### Abstract

We present a particular formulation of optimal transport for matrix-valued density functions. Our aim is to devise a geometry which is suitable for comparing power spectral densities of multivariable time series. More specifically, the value of a power spectral density at a given frequency, which in the matricial case encodes power as well as directionality, is thought of as a proxy for a 'matrix-valued mass density.' Optimal transport aims at establishing a natural metric in the space of such matrix-valued densities which takes into account differences between power across frequencies as well as misalignment of the corresponding principle axes. Thus, our transportation cost includes a cost of transference of power between frequencies together with a cost of rotating the principle directions of matrix densities. The two end-point matrix-valued densities can be thought of as marginals of a joint matrix-valued density on a tensor product space. This joint density, very much as in the classical Monge-Kantorovich setting, can be thought to specify the transportation plan. Contrary to the classical setting, the optimal transport plan for matrices is no longer supported on a thin zero-measure set.

Original language | English (US) |
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Article number | 6881695 |

Pages (from-to) | 373-382 |

Number of pages | 10 |

Journal | IEEE Transactions on Automatic Control |

Volume | 60 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2015 |

### Keywords

- Convex optimization
- matrix-valued density functions
- optimal mass-transport

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## Cite this

*IEEE Transactions on Automatic Control*,

*60*(2), 373-382. [6881695]. https://doi.org/10.1109/TAC.2014.2350171