6,173 research outputs found
Principal arc analysis on direct product manifolds
We propose a new approach to analyze data that naturally lie on manifolds. We
focus on a special class of manifolds, called direct product manifolds, whose
intrinsic dimension could be very high. Our method finds a low-dimensional
representation of the manifold that can be used to find and visualize the
principal modes of variation of the data, as Principal Component Analysis (PCA)
does in linear spaces. The proposed method improves upon earlier manifold
extensions of PCA by more concisely capturing important nonlinear modes. For
the special case of data on a sphere, variation following nongeodesic arcs is
captured in a single mode, compared to the two modes needed by previous
methods. Several computational and statistical challenges are resolved. The
development on spheres forms the basis of principal arc analysis on more
complicated manifolds. The benefits of the method are illustrated by a data
example using medial representations in image analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS370 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Low-Dimensional Representation for Robust Partial Isometric Correspondences Computation
Intrinsic isometric shape matching has become the standard approach for pose
invariant correspondence estimation among deformable shapes. Most existing
approaches assume global consistency, i.e., the metric structure of the whole
manifold must not change significantly. While global isometric matching is well
understood, only a few heuristic solutions are known for partial matching.
Partial matching is particularly important for robustness to topological noise
(incomplete data and contacts), which is a common problem in real-world 3D
scanner data. In this paper, we introduce a new approach to partial, intrinsic
isometric matching. Our method is based on the observation that isometries are
fully determined by purely local information: a map of a single point and its
tangent space fixes an isometry for both global and the partial maps. From this
idea, we develop a new representation for partial isometric maps based on
equivalence classes of correspondences between pairs of points and their
tangent spaces. From this, we derive a local propagation algorithm that find
such mappings efficiently. In contrast to previous heuristics based on RANSAC
or expectation maximization, our method is based on a simple and sound
theoretical model and fully deterministic. We apply our approach to register
partial point clouds and compare it to the state-of-the-art methods, where we
obtain significant improvements over global methods for real-world data and
stronger guarantees than previous heuristic partial matching algorithms.Comment: 17 pages, 12 figure
Chaotic saddles in nonlinear modulational interactions in a plasma
A nonlinear model of modulational processes in the subsonic regime involving
a linearly unstable wave and two linearly damped waves with different damping
rates in a plasma is studied numerically. We compute the maximum Lyapunov
exponent as a function of the damping rates in a two-parameter space, and
identify shrimp-shaped self-similar structures in the parameter space. By
varying the damping rate of the low-frequency wave, we construct bifurcation
diagrams and focus on a saddle-node bifurcation and an interior crisis
associated with a periodic window. We detect chaotic saddles and their stable
and unstable manifolds, and demonstrate how the connection between two chaotic
saddles via coupling unstable periodic orbits can result in a crisis-induced
intermittency. The relevance of this work for the understanding of modulational
processes observed in plasmas and fluids is discussed.Comment: Physics of Plasmas, in pres
Characterizing dynamics with covariant Lyapunov vectors
A general method to determine covariant Lyapunov vectors in both discrete-
and continuous-time dynamical systems is introduced. This allows to address
fundamental questions such as the degree of hyperbolicity, which can be
quantified in terms of the transversality of these intrinsic vectors. For
spatially extended systems, the covariant Lyapunov vectors have localization
properties and spatial Fourier spectra qualitatively different from those
composing the orthonormalized basis obtained in the standard procedure used to
calculate the Lyapunov exponents.Comment: 4 pages, 3 figures, submitted to Physical Review letter
Rank-preserving geometric means of positive semi-definite matrices
The generalization of the geometric mean of positive scalars to positive
definite matrices has attracted considerable attention since the seminal work
of Ando. The paper generalizes this framework of matrix means by proposing the
definition of a rank-preserving mean for two or an arbitrary number of positive
semi-definite matrices of fixed rank. The proposed mean is shown to be
geometric in that it satisfies all the expected properties of a rank-preserving
geometric mean. The work is motivated by operations on low-rank approximations
of positive definite matrices in high-dimensional spaces.Comment: To appear in Linear Algebra and its Application
What is Time? A New Mathematico- Physical and Information Theoretic Approach
A New Mathematico-Physical and Information Theoretic Approach
Examination of the available hard core information to firm up the process of
unification of quantum and gravitational physics leads to the conclusion that
for achieving this synthesis, major paradigm shifts are needed as also the
answering of `What is Time?' The object of this submission is to point out the
means of achieving such a grand synthesis. Currently the main pillars
supporting the edifice of physics are: (i) The geometrical concepts of space-
time-gravitation, (ii) The dynamic concepts involving quantum of action, (iii)
Statistical thermodynamic concepts, heat and entropy, (iv) Mathematical
concepts, tools and techniques serving both as a grand plan and the means of
calculation and last but not least v)Controlled observation, pertinent
experimentation as the final arbiter. In making major changes the author is
following Dirac's dictum "....make changes without sacrificing the existing
superstructure". It is shown that time can be treated as a parameter rather
than an additional dimension. A new entity called "Ekon" having the properties
of both space and momentum is introduced along with a space called
"Chalachala". The requisite connection with Einstein's formulation and
mathematical aperatus required have been formulated which is highly suited for
the purpose. The primacy of the Plancks quantum of action and its
representation geometrically as a twist is introduced. The practical and
numerical estimates have been made and applied to evaluation of the
gravitational constant in a a seperate submission "Estimations of gravitational
constant from CMBR data".Comment: 29 pages, pdf fil
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