# Adaptive Linear Programming Decoding

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2 Throughout the paper, we refer to the constraints of this form as parity-check constraints. In addition, for any element x i of the optimization variable, x, the constraint that 0 x i 1 is also added. III. ADAPTIVE LP DECODING As any odd-sized subset V of the neighborhood N of each check node introduces a unique parity-check constraint, there are 2 dc 1 constraints corresponding to each check node of degree d c. Therefore, the total number of constraints and, hence, the complexity of the problem, is exponential in terms of the maximum check node degree, d max c.thisbecomesmore significant in a high density code where d max c increases with the code length, n. In this section, we show that LP relaxation of linear codes has some properties that allow us to solve the optimization by using a much smaller number of constraints. A. Properties of the Relaxation Constraints Definition 1: Given a constraint of the form a T i x b i, (4) and a vector x 0 R n, we call (4) an active constraint at x 0 if a T i x 0 = b i, (5) and a violated constraint or, equivalently, a cut at x 0 if a T i x 0 >b i. (6) Considering a constraint that generates a cut x i x i > V 1, (7) at point x, we immediately make the following observations: V 1 < x i V, (8) 0 x i <x j j V. (9) The following theorem reveals a special property of the constraints of the LP decoding problem. Theorem 1: At any given point x [0, 1] n, at most one of the constraints introduced by each check node can be a cut. (Proof omitted.) Having a linear (n, k) code with m = n k parity checks, a natural question is how we can find all the cuts defined by the LP relaxation at any given point x R n. For any check node and an odd subset V of its neighborhood that introduces a cut, we know from (9) that the members of V are the variable nodes with the largest values among the neighbors of the check node. Therefore, sorting the elements of x before searching for a cut can simplify the procedure. Consider a check node j. Without loss of generality, assume that variable nodes 1, 2,..., N, are the neighbors of this check node, and they are sorted with respect to their values such that x 1 x 2 x N. The following algorithm is an efficient way to find the cut generated by this check node at x, if it exists. Algorithm 1: Step 1: Set v = 1, V = {1} and V c N\V = {2, 3,..., N }. Step 2: Check the constraint (3). If it is violated, we have found the cut. Exit. Step 3: Set v = v +2.Ifv N, movex v 1 and x v (the two largest members of V c )fromv c to V Step 4: If v N and (8) is satisfied, go to Step 2; otherwise, the check node does not provide a cut at x. If redundant calculations are avoided, this algorithm can find the cut generated by the check node, if it exists, in O(d c ) time, where d c = N is the degree of the check node. Repeating the procedure for each check node, and considering O(n log n) complexity for sorting x, the time required to find all the cuts at point x becomes O(md max c + n log n) 1. B. The Adaptive Procedure The Simplex LP algorithm starts from a vertex of the problem polytope and visits different vertices of the polytope by traveling through the edges until it finds the optimum vertex. The time required to find the solution is approximately proportional to the number of vertices that have been visited, and this, in turn, is determined by the number and properties of the constraints in the problem. Hence, if we eliminate some of the intermediate vertices and only keep those which are close to the optimum point, we can reduce the complexity of the algorithm. To implement this idea in the adaptive LP decoding scheme, we run the LP solver with a minimal number of constraints to ensure boundedness of the solution, and depending on the LP solution, we add only the useful constraints that cut the current solution from the feasible region. This procedure is repeated until no further cut exists. To start the procedure, we need at least n constraints so that the problem has a vertex that can become the solution of the first round. Using the condition that 0 x i 1, we add one side of these inequalities for each i, depending on whether increasing x i increases or decreases the objective function. In other words, for each i {1, 2,...,n}, we initially have the constraint 0 x i if γ i > 0, x i 1 if γ i < 0. (10) The optimum (and only) vertex of this initial problem corresponds to the result of (uncoded) hard decision based on the received vector. Now we proceed with the following algorithm: Algorithm 2: Step 1: Setup the initial problem according to (10). Step 2: Run the LP solver. Step 3: Search for all the cuts for the current solution. Step 4: If one or more cuts are found, add them to the problem constraints and go to Step 2. Claim 1: If at any iteration of Algorithm 2 no cut is found, the current solution is the solution of the LP decoder with all the relaxation constraints given in Section II. 1 For low density codes, it is better to sort the neighbors of each check node separately, so the total complexity becomes O(md max c +md max c log d max c ). 1375

