Pulling back symmetric Riemannian geometry for data analysis
- URL: http://arxiv.org/abs/2403.06612v1
- Date: Mon, 11 Mar 2024 10:59:55 GMT
- Title: Pulling back symmetric Riemannian geometry for data analysis
- Authors: Willem Diepeveen
- Abstract summary: Ideal data analysis tools for data sets should account for non-linear geometry.
A rich mathematical structure to account for non-linear geometries has been shown to be able to capture the data geometry.
Many standard data analysis tools initially developed for data in Euclidean space can be generalised efficiently to data on a symmetric Riemannian manifold.
- Score: 0.0
- License: http://creativecommons.org/licenses/by-nc-nd/4.0/
- Abstract: Data sets tend to live in low-dimensional non-linear subspaces. Ideal data
analysis tools for such data sets should therefore account for such non-linear
geometry. The symmetric Riemannian geometry setting can be suitable for a
variety of reasons. First, it comes with a rich mathematical structure to
account for a wide range of non-linear geometries that has been shown to be
able to capture the data geometry through empirical evidence from classical
non-linear embedding. Second, many standard data analysis tools initially
developed for data in Euclidean space can also be generalised efficiently to
data on a symmetric Riemannian manifold. A conceptual challenge comes from the
lack of guidelines for constructing a symmetric Riemannian structure on the
data space itself and the lack of guidelines for modifying successful
algorithms on symmetric Riemannian manifolds for data analysis to this setting.
This work considers these challenges in the setting of pullback Riemannian
geometry through a diffeomorphism. The first part of the paper characterises
diffeomorphisms that result in proper, stable and efficient data analysis. The
second part then uses these best practices to guide construction of such
diffeomorphisms through deep learning. As a proof of concept, different types
of pullback geometries -- among which the proposed construction -- are tested
on several data analysis tasks and on several toy data sets. The numerical
experiments confirm the predictions from theory, i.e., that the diffeomorphisms
generating the pullback geometry need to map the data manifold into a geodesic
subspace of the pulled back Riemannian manifold while preserving local isometry
around the data manifold for proper, stable and efficient data analysis, and
that pulling back positive curvature can be problematic in terms of stability.
Related papers
- Score-based pullback Riemannian geometry [10.649159213723106]
We propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning.
We produce high-quality geodesics through the data support and reliably estimates the intrinsic dimension of the data manifold.
Our framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training.
arXiv Detail & Related papers (2024-10-02T18:52:12Z) - Improving embedding of graphs with missing data by soft manifolds [51.425411400683565]
The reliability of graph embeddings depends on how much the geometry of the continuous space matches the graph structure.
We introduce a new class of manifold, named soft manifold, that can solve this situation.
Using soft manifold for graph embedding, we can provide continuous spaces to pursue any task in data analysis over complex datasets.
arXiv Detail & Related papers (2023-11-29T12:48:33Z) - The Fisher-Rao geometry of CES distributions [50.50897590847961]
The Fisher-Rao information geometry allows for leveraging tools from differential geometry.
We will present some practical uses of these geometric tools in the framework of elliptical distributions.
arXiv Detail & Related papers (2023-10-02T09:23:32Z) - Neural Latent Geometry Search: Product Manifold Inference via
Gromov-Hausdorff-Informed Bayesian Optimization [21.97865037637575]
We mathematically define this novel formulation and coin it as neural latent geometry search (NLGS)
We propose a novel notion of distance between candidate latent geometries based on the Gromov-Hausdorff distance from metric geometry.
We then design a graph search space based on the notion of smoothness between latent geometries and employ the calculated as an additional inductive bias.
arXiv Detail & Related papers (2023-09-09T14:29:22Z) - Curvature-Independent Last-Iterate Convergence for Games on Riemannian
Manifolds [77.4346324549323]
We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate.
To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
arXiv Detail & Related papers (2023-06-29T01:20:44Z) - Exploring Data Geometry for Continual Learning [64.4358878435983]
We study continual learning from a novel perspective by exploring data geometry for the non-stationary stream of data.
Our method dynamically expands the geometry of the underlying space to match growing geometric structures induced by new data.
Experiments show that our method achieves better performance than baseline methods designed in Euclidean space.
arXiv Detail & Related papers (2023-04-08T06:35:25Z) - Parametrizing Product Shape Manifolds by Composite Networks [5.772786223242281]
We show that it is possible to learn an efficient neural network approximation for shape spaces with a special product structure.
Our proposed architecture leverages this structure by separately learning approximations for the low-dimensional factors and a subsequent combination.
arXiv Detail & Related papers (2023-02-28T15:31:23Z) - Towards a mathematical understanding of learning from few examples with
nonlinear feature maps [68.8204255655161]
We consider the problem of data classification where the training set consists of just a few data points.
We reveal key relationships between the geometry of an AI model's feature space, the structure of the underlying data distributions, and the model's generalisation capabilities.
arXiv Detail & Related papers (2022-11-07T14:52:58Z) - Study of Manifold Geometry using Multiscale Non-Negative Kernel Graphs [32.40622753355266]
We propose a framework to study the geometric structure of the data.
We make use of our recently introduced non-negative kernel (NNK) regression graphs to estimate the point density, intrinsic dimension, and the linearity of the data manifold (curvature)
arXiv Detail & Related papers (2022-10-31T17:01:17Z) - Bayesian Quadrature on Riemannian Data Manifolds [79.71142807798284]
A principled way to model nonlinear geometric structure inherent in data is provided.
However, these operations are typically computationally demanding.
In particular, we focus on Bayesian quadrature (BQ) to numerically compute integrals over normal laws.
We show that by leveraging both prior knowledge and an active exploration scheme, BQ significantly reduces the number of required evaluations.
arXiv Detail & Related papers (2021-02-12T17:38:04Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.