| Summary: | We consider the numerical solution of the discrete multi-marginal optimal transport (MOT) by means of the Sinkhorn algorithm. In general, the Sinkhorn algorithm suffers from the curse of dimensionality with respect to the number of marginals. If the MOT cost function decouples according to a tree or circle, its complexity is linear in the number of marginal measures. In this case, we speed up the convolution with the radial kernel required in the Sinkhorn algorithm via non-uniform fast Fourier methods. Each step of the proposed accelerated Sinkhorn algorithm with a tree-structured cost function has a complexity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mi>K</mi><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula> instead of the classical <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mi>K</mi><msup><mi>N</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> for straightforward matrix–vector operations, where <i>K</i> is the number of marginals and each marginal measure is supported on, at most, <i>N</i> points. In the case of a circle-structured cost function, the complexity improves from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mi>K</mi><msup><mi>N</mi><mn>3</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mi>K</mi><msup><mi>N</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula>. This is confirmed through numerical experiments.
|
|---|