5 WER Fig Regular (3,4) code of length LP C max =20 C max =100 C max =500 ML Lower Bound SNR, db WER of cutting-plane LP versus SNR for different values of C max. Step 4: If this RPC does not introduce a cut, go to Step 2. Our experiments indicate that by exploiting some of the properties of the problem as described above, the search for each cut becomes faster by orders of magnitude than the naive search, in which RPCs are selected in a completely random way. However, this procedure is still the complexity bottleneck of the decoding, and it becomes prohibitively complex for codes of length on the order of several hundred bits and above. On the other hand, experiments demonstrate that the number of cuts required to converge to the ML codeword (if the convergence is possible) does not grow very rapidly with length. This motivates more study of the properties of RPC cuts and efficient schemes to find them. B. Numerical Results To demonstrate the performance improvement achieved by using the RPC cutting-plane technique, we present simulation results for a random regular (3, 4) LDPC code of length 32. The adaptive nature of the algorithm allows us to smoothly trade complexity for performance by changing the number of trials in the search for each RPC cut, i.e. the number of iterations in Algorithm 3, as well as the total number of calls of the LP solver. In this experiment, we fix the total number of iterations of Algorithm 3 to C max, and declare decoding failure if no cut is found after C max trials. The word error rate (WER) of the algorithm is plotted versus SNR in Fig. 4 for different values of C max. For comparison, the WER of pure LP decoding, i.e. with no RPC cut, and a lower bound on the WER of the ML decoder have been included, as well. In order to obtain this lower bound, we count the number of times that the cutting-plane LP algorithm with a large value of C max converges to a codeword other than the transmitted codeword, and divide that by the number of blocks. Due to the ML certificate property of LP decoding, we know that ML decoding would fail in those cases, as well. On the other hand, ML decoding may also fail in some of the cases where LP decoding does not converge to an integral solution. Therefore, this estimate gives a lower bound on the WER of ML decoding. The numerical results suggest that the cutting-plane LP decoding with RPC cuts can significantly outperform the pure LP decoding, at the cost of increased complexity. V. CONCLUSION In this work, we studied the potential for improving LP decoding, both in complexity and error correction capability, by using an adaptive approach. The key idea was to use the fact that we can always recognize the failure of LP decoding to find the ML codeword, a property that message-passing algorithms only have in specific cases such as the erasure channel. This feature allows us to add only constraints that are useful, depending on the current status of the algorithm. In the algorithm proposed in Section III, the complexity is significantly reduced and becomes independent of the degree distributions, making it possible to apply LP decoding to parity-check codes of arbitrary densities. However, since general purpose LP solvers are used at each iteration, the complexity in terms of the block length is still super-linear, as opposed to linear as in the message-passing algorithms. An interesting question is whether we can design special LP solvers for decoding of LDPC codes that can take advantage of the sparsity of the constraints and other properties of the problem to converge in linear time. Section IV serves as a first step to explore the application of cutting-plane techniques in LP decoding. A desirable feature of this approach is that by changing the parameters of the algorithm we can smoothly trade complexity for performance. In contrast, if we want to get the same performance gains by tightening the relaxation in a non-adaptive setting, the required complexity increases much faster. We showed that redundant parity checks provide strong cuts, even though they may not guarantee ML performance. A major open problem is to find efficient ways to search for these cuts by exploiting their properties, and to determine specific classes of codes for which RPC cuts are more effective. Furthermore, the effectiveness of cuts generated by other techniques, such as lift-and-project cuts and Gomory cuts, as well as specially designed cuts, needs further study. REFERENCES [1] J. Feldman, M. J. Wainwright, and D. Karger, Using linear programming to decode binary linear codes, IEEE Trans. Inform. Theory, vol. 51, no. 3, pp , Mar [2] P. O. Vontobel and R. Koetter, On the relationship between linear programming decoding and min-sum algorithm decoding, Proc. Int l Symp. on Inform. Theory and Applications, pp , Parma, Italy, Oct [3] P. O. Vontobel and R. Koetter, Graph-cover decoding and finitelength analysis of message-passing iterative decoding of LDPC codes, Sumbitted to IEEE Trans. Inform. Theory. [4] M. H. Taghavi N. and P. H. Siegel, Adaptive linear programming decoding, in preparation, a draft containing the proofs is available at mtaghavi/publications. [5] GNU Linear Programming Kit, Available at [6] M. Laurent, A comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre relaxations for 0-1 programming, Mathematics of Operations Research, vol. 28, no. 3, pp , [7] R. Gomory, An algorithm for integer solutions to linear programs, Princeton IBM Mathematical Research Report, Nov

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