---
license: mit
tags:
  - icml2026-repro
  - optimal-transport
  - reproduction-bundle
---

# Reproduction bundle — Quadratically Regularized Optimal Transport: Localization Bounds and Affine Case Analysis

Paper: *Quadratically Regularized Optimal Transport: Localization Bounds and Affine Case Analysis*
(ICML 2026, OpenReview `kcnuX4xEpL`, arXiv `2605.24644`).

This bundle contains everything needed to re-run the reproduction described in the Trackio logbook
[JG1310/repro-quadratically-regularized-optimal-transport-localization-bounds-and-affine](https://huggingface.co/spaces/JG1310/repro-quadratically-regularized-optimal-transport-localization-bounds-and-affine).

Both extracted claims are **proven theorems**. The reproduction is therefore an **independent
numerical audit** (per the challenge guide's theory-paper clause), not a GPU benchmark:

- **Claim 1 (Theorem 3.3):** directed-Hausdorff support cannot concentrate around the Monge graph
  faster than `ε^(1/(d+2))` — a *lower* bound.
- **Claim 2 (Theorem 3.7):** in the affine Brenier regime (incl. Gaussian→Gaussian) a sharp
  pointwise tube bound of order `ε^(1/(d+2))` holds — a matching *upper* bound.

## Contents

| Path | What it is |
|---|---|
| `DERIVATIONS.md` | Step-by-step re-derivation of both theorems (D1–D6c), each paired with an executable check |
| `scripts/derivation_checks.py` | 13 numerical audits of the algebra/lemmas (CHK-D1..D6c), CPU-only, <0.5 s |
| `scripts/exp01_affine_scaling.py` | Full-scale affine-scaling diagnostic (paper §5 / Appendix B.2–B.6) |
| `specs/exp01_affine_scaling.md` | Exact experiment spec (parameters, solvers, acceptance gate) |
| `gates.py` | Structural acceptance gate for `results/exp01.json` |
| `results/exp01.json` | Full-scale run output: 80 records, 8 summary rows (N=M=2000, R=10, d∈{100,200,500,1000}) |
| `results/GATE_REPORT.txt` | `gates.py --full` report (21/21 PASS) |
| `results/derivation_checks.log` | `derivation_checks.py` output (13/13 PASS) |
| `results/DRIVER_REPORT.json` | Driver status (`{"exp01":"PASS"}`) |
| `logs/exp01.log` | Verbatim stderr trace of the 800-solve full run |
| `BRIEF_WRITER.md`, `STATE.md` | Planner's claim→evidence map and in-regime honesty notes |

## How to re-run

```bash
python3 -m venv .venv && . .venv/bin/activate
pip install numpy scipy joblib          # derivation_checks also uses scipy

# 1) Derivation audit (seconds, CPU): re-verifies every algebraic step of both theorems.
python3 scripts/derivation_checks.py     # expect "13/13 PASS", exit 0

# 2) Full-scale empirical diagnostic (~8.5 h on 8 cores; d-independent 2000x2000 solves).
JOB_CORES=8 python3 scripts/exp01_affine_scaling.py     # writes results/exp01.json (+ work/ checkpoints)

# 3) Structural gate on the produced results file.
python3 gates.py --full                  # expect "ALL PASS (21/21)"

# Smoke test only (minutes): a d=10, N=M=200, R=2 toy of the same code path.
python3 scripts/exp01_affine_scaling.py --toy
```

`exp01` is checkpointed per `(d, seed)` unit under `work/`; a restart skips completed units, so
a kill loses at most one in-flight unit.

## Outcome

Both claims **VERIFIED** via the derivation audit (13/13 machine-precision checks, incl. an
in-regime confirmation of the shared `ε^(2/(d+2))` value-gap rate at d=1). The full-scale `exp01`
diagnostic reproduces the paper's qualitative Figure-1 pattern (β̂ decreasing with d, tight
two-solver agreement, correct order of magnitude, monotonically increasing RelErr) with **one
honestly documented quantitative discrepancy**: the RelErr zero-crossing occurs between d=100 and
d=200 here, versus the paper's reported crossing between d=500 and d=1000. Because the theorem's
strict asymptotic regime (`ε ≤ ε₀ ≈ 10^(−152)` at d=100) is unreachable in double precision for any
tested d, the entire β̂(d) curve is pre-asymptotic — the discrepancy reflects how far pre-asymptotic
effects push each implementation, not a violation of the theorems (whose correctness is established
by the derivation audit).