{
 "artifacts": null,
 "category": "sampling",
 "description": "Replace nearest-center or k-modes assignment on a pool of binary neural-network samples with responsibility thresholding followed by a coordinate-wise dominance screen. Only retain a candidate mode when its assigned samples are sufficiently explained by that mode and, at every bit position, the responsibility-weighted majority agrees with the proposed center; otherwise mark the mode unreliable or discard it.",
 "formulas_latex": [
  "$$P_{\\theta}(x)=\\frac{\\alpha_{0}}{2^{n}}+\\sum_{k=1}^{K}\\alpha_{k}\\prod_{i=1}^{n}\\epsilon_{k,i}^{\\,x_{i}\\oplus(c_{k})_{i}}(1-\\epsilon_{k,i})^{1-x_{i}\\oplus(c_{k})_{i}}.$$",
  "$$r_k(x)=\\frac{\\alpha_k q_k(x)}{P_\\theta(x)},\\qquad q_k(x)=\\prod_{i=1}^{n}\\epsilon_{k,i}^{x_i\\oplus(c_k)_i}(1-\\epsilon_{k,i})^{1-x_i\\oplus(c_k)_i}.$$",
  "$$D_{j,i}=\\frac{\\sum_{b=1}^{B}r_j(x^{(b)})\\mathbf{1}\\{x_i=(c_j)_i\\}\\mathbf{1}\\{r_j(x^{(b)})\\ge\\tau\\}}{\\sum_{b=1}^{B}r_j(x^{(b)})\\mathbf{1}\\{r_j(x^{(b)})\\ge\\tau\\}},\\qquad \\text{accept }j\\text{ iff }\\min_iD_{j,i}\\ge\\delta.$$",
  "$$(c_j)_i\\leftarrow\\mathbf{1}\\!\\left\\{\\sum_{b:r_j(x^{(b)})\\ge\\tau}r_j(x^{(b)})x_i^{(b)}\\ge\\frac12\\sum_{b:r_j(x^{(b)})\\ge\\tau}r_j(x^{(b)})\\right\\}.$$"
 ],
 "id": 3023,
 "implementation": "Integrate this at the inference-time output stage of a neural combinatorial sampler, such as a GNN producing candidate MaxCut bitstrings, a binary VAE decoder, or a discrete diffusion model. Generate B binary samples x^(1),...,x^(B) for one input instance, initialize K centers using k-modes or the K most frequent unique samples, and fit alpha_k and epsilon_{k,i} with a few EM iterations. In the E-step compute q_k(x) and r_k(x) using the mixture formula above; include the uniform background source k=0 with q_0(x)=2^{-n}. In the M-step update alpha_k proportional to summed responsibilities and update epsilon_{k,i} as the responsibility-weighted disagreement rate with center bit (c_k)_i, clipped to [1e-3, 0.49]. After each EM iteration, retain only samples with r_j(x)\u003e=tau, update each center by the weighted-majority formula, and compute every D_{j,i}. Return accepted centers ranked by summed responsibility or estimated objective value, and attach a confidence score min_i D_{j,i}. Estimate uncertainty by bootstrapping the B samples and reject a center if its lower 95% confidence bound for min_i D_{j,i} is below delta. The math determines responsibilities, weighted centers, and the dominance criterion; tau, delta, K, and m_min are empirical hyperparameters. The first cheap experiment should use a small GNN or MLP sampler on 20- to 50-node synthetic MaxCut graphs, comparing raw best-of-B samples, k-modes, nearest-center refinement, and dominance-aware refinement at equal B. Measure the probability that the returned center is within a chosen objective gap of the exact optimum, unique-mode precision, and performance as B varies. Success is higher high-quality-mode precision at the same number of samples, especially when two modes overlap, with no neural forward-pass overhead.",
 "math_summary": "The paper models a binary sample x in {0,1}^n as a mixture of a uniform background source and K independent Bernoulli components centered at bitstrings c_k. In the model P_theta(x) = alpha_0/2^n + sum_{k=1}^K alpha_k q_k(x), q_k(x) = product_{i=1}^n epsilon_{k,i}^{x_i xor (c_k)_i}(1-epsilon_{k,i})^{1-x_i xor (c_k)_i}; c_k is the kth binary center, alpha_k is its mixture weight, epsilon_{k,i}\u003c1/2 is its bit-flip probability, and alpha_0 is the background weight. For a fitted parameter tuple theta, define the posterior responsibility r_k(x)=alpha_k q_k(x)/P_theta(x), the probability that sample x came from source k. For a candidate center c_j and threshold tau, retain R_j={x: r_j(x)\u003e=tau}. The adapted local dominance score at coordinate i is D_{j,i}=sum_{x in R_j} r_j(x) 1[x_i=(c_j)_i] / sum_{x in R_j} r_j(x). Accept the mode only if |R_j|\u003e=m_min and min_i D_{j,i}\u003e=delta, with delta\u003e1/2. The center can be updated by weighted majority, (c_j)_i \u003c- 1[sum_{x in R_j} r_j(x)x_i \u003e= 0.5 sum_{x in R_j}r_j(x)]. The mathematical guarantee being operationalized is that coordinate-wise source dominance above one half makes majority recovery increasingly reliable as the retained sample count grows; the implementation should estimate reliability with bootstrap confidence intervals rather than assume exact mixture parameters.",
 "math_tags": [
  "probability",
  "statistics",
  "combinatorics",
  "information-theory"
 ],
 "ml_areas": [
  "diffusion-sampling",
  "graph-nn",
  "vae",
  "inference-speedup"
 ],
 "paper": {
  "arxiv_id": "2609.01744",
  "arxiv_url": "https://arxiv.org/abs/2609.01744",
  "summary_what_math_gives_to_ml": "The paper provides a concrete post-processing principle for recovering discrete modes from noisy samples: a cluster is trustworthy only when its retained probability mass is dominated coordinate-by-coordinate by one latent source. This is stronger than nearest-center assignment, which can absorb samples from overlapping components and produce apparently good but unreliable prototypes. The transferable asset is a responsibility-weighted local purity test for binary samples, requiring no additional model evaluations. A useful neural-network application is extracting reliable modes from stochastic combinatorial predictors, binary latent models, or discrete diffusion samplers before selecting final outputs.",
  "title": "Reliable Sample-Level Quantum Error Mitigation via Dominance-Aware Clustering",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 7,
  "usefulness": 5
 },
 "solves": [
  "accuracy",
  "sample-efficiency"
 ],
 "title": "Dominance-aware mode extraction for binary neural samplers",
 "url": "https://synthcore.org/idea/3023/dominance-aware-mode-extraction-for-binary-neural-samplers",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}