Averaging Symmetric Positive-Definite Matrices

Yuan, Xinru; Huang, Wen; Absil, Pierre-Antoine; Gallivan, Kyle A.

doi:10.1007/978-3-030-31351-7_20

Cited by 1 publication

(1 citation statement)

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“…mislabeled data) since it uses the SCM as an estimator. Moreover, other mean computation can be used, such as the Riemannian mean which benefits from many properties compared to its Euclidean counterpart [5]. We propose a metric learning framework that jointly estimates regularized covariance matrices, in a robust manner, while computing their Riemannian mean.…”

Section: Motivations and Contributionsmentioning

confidence: 99%

Robust Geometric Metric Learning

Collas¹,

Breloy²,

Ginolhac³

et al. 2022

Preprint

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This paper proposes new algorithms for the metric learning problem. We start by noticing that several classical metric learning formulations from the literature can be viewed as modified covariance matrix estimation problems. Leveraging this point of view, a general approach, called Robust Geometric Metric Learning (RGML), is then studied. This method aims at simultaneously estimating the covariance matrix of each class while shrinking them towards their (unknown) barycenter. We focus on two specific costs functions: one associated with the Gaussian likelihood (RGML Gaussian), and one with Tyler's M -estimator (RGML Tyler). In both, the barycenter is defined with the Riemannian distance, which enjoys nice properties of geodesic convexity and affine invariance. The optimization is performed using the Riemannian geometry of symmetric positive definite matrices and its submanifold of unit determinant. Finally, the performance of RGML is asserted on real datasets. Strong performance is exhibited while being robust to mislabeled data.

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Section: Motivations and Contributionsmentioning

confidence: 99%