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Abstract
The optimal transport problem seeks a function y(x), with x and y in R^n, which maps a probability density rho(x) into another, mu(y), while minimizing a cost function C[y(x)]. We will discuss the frequently occurring situation in which rho(x) and mu(y) are only known through samples {x_i}, {y_j}, as well as extensions to more general scenarios where a plan pi(x_1,x_2,x_3,...) is sought with marginals rho_j(x_j) (here each x_j is an n-dimensional vector and the marginals are only known through samples) that minimizes a cost C[pi]. Applications include resource allocation, density estimation, medical diagnosis and treatment, portfolio optimization and fluid-flow reconstruction from tracers. |
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