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.2015 Feb;60(2):373-382.
doi: 10.1109/TAC.2014.2350171. Epub 2014 Aug 21.

On Matrix-Valued Monge-Kantorovich Optimal Mass Transport

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On Matrix-Valued Monge-Kantorovich Optimal Mass Transport

Lipeng Ning et al. IEEE Trans Automat Contr.2015 Feb.

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 endpoint 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.

Keywords: Convex optimization; matrix-valued density functions; optimal mass-transport.

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Figures

Fig. 1
Fig. 1
Subplots (1,1), (1,2), and (2,2) showμi(1, 1), |μi(1, 2)| (same as |μi(2, 1)|) andμi(2, 2). Subplot (2,1) shows ∠(μi(2, 1)) fori ∈ {0, 1} in blue and red, respectively.
Fig. 2
Fig. 2
Interpolated resultsμτk fork = 0, …, 8 computed from (17) withμ0 andμ1 as the two boundary points: subplots (1, 1), (1, 2), and (2, 2) showμτk (1, 1), |μτk (1, 2)| (same as |μτk (2, 1)|) andμτk (2, 2), subplot (2, 1) shows ∠(μτk (2, 1)).
Fig. 3
Fig. 3
(a) Shows the estimated spectrogram of the observed time series and (b) corresponds to the geodesic-fitted spectrogram.
Fig. 4
Fig. 4
In (a), the trajectories of the dominant eigenvector ofμ̂τk (θ1) andμ̂τk (θ2) are shown in red and blue, respectively. The corresponding trajectories ofμτk (θ1) andμτk (θ2) are shown in (b).
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References

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