Differential Private Optimal Transport Estimation

---
name: differential-private-optimal-transport-estimation
description: "Differentially private estimation of smooth optimal transport maps using wavelet density estimators and stability bounds. Privacy-preserving statistical methodology for OT map estimation. Activation: differential privacy, optimal transport, private estimation, wavelet density, minimax estimation."
---

## Context
Estimating smooth OT maps between probability distributions under differential privacy constraints. Uses wavelet-based density estimators and stability bounds.
Source: arXiv:2606.04683v1

## Core Methodology
1. Construct wavelet-based density estimators for source/target distributions
2. Apply DP noise to wavelet coefficients at appropriate scales
3. Leverage stability bounds for smooth OT maps to bound estimation error
4. Achieve minimax optimal rates for private OT estimation
5. Decompose error into bias, variance, and privacy noise

## Implementation
1. Choose wavelet basis for distribution smoothness class
2. Estimate wavelet coefficients from samples
3. Add calibrated Laplace/Gaussian noise for DP
4. Reconstruct private density estimates
5. Compute OT map between private densities
6. Validate against theoretical bounds

## Pitfalls
- Privacy budget allocation across scales is critical
- Smoothness assumptions may not hold universally
- Wavelet basis choice affects quality
- High-dimensional OT suffers from curse of dimensionality

## Verification
1. Verify ε-DP guarantee holds
2. Compare error against minimax lower bounds
3. Test on synthetic distributions with known OT maps
4. Robustness to varying sample sizes and privacy budgets

## Activation
differential privacy, optimal transport, private estimation, wavelet density, minimax estimation, smooth OT maps, privacy-preserving statistics