# Static Auxiliary-Graph Ensemble

- ID: 788
- Canonical URL: https://synthcore.org/idea/788/static-auxiliary-graph-ensemble
- API JSON: https://synthcore.org/api/idea/788.json
- API Markdown: https://synthcore.org/api/idea/788.md
- Verification status: mechanism_failed
- Source: [arXiv:2607.12187](https://arxiv.org/abs/2607.12187)
- Category: architecture
- Solves: accuracy, stability
- ML areas: graph-nn, training, inference-speedup
- Math tags: linear-algebra, graph-theory, probability
- Ratings: usefulness 7/10; difficulty 5/10; novelty 7/10

## Idea description

Run the same neural decoder over several algebraically equivalent augmented graphs and aggregate their variable-level predictions. Each graph exposes different cycle structure and message routes, providing structured architectural diversity rather than ordinary random-seed ensembling.

## Mathematical statement

For each admissible binary transformation A_q, the augmented pair satisfies H_q = A_q H and s_q = A_q s, where H is the original parity-check matrix, s is the syndrome, and q indexes graph variants. Thus H_q e = s_q whenever H e = s. Let z_qj be the decoder logit for variable j from graph q and let w_q be a nonnegative graph weight with sum_q w_q = 1. The ensemble logit is z_j = sum_q w_q z_qj and p_j = sigmoid(z_j). A residual-weighted variant uses r_q as the number or soft magnitude of violated augmented checks and sets w_q proportional to exp(-beta r_q), where beta is a tunable reliability coefficient. The paper's repeated-round evaluation uses p_L = 1 - (1 - LER)^(1/r), where LER is the logical error rate and r is the number of syndrome rounds.

## Key formulas

- $$\mathbf{H}_{A}=\mathbf{A}\mathbf{H},\qquad \mathbf{s}_{A}=\mathbf{A}\mathbf{s}$$
- $$\mathbf{H}_{A}\mathbf{e}=\mathbf{A}\mathbf{H}\mathbf{e}=\mathbf{A}\mathbf{s}=\mathbf{s}_{A}$$
- $$z_j=\sum_{q=1}^{E}w_q z_{qj},\qquad p_j=\sigma(z_j),\qquad w_q\ge 0,\quad \sum_{q=1}^{E}w_q=1$$
- $$p_{L}=1-(1-\mathrm{LER})^{1/r}$$

## Implementation notes

Integrate this as an inference-time ensemble wrapper around a neural BP or GNN decoder, using shared weights to limit parameter and memory overhead. Generate E sparse transformations A_q offline. Each should contain the identity rows plus independently selected low-weight combinations of existing checks, chosen to reduce length-4 cycles while keeping H_q sparse. Precompute H_q = (A_q H) mod 2 and compute s_q = (A_q s) mod 2 for every input syndrome. Pseudocode: for q in 1..E, run the shared decoder f_theta(channel, H_q, s_q) for T iterations and obtain variable logits z_q; set z = mean_q z_q and decode sign(z). Optionally compute a soft residual r_q from H_q and sigmoid(z_q), set w_q = softmax(-beta r_q), and use z = sum_q w_q z_q. Train with per-graph binary cross-entropy plus an ensemble loss and syndrome-consistency penalty. The paper-derived component is the algebraically equivalent graph family; logit aggregation and residual weighting are the neural adaptation. Begin with E=4 and E=8 on a small LDPC or quantum-DEM-inspired sparse binary benchmark. Compare against one decoder, a wider decoder with equal FLOPs, and ordinary seed ensembling. Measure block or logical error rate, calibration, error versus iteration count, and latency. Success is at least a 20% error reduction at fixed total message evaluations or equal error with fewer iterations per graph.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a readable MVP for the Static Auxiliary-Graph Ensemble in experiment.py, including GF(2) row-transform generation, exact syndrome-equivalence checks, logit aggregation, and a sparse parity-decoding mini-experiment. The mechanism predictions were confirmed: transformed syndromes were exact, averaged independent-logit variance matched sigma^2/E within about 0.6%, and repeated-round LER inversion was numerically exact. However, the practical decoder gain was weak: BER was 0.06092 for one graph, 0.05779 for four auxiliary graphs, 0.05729 for eight, and 0.05823 for four ordinary seeds, so the stated 20% target and clear structured-diversity advantage were not observed.

### Mechanism check

- Verdict: Built a readable MVP for the Static Auxiliary-Graph Ensemble in experiment.py, including GF(2) row-transform generation, exact syndrome-equivalence checks, logit aggregation, and a sparse parity-decoding mini-experiment. The mechanism predictions were confirmed: transformed syndromes were exact, averaged independent-logit variance matched sigma^2/E within about 0.6%, and repeated-round LER inversion was numerically exact. However, the practical decoder gain was weak: BER was 0.06092 for one graph, 0.05779 for four auxiliary graphs, 0.05729 for eight, and 0.05823 for four ordinary seeds, so the stated 20% target and clear structured-diversity advantage were not observed.
- Confidence: 9/10
- Limitations: This is a synthetic, non-trained message-voting decoder rather than a trained BP/GNN decoder; no LDPC benchmark, calibration, latency/FLOP measurement, residual weighting, wider equal-FLOP model, or multi-round logical-error experiment was tested. The toy uses an equivalent transformed syndrome computed directly from the sampled error, and graph diversity was not optimized for cycle reduction.

## Artifacts

- [experiment.py](https://synthcore.org/code/322/experiment.py)
- [report.md](https://synthcore.org/code/322/report.md)
- [Download all files as ZIP](https://synthcore.org/download/322)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.