy [5] M. J. Wainwright, T. S. Jaakkoa, and A. S. Wisky, A new cass of upper bounds on the og partition function, IEEE Transactions on Information Theory, vo. 51, no. 7, pp , Juy [6] T. Minka, Divergence measures and message passing, Microsoft Research Technica Report MSR-TR , Dec [7] V. Savic, H. Wymeersch, F. Penna, and S. Zazo, Optimized edge appearance probabiity for cooperative ocaization based on treereweighted nonparametric beief propagation, in Proc. IEEE Internationa Conference on Acoustics, Speech and Signa Processing (ICASSP), Prague, May [8] F. Penna, V. Savic, and H. Wymeersch, Uniformy reweighted beief propagation for distributed bayesian hypothesis testing, in Proc. IEEE Internationa Workshop on Statistica Signa Processing (SSP), June [9] J. S. Yedidia, Yige Wang, and S. C. Draper, Divide and concur and difference-map BP decoders for LDPC codes, IEEE Transactions on Information Theory, vo. 57, no. 2, pp , Feb [10] M. J. Wainwright and M. I. Jordan, Graphica modes, exponentia famiies, and variationa inference, Foundations and Trends in Machine Learning. Now Pubishers Inc., 2007.