4 The ADMM penalized decoder for LDPC codes∗
Bibliographic record
Abstract
Linear programming (LP) decoding for low-density parity-check codes was introduced by Feldman et al. and has been shown to have theoretical guarantees in several regimes. Furthermore, it has been reported in the literature-via simulation and via instanton analysis-that LP decoding displays better error rate performance at high signal-to-noise ratios (SNR) than does belief propagation (BP) decoding. However, at low SNRs, LP decoding is observed to have worse performance than BP. In this paper, we seek to improve LP decoding at low SNRs while maintaining LP decoding's high SNR performance. Our main contribution is a new class of decoders obtained by applying the alternating direction method of multipliers (ADMM) algorithm to a set of non-convex optimization problems. These non-convex problems are constructed by adding a penalty term to the objective of LP decoding. The goal of the penalty is to make pseudocodewords, which are non-integer vertices of the LP relaxation, more costly. We name this class of decoders-ADMM penalized decoders. For low and moderate SNRs, we simulate ADMM penalized decoding with ℓ 1 and ℓ 2 penalties. We find that these decoders can outperform both BP and LP decoding. For high SNRs, where it is difficult to obtain data via simulation, we use an instanton analysis and find that, asymptotically, ADMM penalized decoding performs better than BP but not as well as LP. Unfortunately, since ADMM penalized decoding is not a convex program, we have not been successful in developing theoretical guarantees. However, the non-convex program can be approximated using a sequence of linear programs; an approach that yields a reweighted LP decoder. We show that a two-round reweighted LP decoder has an improved theoretical recovery threshold when compared with LP decoding. In addition, we find via simulation that reweighted LP decoding significantly attains lower error rates than LP decoding at low SNRs.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".