Nissim Halabi

LP Decoding of Regular LDPC Codes in Memoryless Channels

We study error bounds for linear programming decoding of regular low-density parity-check (LDPC) codes. For memoryless binary-input output-symmetric channels, we prove bounds on the word error probability that are inverse doubly exponential in the girth of the factor graph. For memoryless binary-input AWGN channel, we prove lower bounds on the threshold for regular LDPC codes whose factor graphs have logarithmic girth under LP-decoding. Specifically, we prove a lower bound of σ = 0.735 (upper bound of [(Eb)/(N0)]=2.67 dB) on the threshold of (3, 6)-regular LDPC codes whose factor graphs have logarithmic girth. Our proof is an extension of a recent paper of Arora, Daskalakis, and Steurer [STOC 2009] who presented a novel probabilistic analysis of LP decoding over a binary symmetric channel. Their analysis is based on the primal LP representation and has an explicit connection to message passing algorithms. We extend this analysis to any MBIOS channel.

On Decoding Irregular Tanner Codes With Local-Optimality Guarantees

We consider decoding of binary linear Tanner codes using message-passing iterative decoding and linear-programming (LP) decoding in memoryless binary-input output-symmetric (MBIOS) channels. We present new certificates that are based on a combinatorial characterization for the local optimality of a codeword in irregular Tanner codes with respect to any MBIOS channel. This characterization is a generalization of (Arora , Proc. ACM Symp. Theory of Computing, 2009) and (Vontobel, Proc. Inf. Theory and Appl. Workshop, 2010) and is based on a conical combination of normalized weighted subtrees in the computation trees of the Tanner graph. These subtrees may have any finite height h (even equal or greater than half of the girth of the Tanner graph). In addition, the degrees of local-code nodes in these subtrees are not restricted to two (i.e., these subtrees are not restricted to skinny trees). We prove that local optimality in this new characterization implies maximum-likelihood (ML) optimality and LP optimality, and show that a certificate can be computed efficiently. We also present a new message-passing iterative decoding algorithm, called normalized weighted min-sum (NWMS). NWMS decoding is a belief-propagation (BP) type algorithm that applies to any irregular binary Tanner code with single parity-check local codes (e.g., low-density and high-density parity-check codes). We prove that if a locally optimal codeword with respect to height parameter h exists (whereby notably h is not limited by the girth of the Tanner graph), then NWMS decoding finds this codeword in h iterations. The decoding guarantee of the NWMS decoding algorithm applies whenever there exists a locally optimal codeword. Because local optimality of a codeword implies that it is the unique ML codeword, the decoding guarantee also provides an ML certificate for this codeword. Finally, we apply the new local-optimality characterization to regular Tanner codes, and prove lower bounds on the noise thresholds of LP decoding in MBIOS channels. When the noise is below these lower bounds, the probability that LP decoding fails to decode the transmitted codeword decays doubly exponentially in the girth of the Tanner graph.

LP decoding of regular LDPC codes in memoryless channels

We study error bounds for linear programming decoding of regular LDPC codes. For memoryless binary-input output-symmetric channels, we prove bounds on the word error probability that are inverse doubly-exponential in the girth of the factor graph. For memoryless binary-input AWGN channels, we derive lower bounds on the thresholds for regular LDPC codes under LP decoding. Specifically, we prove a lower bound of σ = 0.735 on the threshold of (3, 6)-regular LDPC codes with logarithmic girth.

Linear-programming decoding of Tanner codes with local-optimality certificates

Given a channel observation y and a codeword x, we are interested in a one-sided error test that answers the questions: is x optimal with respect to y? is it unique? A positive answer for such a test is called a certificate for the optimality of a codeword. We present new certificates that are based on combinatorial characterization for local-optimality of a codeword in irregular Tanner codes. The certificate is based on weighted normalized trees in computation trees of the Tanner graph. These trees may have any finite height h (even greater than the girth of the Tanner graph). In addition, the degrees of local-code nodes are not restricted to two (i.e., skinny trees). We prove that local-optimality in this new characterization implies ML-optimality and LP-optimality, and show that a certificate can be computed efficiently. We apply the new local-optimality characterization to regular Tanner codes, and prove lower bounds on the noise thresholds of LP-decoding in MBIOS channels. When the noise is below these lower bounds, the probability that LP-decoding fails decays doubly exponentially in the girth of the Tanner graph.

