671 research outputs found
A variational model for data fitting on manifolds by minimizing the acceleration of a B\'ezier curve
We derive a variational model to fit a composite B\'ezier curve to a set of
data points on a Riemannian manifold. The resulting curve is obtained in such a
way that its mean squared acceleration is minimal in addition to remaining
close the data points. We approximate the acceleration by discretizing the
squared second order derivative along the curve. We derive a closed-form,
numerically stable and efficient algorithm to compute the gradient of a
B\'ezier curve on manifolds with respect to its control points, expressed as a
concatenation of so-called adjoint Jacobi fields. Several examples illustrate
the capabilites and validity of this approach both for interpolation and
approximation. The examples also illustrate that the approach outperforms
previous works tackling this problem
Warped Riemannian metrics for location-scale models
The present paper shows that warped Riemannian metrics, a class of Riemannian
metrics which play a prominent role in Riemannian geometry, are also of
fundamental importance in information geometry. Precisely, the paper features a
new theorem, which states that the Rao-Fisher information metric of any
location-scale model, defined on a Riemannian manifold, is a warped Riemannian
metric, whenever this model is invariant under the action of some Lie group.
This theorem is a valuable tool in finding the expression of the Rao-Fisher
information metric of location-scale models defined on high-dimensional
Riemannian manifolds. Indeed, a warped Riemannian metric is fully determined by
only two functions of a single variable, irrespective of the dimension of the
underlying Riemannian manifold. Starting from this theorem, several original
contributions are made. The expression of the Rao-Fisher information metric of
the Riemannian Gaussian model is provided, for the first time in the
literature. A generalised definition of the Mahalanobis distance is introduced,
which is applicable to any location-scale model defined on a Riemannian
manifold. The solution of the geodesic equation is obtained, for any Rao-Fisher
information metric defined in terms of warped Riemannian metrics. Finally,
using a mixture of analytical and numerical computations, it is shown that the
parameter space of the von Mises-Fisher model of -dimensional directional
data, when equipped with its Rao-Fisher information metric, becomes a Hadamard
manifold, a simply-connected complete Riemannian manifold of negative sectional
curvature, for . Hopefully, in upcoming work, this will be
proved for any value of .Comment: first version, before submissio
The geometry of nonlinear least squares with applications to sloppy models and optimization
Parameter estimation by nonlinear least squares minimization is a common
problem with an elegant geometric interpretation: the possible parameter values
of a model induce a manifold in the space of data predictions. The minimization
problem is then to find the point on the manifold closest to the data. We show
that the model manifolds of a large class of models, known as sloppy models,
have many universal features; they are characterized by a geometric series of
widths, extrinsic curvatures, and parameter-effects curvatures. A number of
common difficulties in optimizing least squares problems are due to this common
structure. First, algorithms tend to run into the boundaries of the model
manifold, causing parameters to diverge or become unphysical. We introduce the
model graph as an extension of the model manifold to remedy this problem. We
argue that appropriate priors can remove the boundaries and improve convergence
rates. We show that typical fits will have many evaporated parameters. Second,
bare model parameters are usually ill-suited to describing model behavior; cost
contours in parameter space tend to form hierarchies of plateaus and canyons.
Geometrically, we understand this inconvenient parametrization as an extremely
skewed coordinate basis and show that it induces a large parameter-effects
curvature on the manifold. Using coordinates based on geodesic motion, these
narrow canyons are transformed in many cases into a single quadratic, isotropic
basin. We interpret the modified Gauss-Newton and Levenberg-Marquardt fitting
algorithms as an Euler approximation to geodesic motion in these natural
coordinates on the model manifold and the model graph respectively. By adding a
geodesic acceleration adjustment to these algorithms, we alleviate the
difficulties from parameter-effects curvature, improving both efficiency and
success rates at finding good fits.Comment: 40 pages, 29 Figure
Efficient Rank Reduction of Correlation Matrices
Geometric optimisation algorithms are developed that efficiently find the
nearest low-rank correlation matrix. We show, in numerical tests, that our
methods compare favourably to the existing methods in the literature. The
connection with the Lagrange multiplier method is established, along with an
identification of whether a local minimum is a global minimum. An additional
benefit of the geometric approach is that any weighted norm can be applied. The
problem of finding the nearest low-rank correlation matrix occurs as part of
the calibration of multi-factor interest rate market models to correlation.Comment: First version: 20 pages, 4 figures Second version [changed content]:
21 pages, 6 figure
On the Shape of Things: From holography to elastica
We explore the question of which shape a manifold is compelled to take when
immersed in another one, provided it must be the extremum of some functional.
We consider a family of functionals which depend quadratically on the extrinsic
curvatures and on projections of the ambient curvatures. These functionals
capture a number of physical setups ranging from holography to the study of
membranes and elastica. We present a detailed derivation of the equations of
motion, known as the shape equations, placing particular emphasis on the issue
of gauge freedom in the choice of normal frame. We apply these equations to the
particular case of holographic entanglement entropy for higher curvature three
dimensional gravity and find new classes of entangling curves. In particular,
we discuss the case of New Massive Gravity where we show that non-geodesic
entangling curves have always a smaller on-shell value of the entropy
functional. Then we apply this formalism to the computation of the entanglement
entropy for dual logarithmic CFTs. Nevertheless, the correct value for the
entanglement entropy is provided by geodesics. Then, we discuss the importance
of these equations in the context of classical elastica and comment on terms
that break gauge invariance.Comment: 54 pages, 8 figures. Significantly improved version, accepted for
publication in Annals of Physics. New section on logarithmic CFTs. Detailed
derivation of the shape equations added in appendix B. Typos corrected,
clarifications adde
Representation Learning via Manifold Flattening and Reconstruction
This work proposes an algorithm for explicitly constructing a pair of neural
networks that linearize and reconstruct an embedded submanifold, from finite
samples of this manifold. Our such-generated neural networks, called Flattening
Networks (FlatNet), are theoretically interpretable, computationally feasible
at scale, and generalize well to test data, a balance not typically found in
manifold-based learning methods. We present empirical results and comparisons
to other models on synthetic high-dimensional manifold data and 2D image data.
Our code is publicly available.Comment: 44 pages, 19 figure
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