1,243 research outputs found
Principal components analysis in the space of phylogenetic trees
Phylogenetic analysis of DNA or other data commonly gives rise to a
collection or sample of inferred evolutionary trees. Principal Components
Analysis (PCA) cannot be applied directly to collections of trees since the
space of evolutionary trees on a fixed set of taxa is not a vector space. This
paper describes a novel geometrical approach to PCA in tree-space that
constructs the first principal path in an analogous way to standard linear
Euclidean PCA. Given a data set of phylogenetic trees, a geodesic principal
path is sought that maximizes the variance of the data under a form of
projection onto the path. Due to the high dimensionality of tree-space and the
nonlinear nature of this problem, the computational complexity is potentially
very high, so approximate optimization algorithms are used to search for the
optimal path. Principal paths identified in this way reveal and quantify the
main sources of variation in the original collection of trees in terms of both
topology and branch lengths. The approach is illustrated by application to
simulated sets of trees and to a set of gene trees from metazoan (animal)
species.Comment: Published in at http://dx.doi.org/10.1214/11-AOS915 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Compression for Smooth Shape Analysis
Most 3D shape analysis methods use triangular meshes to discretize both the
shape and functions on it as piecewise linear functions. With this
representation, shape analysis requires fine meshes to represent smooth shapes
and geometric operators like normals, curvatures, or Laplace-Beltrami
eigenfunctions at large computational and memory costs.
We avoid this bottleneck with a compression technique that represents a
smooth shape as subdivision surfaces and exploits the subdivision scheme to
parametrize smooth functions on that shape with a few control parameters. This
compression does not affect the accuracy of the Laplace-Beltrami operator and
its eigenfunctions and allow us to compute shape descriptors and shape
matchings at an accuracy comparable to triangular meshes but a fraction of the
computational cost.
Our framework can also compress surfaces represented by point clouds to do
shape analysis of 3D scanning data
Curvature weighted metrics on shape space of hypersurfaces in -space
Let be a compact connected oriented dimensional manifold without
boundary. In this work, shape space is the orbifold of unparametrized
immersions from to . The results of \cite{Michor118}, where
mean curvature weighted metrics were studied, suggest incorporating Gau{\ss}
curvature weights in the definition of the metric. This leads us to study
metrics on shape space that are induced by metrics on the space of immersions
of the form G_f(h,k) = \int_{M} \Phi . \bar g(h, k) \vol(f^*\bar{g}). Here
f \in \Imm(M,\R^n) is an immersion of into and are tangent vectors at . is the standard
metric on , is the induced metric on ,
\vol(f^*\bar g) is the induced volume density and is a suitable smooth
function depending on the mean curvature and Gau{\ss} curvature. For these
metrics we compute the geodesic equations both on the space of immersions and
on shape space and the conserved momenta arising from the obvious symmetries.
Numerical experiments illustrate the behavior of these metrics.Comment: 12 pages 3 figure
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
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