On the Geometry of Covariance Matrices

Ning, Lipeng; Jiang, Xianhua; Georgiou, Tryphon T.

doi:10.1109/lsp.2013.2266273

Cited by 26 publications

(19 citation statements)

References 19 publications

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“…First, it is clearly (9) implies (8). We show that (8) also implies (9). The proof below follows that in [7, page 216] for the scalar-valued Wasserstein metric.…”

Section: Appendix a Proof Of Propositionmentioning

confidence: 53%

“…This is clearly a desirable feature in any experimental engineering quantification of distances between distributions, whether these represent probability, power, spectral power or other entities. For this precise reason, the Wasserstein metric has turned out to be a useful tool in modeling of slowly varying time-series [8] and for comparing covariance matrices [9], among many other applications [6].…”

Section: Introductionmentioning

confidence: 99%

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Metrics for Matrix-valued Measures via Test Functions

Ning

Georgiou

2014

53rd IEEE Conference on Decision and Control

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It is perhaps not widely recognized that certain common notions of distance between probability measures have an alternative dual interpretation which compares corresponding functionals against suitable families of test functions. This dual viewpoint extends in a straightforward manner to suggest metrics between matrix-valued measures. Our main interest has been in developing weakly-continuous metrics that are suitable for comparing matrix-valued power spectral density functions. To this end, and following the suggested recipe of utilizing suitable families of test functions, we develop a weakly-continuous metric that is analogous to the Wasserstein metric and applies to matrix-valued densities. We use a numerical example to compare this metric to certain standard alternatives including a different version of a matricial Wasserstein metric developed in [1], [2]. I |dµ 1 (x) − dµ 2 (x)|, L. Ning is with Brigham and Women'

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“…First, it is clearly (9) implies (8). We show that (8) also implies (9). The proof below follows that in [7, page 216] for the scalar-valued Wasserstein metric.…”

Section: Appendix a Proof Of Propositionmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

Metrics for Matrix-valued Measures via Test Functions

Ning

Georgiou

2014

53rd IEEE Conference on Decision and Control

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“…The much lower estimation error corresponding to P wls t is because of its capability in tracking rotations in the covariance. The general theme of this paper is closely related to the work in [12], [18]- [22] which all focused on investigating smooth paths connecting positive definite matrices. The proposed approach is based on different regularizations functions of system matrices which distinguishes this paper from early work.…”

Section: Draftmentioning

confidence: 99%

Regularization of time-varying covariance matrices using linear stochastic systems

Ning

2018

Preprint

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This work focuses on modeling of time-varying covariance matrices using the state covariance of linear stochastic systems. Following concepts from optimal mass transport and the Schrödinger bridge problem (SBP), we investigate several covariance paths induced by different regularizations on the system matrices. Specifically, one of the proposed covariance path generalizes the geodesics based on the Fisher-Rao metric to the situation with stochastic input. Another type of covariance path is generated by linear system matrices with rotating eigenspace in the noiseless situation. The main contributions of this paper include the differential equations for these covariance paths and the closed-form expressions for scalar-valued covariance. We also compare these covariance paths using several examples.

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“…DC has a closed form expression in terms of R and R g , (see e.g. [33], [34]) and is given by:

DC = trace (R + R_{g} - 2 {false(R_{g}^{\frac{1}{2}} R R_{g}^{\frac{1}{2}} false)}^{\frac{1}{2}}) .

…”

Section: Scalar Indices Derived From the Eapmentioning

confidence: 99%

Estimating Diffusion Propagator and Its Moments Using Directional Radial Basis Functions

Ning

Westin

Rathi

2015

IEEE Trans. Med. Imaging

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The ensemble average diffusion propagator (EAP) obtained from diffusion MRI (dMRI) data captures important structural properties of the underlying tissue. As such, it is imperative to derive an accurate estimate of the EAP from the acquired diffusion data. In this work, we propose a novel method for estimating the EAP by representing the diffusion signal as a linear combination of directional radial basis functions scattered in q-space. In particular, we focus on a special case of anisotropic Gaussian basis functions and derive analytical expressions for the diffusion orientation distribution function (ODF), the return-to-origin probability (RTOP), and mean-squared-displacement (MSD). A significant advantage of the proposed method is that the second and the fourth order moment tensors of the EAP can be computed explicitly. This allows for computing several novel scalar indices (from the moment tensors) such as mean-fourth-order-displacement (MFD) and generalized kurtosis (GK) – which is a generalization of the mean kurtosis measure used in diffusion kurtosis imaging. Additionally, we also propose novel scalar indices computed from the signal in q-space, called the q-space mean-squared-displacement (QMSD) and the q-space mean-fourth-order-displacement (QMFD), which are sensitive to short diffusion time scales. We validate our method extensively on data obtained from a physical phantom with known crossing angle as well as on in-vivo human brain data. Our experiments demonstrate the robustness of our method for different combinations of b-values and number of gradient directions.

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On the Geometry of Covariance Matrices

Cited by 26 publications

References 19 publications

Metrics for Matrix-valued Measures via Test Functions

Metrics for Matrix-valued Measures via Test Functions

Regularization of time-varying covariance matrices using linear stochastic systems

Estimating Diffusion Propagator and Its Moments Using Directional Radial Basis Functions

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