Verification Decoding of High-Rate LDPC Codes With Applications in Compressed Sensing

Abstract

This paper considers the performance of (j, k)-regular low-density parity-check (LDPC) codes with messagepassing (MP) decoding algorithms in the high-rate regime. In particular, we derive the high-rate scaling law for MP decoding of LDPC codes on the binary erasure channel (BEC) and the q-ary symmetric channel (q-SC). For the BEC and a fixed j, the density evolution (DE) threshold of iterative decoding scales like Θ(k −1 ) and the critical stopping ratio scales like Θ(k −j/(j−2) ). For the q-SC and a fixed j, the DE threshold of verification decoding depends on the details of the decoder and scales like Θ(k −1 ) for one decoder.Using the fact that coding over large finite alphabets is very similar to coding over the real numbers, the analysis of verification decoding is also extended to the the compressed sensing (CS) of strictly-sparse signals. A DE based approach is used to analyze the CS systems with randomized-reconstruction guarantees. This leads to the result that strictly-sparse signals can be reconstructed efficiently with high-probability using a constant oversampling ratio (i.e., when the number of measurements scales linearly with the sparsity of the signal). A stopping-set based approach is also used to get stronger (e.g., uniform-in-probability) reconstruction guarantees.

Compressed Sensing using sparse binary measurements: A rateless coding perspective

“…For simplicity and convenience of analysis, our work in this paper is based on the noiseless assumption. This is consistent with the nature of recovery process since LM1/LM2 methods cannot cope with noisy coefficients [4] [5]. Noisy extensions have been considered elsewhere [3] [14], however, we believe that there is a room for improvement in their simplicity and efficiency.…”

“…However, CSBP is rather complex both from the analysis and implementation perspective, since the messages exchanged across factor graph represent continuous probability distributions. As lowcomplexity versions of CSBP amenable to rigorous analysis, VB decoding algorithms (such as LM1 and LM2) have been investigated in noiseless CS scenario in [4] [5]. Using the CS setup analogue to LDPC codes, in these works, codingtheoretic tools such as the density evolution (DE) and stopping set analysis are applied to assess the CS recovery performance.…”

Interval-Passing Algorithm for Chemical Mixture Estimation

“…The first class of message-passing algorithms, such as the approximate message-passing algorithm (AMP) presented in [13], are iterative thresholding algorithms. Other messagepassing algorithms exist in the literature such as list decoding and multiple-basis belief propagation [14], sparse matching pursuit [15], verification decoding [16] or interval-passing [8] that we present in the next section.…”

“…Notably, it was shown that the IPA is a good complexity/performance trade-off between LP reconstruction and simple verification decoding [16]. The IPA is an iterative algorithm.…”

Explicit matrices for sparse approximation

“…We note that even though our analysis involves a rigorous density evolution argument, our decoder is always the basis pursuit linear relaxation which is substantially different from the related work on message-passing algorithms for compressed sensing [15], [28].…”