Abstract
We propose an innovative algorithm that iteratively evolves a particle system to approximate the sample-wised Optimal Transport plan for given continuous probability densities. Our algorithm is proposed via the gradient flow of certain functional derived from the Entropy Transport Problem constrained on probability space, which can be understood as a relaxed Optimal Transport problem. We realize our computation by designing and evolving the corresponding interacting particle system. We present theoretical analysis as well as numerical verifications to our method.
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Notes
- 1.
When \(\mu ,\nu \) are absolute continuous with respect to the Lebesgue measure on \(\mathbb {R}^d\), the optimizer of (1) is guaranteed to be unique.
- 2.
In this paper, we choose the Radial Basis Function (RBF) as the kernel: \(K(x,\xi )=\exp (-\frac{|x-\xi |^2}{2\tau ^2})\).
- 3.
Notice that we always use \(\nabla _x K\) to denote the partial derivative of K with respect to the first components.
- 4.
Here \(\mathcal {N}(\mu ,\sigma ^2)\) denotes the Gaussian distribution with mean value \(\mu \) and variace \(\sigma ^2\).
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Liu, S., Sun, H., Zha, H. (2021). A Particle-Evolving Method for Approximating the Optimal Transport Plan. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_94
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