561 research outputs found
Barycentric Subspace Analysis on Manifolds
This paper investigates the generalization of Principal Component Analysis
(PCA) to Riemannian manifolds. We first propose a new and general type of
family of subspaces in manifolds that we call barycentric subspaces. They are
implicitly defined as the locus of points which are weighted means of
reference points. As this definition relies on points and not on tangent
vectors, it can also be extended to geodesic spaces which are not Riemannian.
For instance, in stratified spaces, it naturally allows principal subspaces
that span several strata, which is impossible in previous generalizations of
PCA. We show that barycentric subspaces locally define a submanifold of
dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in
Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy
of properly embedded linear subspaces of increasing dimension). We show that
the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the
subspaces of the flag (AUV). Barycentric subspaces are naturally nested,
allowing the construction of hierarchically nested subspaces. Optimizing the
AUV criterion to optimally approximate data points with flags of affine spans
in Riemannian manifolds lead to a particularly appealing generalization of PCA
on manifolds called Barycentric Subspaces Analysis (BSA).Comment: Annals of Statistics, Institute of Mathematical Statistics, A
Para\^itr
Higher-Order Momentum Distributions and Locally Affine LDDMM Registration
To achieve sparse parametrizations that allows intuitive analysis, we aim to
represent deformation with a basis containing interpretable elements, and we
wish to use elements that have the description capacity to represent the
deformation compactly. To accomplish this, we introduce in this paper
higher-order momentum distributions in the LDDMM registration framework. While
the zeroth order moments previously used in LDDMM only describe local
displacement, the first-order momenta that are proposed here represent a basis
that allows local description of affine transformations and subsequent compact
description of non-translational movement in a globally non-rigid deformation.
The resulting representation contains directly interpretable information from
both mathematical and modeling perspectives. We develop the mathematical
construction of the registration framework with higher-order momenta, we show
the implications for sparse image registration and deformation description, and
we provide examples of how the parametrization enables registration with a very
low number of parameters. The capacity and interpretability of the
parametrization using higher-order momenta lead to natural modeling of
articulated movement, and the method promises to be useful for quantifying
ventricle expansion and progressing atrophy during Alzheimer's disease
Power Euclidean metrics for covariance matrices with application to diffusion tensor imaging
Various metrics for comparing diffusion tensors have been recently proposed
in the literature. We consider a broad family of metrics which is indexed by a
single power parameter. A likelihood-based procedure is developed for choosing
the most appropriate metric from the family for a given dataset at hand. The
approach is analogous to using the Box-Cox transformation that is frequently
investigated in regression analysis. The methodology is illustrated with a
simulation study and an application to a real dataset of diffusion tensor
images of canine hearts
Bures-Wasserstein minimizing geodesics between covariance matrices of different ranks
The set of covariance matrices equipped with the Bures-Wasserstein distance
is the orbit space of the smooth, proper and isometric action of the orthogonal
group on the Euclidean space of square matrices. This construction induces a
natural orbit stratification on covariance matrices, which is exactly the
stratification by the rank. Thus, the strata are the manifolds of symmetric
positive semi-definite (PSD) matrices of fixed rank endowed with the
Bures-Wasserstein Riemannian metric. In this work, we study the geodesics of
the Bures-Wasserstein distance. Firstly, we complete the literature on
geodesics in each stratum by clarifying the set of preimages of the exponential
map and by specifying the injection domain. We also give explicit formulae of
the horizontal lift, the exponential map and the Riemannian logarithms that
were kept implicit in previous works. Secondly, we give the expression of all
the minimizing geodesic segments joining two covariance matrices of any rank.
More precisely, we show that the set of all minimizing geodesics between two
covariance matrices and is parametrized by the closed unit
ball of for the spectral norm, where
are the respective ranks of , , . In
particular, the minimizing geodesic is unique if and only if .
Otherwise, there are infinitely many
Stratified Principal Component Analysis
This paper investigates a general family of models that stratifies the space
of covariance matrices by eigenvalue multiplicity. This family, coined
Stratified Principal Component Analysis (SPCA), includes in particular
Probabilistic PCA (PPCA) models, where the noise component is assumed to be
isotropic. We provide an explicit maximum likelihood and a geometric
characterization relying on flag manifolds. A key outcome of this analysis is
that PPCA's parsimony (with respect to the full covariance model) is due to the
eigenvalue-equality constraint in the noise space and the subsequent inference
of a multidimensional eigenspace. The sequential nature of flag manifolds
enables to extend this constraint to the signal space and bring more
parsimonious models. Moreover, the stratification and the induced partial order
on SPCA yield efficient model selection heuristics. Experiments on simulated
and real datasets substantiate the interest of equalising adjacent sample
eigenvalues when the gaps are small and the number of samples is limited. They
notably demonstrate that SPCA models achieve a better
complexity/goodness-of-fit tradeoff than PPCA
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