{
 "artifacts": null,
 "category": "training",
 "description": "Use a small set of known or trusted pairwise costs to estimate the effective entropic temperature of a Sinkhorn attention or mixture-of-experts routing layer directly from its observed transport plan. This provides a calibration controller that can detect over-concentrated routing and adjust epsilon without backpropagating through a costly temperature search.",
 "formulas_latex": [
  "$$B=-H_m\\log(W+\\delta)H_n=\\frac{H_mCH_n}{\\varepsilon}$$",
  "$$\\widehat{\\varepsilon}=\\frac{\\sum_{(i,j)\\in\\Omega}B_{ij}C_{\\mathrm{centered},ij}}{\\sum_{(i,j)\\in\\Omega}B_{ij}^{2}},\\qquad C_{\\mathrm{centered}}=H_mCH_n$$",
  "$$R=\\frac{\\sum_{(i,j)\\in\\Omega}\\left(\\widehat{\\varepsilon}B_{ij}-C_{\\mathrm{centered},ij}\\right)^2}{\\sum_{(i,j)\\in\\Omega}C_{\\mathrm{centered},ij}^{2}}$$",
  "$$\\varepsilon_{t+1}=\\operatorname{clip}\\left(\\varepsilon_t\\left(\\frac{\\widehat{\\varepsilon}_t}{\\varepsilon_t}\\right)^\\alpha,\\varepsilon_{\\min},\\varepsilon_{\\max}\\right),\\qquad 0\u003c\\alpha\\leq 1$$"
 ],
 "id": 3001,
 "implementation": "Apply this controller to the temperature of a Sinkhorn attention head or an MoE routing matrix. Trusted costs can come from synthetic teacher geometry, distances between token metadata, class relations, or a small routing table; only a sparse index set Omega is required. Every T training steps, compute W, form B = -H_m log(W plus delta) H_n, gather B and centered reference costs on Omega, and calculate epsilon_hat using the closed-form regression formula. Compute R as the normalized residual. If R is below a threshold, update the live temperature with the damped geometric rule and clip it to safe limits. If R is high, freeze epsilon, increase Sinkhorn iterations, and log a model-mismatch warning rather than applying an unreliable update. The closed-form estimator and residual are the paper mechanism; damping, clipping, threshold, and update frequency are empirical design choices. First test on a small Transformer for synthetic sequence classification in which pairwise costs are Euclidean distances and 5 percent of costs are exposed. Compare fixed epsilon, gradient-learned epsilon, and inverse-OT calibration. The prediction is epsilon_hat divided by true epsilon within 10 percent when R is below 0.01. With independent observation noise, estimation error should decrease approximately as the inverse square root of the number of trusted entries. R should increase before routing entropy collapses when the learned attention no longer follows an entropic OT model.",
 "math_summary": "Let W be the observed transport plan, delta a positive numerical floor, and B = -H_m log(W plus delta) H_n. If a subset Omega of cost entries is known, the inverse-transport relation implies B_ij = C_centered,ij divided by epsilon, where C_centered = H_m C H_n. The least-squares temperature estimate minimizes the squared discrepancy between epsilon B_ij and C_centered,ij over Omega. Its closed form is epsilon_hat = sum over Omega of B_ij C_centered,ij divided by sum over Omega of B_ij squared. A normalized residual R measures whether the observed plan is consistent with the entropic model. A low R indicates reliable calibration; a high R indicates model mismatch, insufficient Sinkhorn convergence, or noisy cost references.",
 "math_tags": [
  "optimal-transport",
  "statistics",
  "linear-algebra",
  "information-theory"
 ],
 "ml_areas": [
  "attention",
  "transformer",
  "moe-routing",
  "scheduler"
 ],
 "paper": {
  "arxiv_id": "2609.01278",
  "arxiv_url": "https://arxiv.org/abs/2609.01278",
  "summary_what_math_gives_to_ml": "The paper provides an exact inverse map for entropic optimal transport: from an observed coupling W, the cost matrix can be recovered up to the unavoidable row-plus-column gauge by applying double-centering to log W. This removes the Sinkhorn scaling potentials without iterative optimization, and the same structure permits estimating the entropic temperature when some true cost entries are known. The strongest neural-network transfer is a Sinkhorn-constrained attention or routing layer whose observed transport plan is converted into a gauge-fixed cost matrix for initialization, distillation, monitoring, or adaptive temperature control. The key falsifiable signature is exact affine recovery of centered costs, with degradation only when entries of W approach numerical zero.",
  "title": "An exact and fast solution of the inverse Regularized Optimal Transport problem",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 8,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Sparse-Cost Temperature Calibration",
 "url": "https://synthcore.org/idea/3001/sparse-cost-temperature-calibration",
 "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
   }
  }
 }
}