Scale and curvature effects in principal geodesic analysis

Lazar, Drew; Lin, Lizhen

doi:10.1016/j.jmva.2016.09.009

Cited by 6 publications

(4 citation statements)

References 21 publications

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“…The variation of X around the intrinsic mean is summarized by the covariance operator, which is a linear characterization of the intrinsic variation that has been applied to generalize the principal component analysis to Riemannian manifolds (Fletcher et al, 2004;Lazar and Lin, 2017). The population and empirical covariance operators are defined as…”

Section: Tangent Vectors and The Covariance Operatormentioning

confidence: 99%

Statistical inference on the Hilbert sphere with application to random densities

Dai

2022

Electron. J. Statist.

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The infinite-dimensional Hilbert sphere S ∞ has been widely employed to model density functions and shapes, extending the finitedimensional counterpart. We consider the Fréchet mean as an intrinsic summary of the central tendency of data lying on S ∞ . For sound statistical inference, we derive properties of the Fréchet mean on S ∞ by establishing its existence and uniqueness as well as a root-n central limit theorem (CLT) for the sample version, overcoming obstructions from infinite-dimensionality and lack of compactness on S ∞ . Intrinsic CLTs for the estimated tangent vectors and covariance operator are also obtained. Asymptotic and bootstrap hypothesis tests for the Fréchet mean based on projection and norm are then proposed and are shown to be consistent. The proposed twosample tests are applied to make inference for daily taxi demand patterns over Manhattan, modeled as densities, of which the square root densities are analyzed on the Hilbert sphere. Numerical properties of the proposed hypothesis tests which utilize the spherical geometry are studied in the real data application and simulations, where we demonstrate that the tests based on the intrinsic geometry compare favorably to those based on an extrinsic or flat geometry.

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Section: Tangent Vectors and The Covariance Operatormentioning

confidence: 99%

Statistical inference on the Hilbert sphere with application to random densities

Dai

2022

Electron. J. Statist.

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show abstract

“…The variation of X around the intrinsic mean is summarized by the covariance operator, which is a linear characterization of the intrinsic variation and has been applied to generalize the principal component analysis to Riemannian manifolds (Fletcher et al, 2004;Lazar and Lin, 2017). The population and empirical covariances are…”

Section: Tangent Vectors and The Covariance Operatormentioning

confidence: 99%

Statistical Inference on the Hilbert Sphere with Application to Random Densities

Dai¹

2021

Preprint

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The infinite-dimensional Hilbert sphere S ∞ has been widely employed to model density functions and shapes, extending the finite-dimensional counterpart. We consider the Fréchet mean as an intrinsic summary of the central tendency of data lying on S ∞ . To break a path for sound statistical inference, we derive properties of the Fréchet mean on S ∞ by establishing its existence and uniqueness as well as a root-n central limit theorem (CLT) for the sample version, overcoming obstructions from infinite-dimensionality and lack of compactness on S ∞ . Intrinsic CLTs for the estimated tangent vectors and covariance operator are also obtained. Asymptotic and bootstrap hypothesis tests for the Fréchet mean based on projection and norm are then proposed and are shown to be consistent. The proposed two-sample tests are applied to make inference for daily taxi demand patterns over Manhattan modeled as densities, of which the square roots are analyzed on the Hilbert sphere. Numerical properties of the proposed hypothesis tests which utilize the spherical geometry are studied in the real data application and simulations, where we demonstrate that the tests based on the intrinsic geometry compare favorably to those based on an extrinsic or flat geometry.

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“…Robust Principal Geodesic Analysis (RPGA). Principal Geodesic Analysis (PGA) as in [18] is a two-step procedure which involves 1) computing a center of the data and 2) successively finding orthogonal tangent vectors at that center so that their exponentiated span best fits the data according to intrinsic sum-of-squared residuals.…”

Section: 22mentioning

confidence: 99%

Robust Optimization and Inference on Manifolds

Lin¹,

Lazar²,

Sarpabayeva³

et al. 2020

Preprint

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We propose a robust and scalable procedure for general optimization and inference problems on manifolds leveraging the classical idea of 'median-of-means' estimation. This is motivated by ubiquitous examples and applications in modern data science in which a statistical learning problem can be cast as an optimization problem over manifolds. Being able to incorporate the underlying geometry for inference while addressing the need for robustness and scalability presents great challenges. We address these challenges by first proving a key lemma that characterizes some crucial properties of geometric medians on manifolds. In turn, this allows us to prove robustness and tighter concentration of our proposed final estimator in a subsequent theorem. This estimator aggregates a collection of subset estimators by taking their geometric median over the manifold. We illustrate bounds on this estimator via calculations in explicit examples. The robustness and scalability of the procedure is illustrated in numerical examples on both simulated and real data sets.

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Scale and curvature effects in principal geodesic analysis

Cited by 6 publications

References 21 publications

Statistical inference on the Hilbert sphere with application to random densities

Statistical inference on the Hilbert sphere with application to random densities

Statistical Inference on the Hilbert Sphere with Application to Random Densities

Robust Optimization and Inference on Manifolds

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