86 research outputs found
Principal subbundles for dimension reduction
In this paper we demonstrate how sub-Riemannian geometry can be used for
manifold learning and surface reconstruction by combining local linear
approximations of a point cloud to obtain lower dimensional bundles. Local
approximations obtained by local PCAs are collected into a rank tangent
subbundle on , , which we call a principal subbundle. This
determines a sub-Riemannian metric on . We show that
sub-Riemannian geodesics with respect to this metric can successfully be
applied to a number of important problems, such as: explicit construction of an
approximating submanifold , construction of a representation of the
point-cloud in , and computation of distances between
observations, taking the learned geometry into account. The reconstruction is
guaranteed to equal the true submanifold in the limit case where tangent spaces
are estimated exactly. Via simulations, we show that the framework is robust
when applied to noisy data. Furthermore, the framework generalizes to
observations on an a priori known Riemannian manifold
Geodesics in Heat
We introduce the heat method for computing the shortest geodesic distance to
a specified subset (e.g., point or curve) of a given domain. The heat method is
robust, efficient, and simple to implement since it is based on solving a pair
of standard linear elliptic problems. The method represents a significant
breakthrough in the practical computation of distance on a wide variety of
geometric domains, since the resulting linear systems can be prefactored once
and subsequently solved in near-linear time. In practice, distance can be
updated via the heat method an order of magnitude faster than with
state-of-the-art methods while maintaining a comparable level of accuracy. We
provide numerical evidence that the method converges to the exact geodesic
distance in the limit of refinement; we also explore smoothed approximations of
distance suitable for applications where more regularity is required
Estimating the Reach of a Manifold
Various problems in manifold estimation make use of a quantity called the
reach, denoted by , which is a measure of the regularity of the
manifold. This paper is the first investigation into the problem of how to
estimate the reach. First, we study the geometry of the reach through an
approximation perspective. We derive new geometric results on the reach for
submanifolds without boundary. An estimator of is
proposed in a framework where tangent spaces are known, and bounds assessing
its efficiency are derived. In the case of i.i.d. random point cloud
, is showed to achieve uniform
expected loss bounds over a -like model. Finally, we obtain
upper and lower bounds on the minimax rate for estimating the reach
RSA-INR:Riemannian Shape Autoencoding via 4D Implicit Neural Representations
Shape encoding and shape analysis are valuable tools for comparing shapes and for dimensionality reduction. A specific framework for shape analysis is the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, which is capable of shape matching and dimensionality reduction. Researchers have recently introduced neural networks into this framework. However, these works can not match more than two objects simultaneously or have suboptimal performance in shape variability modeling. The latter limitation occurs as the works do not use state-of-the-art shape encoding methods. Moreover, the literature does not discuss the connection between the LDDMM Riemannian distance and the Riemannian geometry for deep learning literature. Our work aims to bridge this gap by demonstrating how LDDMM can integrate Riemannian geometry into deep learning. Furthermore, we discuss how deep learning solves and generalizes shape matching and dimensionality reduction formulations of LDDMM. We achieve both goals by designing a novel implicit encoder for shapes. This model extends a neural network-based algorithm for LDDMM-based pairwise registration, results in a nonlinear manifold PCA, and adds a Riemannian geometry aspect to deep learning models for shape variability modeling. Additionally, we demonstrate that the Riemannian geometry component improves the reconstruction procedure of the implicit encoder in terms of reconstruction quality and stability to noise. We hope our discussion paves the way to more research into how Riemannian geometry, shape/image analysis, and deep learning can be combined
Rda-inr:Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations
Diffeomorphic registration frameworks such as Large Deformation Diffeomorphic Metric Mapping (LDDMM) are used in computer graphics and the medical domain for atlas building, statistical latent modeling, and pairwise and groupwise registration. In recent years, researchers have developed neural network-based approaches regarding diffeomorphic registration to improve the accuracy and computational efficiency of traditional methods. In this work, we focus on a limitation of neural network-based atlas building and statistical latent modeling methods, namely that they either are (i) resolution dependent or (ii) disregard any data/problem-specific geometry needed for proper mean-variance analysis. In particular, we overcome this limitation by designing a novel encoder based on resolution-independent implicit neural representations. The encoder achieves resolution invariance for LDDMM-based statistical latent modeling. Additionally, the encoder adds LDDMM Riemannian geometry to resolution-independent deep learning models for statistical latent modeling. We showcase that the Riemannian geometry aspect improves latent modeling and is required for a proper mean-variance analysis. Furthermore, to showcase the benefit of resolution independence for LDDMM-based data variability modeling, we show that our approach outperforms another neural network-based LDDMM latent code model. Our work paves a way to more research into how Riemannian geometry, shape/image analysis, and deep learning can be combined
Riemannian Multi-Manifold Modeling
This paper advocates a novel framework for segmenting a dataset in a
Riemannian manifold into clusters lying around low-dimensional submanifolds
of . Important examples of , for which the proposed clustering algorithm
is computationally efficient, are the sphere, the set of positive definite
matrices, and the Grassmannian. The clustering problem with these examples of
is already useful for numerous application domains such as action
identification in video sequences, dynamic texture clustering, brain fiber
segmentation in medical imaging, and clustering of deformed images. The
proposed clustering algorithm constructs a data-affinity matrix by thoroughly
exploiting the intrinsic geometry and then applies spectral clustering. The
intrinsic local geometry is encoded by local sparse coding and more importantly
by directional information of local tangent spaces and geodesics. Theoretical
guarantees are established for a simplified variant of the algorithm even when
the clusters intersect. To avoid complication, these guarantees assume that the
underlying submanifolds are geodesic. Extensive validation on synthetic and
real data demonstrates the resiliency of the proposed method against deviations
from the theoretical model as well as its superior performance over
state-of-the-art techniques
Density estimation on an unknown submanifold
We investigate density estimation from a -sample in the Euclidean space , when the data is supported by an unknown submanifold of possibly unknown dimension under a reach condition. We study nonparametric kernel methods for pointwise and integrated loss, with data-driven bandwidths that incorporate some learning of the geometry via a local dimension estimator. When has H\"older smoothness and has regularity in a sense to be defined, our estimator achieves the rate and does not depend on the ambient dimension and is asymptotically minimax for . Following Lepski's principle, a bandwidth selection rule is shown to achieve smoothness adaptation. We also investigate the case : by estimating in some sense the underlying geometry of , we establish in dimension that the minimax rate is proving in particular that it does not depend on the regularity of . Finally, a numerical implementation is conducted on some case studies in order to confirm the practical feasibility of our estimators
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