Direct Estimation of Schrödinger Bridge Time-Series Drifts

This is the official experimental repository for:

Othmane Mazhar and Huyên Pham,
Direct Estimation of Schrödinger Bridge Time-Series Drifts: Finite-Sample, Asymptotic, and Adaptive Guarantees, 2026.

The paper develops direct nonparametric estimators of time-dependent Schrödinger-bridge drifts and establishes finite-sample, asymptotic, adaptive, and minimax guarantees.

• Paper: https://arxiv.org/abs/2605.05432
• Author ORCID: https://orcid.org/0000-0003-1521-9091

The repository contains:

• synthetic Schrödinger-bridge model families;
• deterministic computation of the population drift;
• kernel estimators of the bridge drift;
• finite-sample rate experiments;
• pointwise central-limit-theorem experiments;
• adaptive bandwidth-selection experiments;
• stress tests;
• YAML experiment configurations;
• saved outputs, figures, and tables used in the paper.

• Python 3.10 or later
• NumPy
• SciPy
• pandas
• Matplotlib
• PyYAML
• pytest for development and testing

Clone the repository:

git clone https://github.com/OthmaneAnalytics/sb-drift-experiments.git
cd sb-drift-experiments

Create and activate a virtual environment.

On Linux or macOS:

python -m venv .venv
source .venv/bin/activate

On Windows:

python -m venv .venv
.venv\Scripts\activate

Install the package and development dependencies:

python -m pip install --upgrade pip
python -m pip install -e ".[dev]"

Run the tests:

pytest -q

Run a preflight check for the one-dimensional Gaussian model:

python scripts/00_preflight.py --config configs/gg_1d.yaml

Run preflight checks for all configured model families:

python scripts/00_preflight.py --config configs/common.yaml --all

A small finite-sample rate experiment can be run with:

python scripts/01_rate.py \
  --config configs/gg_1d.yaml \
  --sample-sizes 200,400 \
  --reps 5 \
  --tag smoke

A small CLT experiment can be run with:

python scripts/02_clt.py \
  --config configs/gg_1d.yaml \
  --sample-sizes 200,400 \
  --reps 20 \
  --out-tag smoke \
  --no-qq

These commands are intended to verify that the installation and experiment pipeline work. They are not the full paper experiments.

Finite-sample rates and adaptive bandwidth selection:

python scripts/01_rate.py --config configs/gg_1d.yaml

Pointwise Gaussian approximation:

python scripts/02_clt.py --config configs/gg_1d.yaml

Stress-test summary:

python scripts/03_stress_summary_raw_only.py

The available configurations are:

• configs/gg_1d.yaml
• configs/gg_2d.yaml
• configs/mm_1d.yaml
• configs/mm_1d_stress_wide_strong.yaml
• configs/mm_2d.yaml

Run-specific parameters, random seeds, and output locations are controlled through the command-line arguments and YAML configuration files.

configs/             experiment configurations
figures/             paper-facing figures
paper_figure_upload/ final figure material
results/             raw and processed experimental outputs
scripts/             experiment drivers and summary scripts
src/sbdrift/         core implementation
tests/               automated tests

The principal processed outputs used for the paper are located in:

results/processed/adapt_final/
results/processed/clt_runs/
results/processed/stress/latest/
figures/

See results/README.md and FINAL_ARTIFACTS.md for additional information about saved outputs and final artifacts.

• All reported experiments were run on CPUs.
• The repository contains both final reported artifacts and exploratory outputs.
• Full experiments may require substantially more time than the smoke tests.
• The supplied random seeds and saved outputs support reproducibility of the paper figures and tables.

Please cite the associated paper when using this repository:

@misc{mazhar2026direct,
  title        = {Direct Estimation of Schrödinger Bridge Time-Series Drifts: Finite-Sample, Asymptotic, and Adaptive Guarantees},
  author       = {Mazhar, Othmane and Pham, Huyên},
  year         = {2026},
  eprint       = {2605.05432},
  archivePrefix = {arXiv},
  primaryClass = {math.ST},
  url          = {https://arxiv.org/abs/2605.05432}
}

This repository is released under the MIT Licence. See LICENSE for details.