On the Private Estimation of Smooth Transport Maps
Authors: Clément Lalanne, Franck Iutzeler, Jean-Michel Loubes, Julien Chhor
ICML 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | This section focuses on how to numerically approximate ˆfpriv and ˆTpriv defined in (20) and presents numerical results to illustrate the proposed method. [...] This is illustrated in Figure 1, where we represent two kernel density estimators constructed on the datasets X1:n and Y1:n, respectively. We then apply the algorithm defined in Section 6.1 where F is a collection of T independent potentials that are generated according to the same distribution as the true potential. The result of the algorithm and the optimal transport map are represented in Figure 2. This simulation confirms that the estimated transport map is close to the optimal one. |
| Researcher Affiliation | Academia | 1Institut de Math ematiques de Toulouse, UMR5219, Universit e de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France 2INRIA, France 3Toulouse School of Economics, Universit e Toulouse Capitole, France. |
| Pseudocode | No | This procedure leads to the following practical algorithm: Start with a finite family of candidate discretized potentials F = f(grid),1, . . . , f(grid),N , (28) then compute ˆi argmin ˆSC (grid)(f(grid),i|X(grid) 1:n , Y (grid) 1:n ) + 4C (29) where for any f, X1:n and Y1:n ˆSC (grid)(f|X1:n, Y1:n) := 1 i=1 Proj[ C,C] f(Xi) i=1 Proj[ C,C] f (grid)(Yi) (30) and where L1, . . . , LN are independent Laplace random variables. By Lemma 3.2, ˆi is ϵ-DP, and by post-processing, so is ˆT := (grid)f(grid),ˆi, which we use as an estimator. |
| Open Source Code | Yes | The code used to generate those results can be found at /https://github.com/clemlal/ Private Smooth Transport Map RNArg Min. |
| Open Datasets | No | We then sample n i.i.d. uniform random variables X1:n over [ 1/2, 1/2]2, and generate the observations Y1:n by applying the gradient of the previously defined potential to n new i.i.d. uniform random variables over [ 1/2, 1/2]2. This is illustrated in Figure 1, where we represent two kernel density estimators constructed on the datasets X1:n and Y1:n, respectively. |
| Dataset Splits | No | The paper describes generating synthetic data by sampling i.i.d. uniform random variables and transforming them. It does not mention any explicit training, validation, or test splits for this generated data. |
| Hardware Specification | No | The paper mentions numerical simulations and experiments but does not provide any specific details about the hardware used, such as CPU or GPU models, or memory. |
| Software Dependencies | No | The paper does not explicitly state any software dependencies with specific version numbers. It mentions implementing an algorithm and generating results but provides no details on the programming languages, libraries, or frameworks used. |
| Experiment Setup | Yes | For the numerical values, Ω(grid) is a uniform grid in [ 1/2, 1/2]2 with 642 points, ϵ = 1., n = 200000, α1 = α2 = 0.005, σ = σ1 = σ2 = 0.1, T = 2000, and C = 0.25. |