Mixture Probabilistic Principal Geodesic Analysis

Zhang, Youshan; Xing, Jiarui; Zhang, Miaomiao

doi:10.1007/978-3-030-33226-6_21

Cited by 29 publications

(43 citation statements)

References 22 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this work, we define a Gaussian distribution on a homogeneous space, which is a more general topological space than the symmetric space. A few others in literature have generalized the Gaussian distribution to all Riemannian manifolds, for instance, in [52], authors defined the Gaussian distribution on a Riemannian manifold without a proof to show that the normalizing factor is a constant. Whereas, in [38], the author defined the normal law on Riemannian manifolds using the concept of entropy maximization for distributions with known mean and covariance.…”

Section: Key Contributionsmentioning

confidence: 99%

Statistics on the Stiefel manifold: Theory and applications

Chakraborty¹,

Vemuri²

2019

Ann. Statist.

View full text Add to dashboard Cite

A Stiefel manifold of the compact type is often encountered in many fields of Engineering including, signal and image processing, machine learning, numerical optimization and others. The Stiefel manifold is a Riemannian homogeneous space but not a symmetric space. In previous work, researchers have defined probability distributions on symmetric spaces and performed statistical analysis of data residing in these spaces.In this paper, we present original work involving definition of Gaussian distributions on a homogeneous space and show that the maximum-likelihood estimate of the location parameter of a Gaussian distribution on the homogeneous space yields the Fréchet mean (FM) of the samples drawn from this distribution. Further, we present an algorithm to sample from the Gaussian distribution on the Stiefel manifold and recursively compute the FM of these samples. We also prove the weak consistency of this recursive FM estimator. Several synthetic and real data experiments are then presented, demonstrating the superior computational performance of this estimator over the gradient descent based non-recursive counter part as well as the stochastic gradient descent based method prevalent in literature.

show abstract

Section: Key Contributionsmentioning

confidence: 99%

Statistics on the Stiefel manifold: Theory and applications

Chakraborty¹,

Vemuri²

2019

Ann. Statist.

View full text Add to dashboard Cite

show abstract

“…Turning to the manifold situation, the probabilistic view implies that the fundamental problem in generalizing PPCA is not to define low-dimensional subspaces as sought by the approaches described in section 2.3 but instead to define a natural generalization of the Euclidean normal distribution to manifolds. PPCA has previously been generalized to manifolds with the probabilistic principal geodesic analysis (PPGA, [27]) procedure. The probability model for the data conditioned on the latent variables is for PPGA a Riemannian normal distribution defined via its density…”

Section: 1mentioning

confidence: 99%

“…The model is based on the Euclidean probabilistic principal component analysis procedure (PPCA, [26]) that interprets PCA as a latent variable model (1.1) with W having low rank k ≤ d. We use the PPCA approach with a probability model based on a notion of infinitesimal covariance and thereby avoid linearizing the nonlinear data space while intrinsically incorporating the effect of data anisotropy, here difference in the principal eigenvalues. The model is related to the probabilistic principal geodesic analysis (PPGA, [27]) procedure, however using the probability model and normal distributions defined in [19,24]. This construction in particular emphasizes the role of the connection on the manifold in linking infinitesimally close tangent spaces.…”

Section: Introductionmentioning

confidence: 99%

An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data

Sommer

2018

Sankhya A

View full text Add to dashboard Cite

We provide a probabilistic and infinitesimal view of how the principal component analysis procedure (PCA) can be generalized to analysis of nonlinear manifold valued data. Starting with the probabilistic PCA interpretation of the Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an intrinsic way that does not resort to linearization of the data space. The underlying probability model is constructed by mapping a Euclidean stochastic process to the manifold using stochastic development of Euclidean semimartingales. The construction uses a connection and bundles of covariant tensors to allow global transport of principal eigenvectors, and the model is thereby an example of how principal fiber bundles can be used to handle the lack of global coordinate system and orientations that characterizes manifold valued statistics. We show how curvature implies non-integrability of the equivalent of Euclidean principal subspaces, and how the stochastic flows provide an alternative to explicit construction of such subspaces. We describe estimation procedures for inference of parameters and prediction of principal components, and we give examples of properties of the model on embedded surfaces. principal component analysis, manifold valued statistics, stochastic development, probabilistic PCA, anisotropic normal distributions, frame bundle

show abstract

“…where f (y) = d(p i , y) 2 . Using (31), ∇π H(v) (pi) f can be computed as the magnitude of the component of −2Logπ…”

Section: Linear Difference Indicatorsmentioning

confidence: 99%

Scale and curvature effects in principal geodesic analysis

Lazar

Lin

2017

Journal of Multivariate Analysis

View full text Add to dashboard Cite

There is growing interest in using the close connection between differential geometry and statistics to model smooth manifold-valued data. In particular, much work has been done recently to generalize principal component analysis (PCA), the method of dimension reduction in linear spaces, to Riemannian manifolds. One such generalization is known as principal geodesic analysis (PGA). This paper, in a novel fashion, obtains Taylor expansions in scaling parameters introduced in the domain of objective functions in PGA. It is shown this technique not only leads to better closed-form approximations of PGA but also reveals the effects that scale, curvature and the distribution of data have on solutions to PGA and on their differences to first-order tangent space approximations. This approach should be able to be applied not only to PGA but also to other generalizations of PCA and more generally to other intrinsic statistics on Riemannian manifolds.

show abstract

Mixture Probabilistic Principal Geodesic Analysis

Cited by 29 publications

References 22 publications

Statistics on the Stiefel manifold: Theory and applications

Statistics on the Stiefel manifold: Theory and applications

An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data

Scale and curvature effects in principal geodesic analysis

Contact Info

Product

Resources

About