Linear optimal partial transport

Example code for linear optimal partial transport as introduced in the paper. The code repdocues the numerical examples in section 4 in the paper.

Required packages

Install numba, pytorch, pythonOT before clone this repository.

Background: Optimal partial transport

Choose open set $\Omega\subset\mathbb{R}^d$, let $M_+(\Omega)$ denote the set of all positive randon measures defined on $\Omega$ and select $\mu^0,\mu^1,\mu^2$. The following are the kantorovich formulation: $$OPT(\mu^0,\mu^1):=\inf_{\gamma\in\Gamma_{\leq}(\mu^0,\mu^1)}\int_{\Omega^2}|x-y|^2d\gamma(x,y)+\lambda(|\mu^0|+|\mu^1|-2|\gamma|) \hspace{1em}(1)$$

where $\Pi_{\leq}(\mu^0,\mu^1)={\gamma\in \mathcal{M}_+(\Omega^2): \gamma_0\leq \mu^0,\gamma_1\leq \mu^1},$ $\gamma_0,\gamma_1$ are the first and second marginals of $\gamma$.

Outline of repository

• lib/lin_lopt.py contains a simple OPT solver based on LP in PythonOT, basic implementations for linearized OT and OPT embeddings, and a OT barycenter solver.
• lib/geodesic.py contains basic implementations for OT geodesic, LOT geodesic, OPT interporlation and LOPT interporlation.
• lib/HK contains the LHK implementations from Linear Hellinger Kantorovich.

Examples:

• Run accuracy_test.ipynb to see the accuracy performance of LOPT distance.
• Run wall_clock_time.ipynb to see the wall clock test of OPT and LOPT.
• Run Interporlation.ipynb to see the comparison between OT geodesic, LOT geodesic, OPT interpolation, LOPT interporlations on MNIST data.
• Run pca.ipynb to see the comparison between LOT embeddings and LOPT embeddings