Hierarchies of local-optimality characterizations in decoding Tanner codes

Recent developments in decoding Tanner codes with maximum-likelihood certificates are based on a sufficient condition called local optimality. We define hierarchies of locally optimal codewords with respect to two parameters. One parameter is related to the minimum distance of the local codes in Tanner codes. The second parameter is related to the finite number of iterations used in iterative decoding. We show that these hierarchies satisfy inclusion properties as these parameters are increased. In particular, this implies that a codeword that is decoded with a certificate using an iterative decoder after h iterations is decoded with a certificate after k·h iterations, for every integer k.

Analysis of the Min-Sum Algorithm for Packing and Covering Problems via Linear Programming

Message-passing algorithms based on belief-propagation (BP) are successfully used in many applications, including decoding error correcting codes and solving constraint satisfaction and inference problems. The BP-based algorithms operate over graph representations, called factor graphs, that are used to model the input. Although in many cases, the BP-based algorithms exhibit impressive empirical results, not much has been proved when the factor graphs have cycles. This paper deals with packing and covering integer programs in which the constraint matrix is zero-one, the constraint vector is integral, and the variables are subject to box constraints. We study the performance of the min-sum algorithm when applied to the corresponding factor graph models of packing and covering linear programmings (LPs). We compare the solutions computed by the min-sum algorithm for packing and covering problems to the optimal solutions of the corresponding LP relaxations. In particular, we prove that if the LP has an optimal fractional solution, then for each fractional component, the minsum algorithm either computes multiple solutions or the solution oscillates below and above the fraction. This implies that the min-sum algorithm computes the optimal integral solution only if the LP has a unique optimal solution that is integral. The converse is not true in general. For a special case of packing and covering problems, we prove that if the LP has a unique optimal solution that is integral and on the boundary of the box constraints, then the min-sum algorithm computes the optimal solution in pseudopolynomial time. Our results unify and extend recent results for the maximum weight matching problem and for the maximum weight independent set problem.

LOCAL-OPTIMALITY GUARANTEES BASED ON PATHS FOR OPTIMAL DECODING ∗

This paper presents a unified analysis framework that captures recent advances in the study of local-optimality characterizations for codes on graphs. These local-optimality characterizations are based on combinatorial structures embedded in the Tanner graph of the code. Local optimality implies both unique maximum likelihood optimality and unique linear programming (LP) decoding optimality. Also, an iterative message-passing decoding algorithm is guaranteed to find the unique locally optimal codeword if one exists. We demonstrate an instance of this proof technique by considering a definition of local optimality that is based on the simplest combinatorial structures in Tanner graphs, namely, paths of length

Local-optimality guarantees for optimal decoding based on paths

h

Message-passing decoding beyond the girth with local-optimality guarantees

. We apply the technique of local optimality to binary Tanner codes (including any low-density parity-check code, and in particular any irregular repeat-accumulate code with both even and odd repetition factors). Inverse polynomial bounds in the code length are proved on the word error probability ...

Improved bounds on the word error probability of RA(2) codes with linear-programming-based decoding

This paper presents a unified analysis framework that captures recent advances in the study of local-optimality characterizations for codes on graphs. These local-optimality characterizations are based on combinatorial structures embedded in the Tanner graph of the code. Local-optimality implies both maximum-likelihood (ML) optimality and linear-programming (LP) decoding optimality. Also, an iterative message-passing decoding algorithm is guaranteed to find the unique locally-optimal codeword, if one exists. We demonstrate this proof technique by considering a definition of local optimality that is based on the simplest combinatorial structures in Tanner graphs, namely, paths of length h. We apply the technique of local optimality to a family of Tanner codes. Inverse polynomial bounds in the code length are proved on the word error probability of LP-decoding for this family of Tanner codes.