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Latent Space Cartography: Generalised Metric-Inspired Measures and Measure-Based Transformations for Generative Models [article]

Max F. Frenzel, Bogdan Teleaga, Asahi Ushio
2019 arXiv   pre-print
We provide a study of latent space geometries and extend and build upon previous results on Riemannian metrics.  ...  Deep generative models are universal tools for learning data distributions on high dimensional data spaces via a mapping to lower dimensional latent spaces.  ...  if this high dimensional space is very sparsely populated as is the case for most high dimensional latent spaces.  ... 
arXiv:1902.02113v1 fatcat:ty2fglq7kraltjchc2cbf5u2s4

Riemannian Manifold Clustering and Dimensionality Reduction for Vision-Based Analysis [chapter]

Alvina Goh
2011 Advances in Computer Vision and Pattern Recognition  
Over the past few years, various techniques have been developed for learning a low-dimensional representation of a nonlinear manifold embedded in a high-dimensional space.  ...  In addition, all these manifold learning algorithms assume that the feature vectors are embedded in a Euclidean space and make use of (at least locally) the Euclidean metric or a variation of it to perform  ...  A smooth manifold is a topological space that is locally diffeomorphic to a Euclidean space smooth function γ (t) : R → M. Figure 1 shows an example of a two-dimensional manifold.  ... 
doi:10.1007/978-0-85729-057-1_2 fatcat:txv4ivisjja7jb4r65eswz32ne

Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery [article]

Dominique Perraul-Joncas, Marina Meila
2013 arXiv   pre-print
We provide an algorithm for estimating the Riemannian metric from data and demonstrate possible application s of our approach in a variety of examples.  ...  Our approach is based on augmenting the output of embedding algorithms with geometric informatio n embodied in the Riemannian metric of the manifold.  ...  One popular idea for Euclidean data is to appeal to the manifold hypothesis, whereby the data is assumed to lie on a low-dimensional smooth manifold embedded in the high dimensional space.  ... 
arXiv:1305.7255v1 fatcat:qqeuwuyjx5hrven3w3ppccxvbu

Geometrically Enriched Latent Spaces [article]

Georgios Arvanitidis, Søren Hauberg, Bernhard Schölkopf
2020 arXiv   pre-print
Instead, we consider the ambient space to be a Riemannian manifold, which allows for encoding domain knowledge through the associated Riemannian metric.  ...  A common assumption in generative models is that the generator immerses the latent space into a Euclidean ambient space.  ...  In particular, assume an embedded d-dimensional manifold M ⊂ X within a Riemannian manifold X = R D with metric M X (·), a Euclidean space Z = R d Z called as latent space and a smooth function g : Z →  ... 
arXiv:2008.00565v1 fatcat:7aaieueko5aj7arbimnfge7soi

Visualization of 2-dimensional Ricci flow

Junfei Dai, Xianfeng Gu, Wei Luo, Shing-Tung Yau, Min Zhang
2013 Pure and Applied Mathematics Quarterly  
The major challenges are to represent the intrinsic Riemannian metric of a surface by extrinsic embedding in the three dimensional Euclidean space and demonstrate the deformation process which preserves  ...  The embedding of convex surfaces with prescribed Riemannian metrics is illustrated by embedding meshes with given edge lengths.  ...  Background Ricci Flow for Smooth Manifolds. Ricci Flow on Surfaces. Suppose S is a smooth surface embedded in R 3 , therefore S has an induced Riemannian metric g.  ... 
doi:10.4310/pamq.2013.v9.n3.a2 fatcat:tlyupvbqerf5dblcxqnkk577hi

Hyperbolic Entailment Cones for Learning Hierarchical Embeddings [article]

Octavian-Eugen Ganea, Gary Bécigneul, Thomas Hofmann
2018 arXiv   pre-print
Learning graph representations via low-dimensional embeddings that preserve relevant network properties is an important class of problems in machine learning.  ...  We prove that these entailment cones admit an optimal shape with a closed form expression both in the Euclidean and hyperbolic spaces, and they canonically define the embedding learning process.  ...  A smooth manifold equipped with a Riemannian metric is called a Riemannian manifold. Subsequently, due to their metric properties, we will only consider such manifolds. Geodesics.  ... 
arXiv:1804.01882v3 fatcat:x5jnsxj5qvgfjnyexi3xxwtcqi

Turing approximations, toric isometric embeddings manifold convolutions [article]

P. Suárez-Serrato
2021 arXiv   pre-print
A result of Alan Turing from 1938 underscores the need for such a toric isometric embedding approach to achieve a global definition of convolution on computable, finite metric space approximations to a  ...  smooth manifold.  ...  We propose a new way to define convolutions on manifolds by first isometrically embedding the manifold into a high dimensional torus and then extending a continuous function from the isometric image of  ... 
arXiv:2110.02279v1 fatcat:uxjjqzjflrbu3ots2scvejdx5m

Weighted averages on surfaces

Daniele Panozzo, Ilya Baran, Olga Diamanti, Olga Sorkine-Hornung
2013 ACM Transactions on Graphics  
From left to right: texture transfer, decal placement, semiregular remeshing and Laplacian smoothing, splines on surfaces.  ...  We address both the forward problem, namely computing an average of given anchor points on the mesh with given weights, and the inverse problem, which is computing the weights given anchor points and a  ...  This work was supported in part by the ERC grant iModel (StG-2012-306877), by an SNF award 200021 137879 and a gift from Adobe Research.  ... 
doi:10.1145/2461912.2461935 fatcat:f3vy7xtgsvhhxkjkh63jd7ubga

Anisotropic geometry-conforming d -simplicial meshing via isometric embeddings

Philip Claude Caplan, Robert Haimes, David L. Darmofal, Marshall C. Galbraith
2017 Procedia Engineering  
The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and a dimension-independent vertex insertion algorithm  ...  The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and a dimension-independent vertex insertion algorithm  ...  Since the numerical solver expects a metric-conforming mesh upon termination of our meshing procedure, we develop an algorithm for isometrically embedding arbitrary mesh-metric pairs to high-dimensional  ... 
doi:10.1016/j.proeng.2017.09.798 fatcat:mq2wojjxwbdl3do3i6gatt5jja

Cluster Analysis of Diffusion Tensor Fields with Application to the Segmentation of the Corpus Callosum

Safa Elsheikh, Andrew Fish, Roma Chakrabarti, Diwei Zhou
2016 Procedia Computer Science  
Then we applied the adapted Hartigan's K-means, using Euclidean, Cholesky, root Euclidean and log Euclidean metrics along with Procrustes and Riemannian metrics (which need numerical solutions for mean  ...  computation), to diffusion tensor images of the brain to provide a segmentation of the CC.  ...  In fact, LLDTC is a generalization of a locally linear embedding method which is a dimensionality reduction method based on embedding high dimensional data on Euclidean space into low dimensional space  ... 
doi:10.1016/j.procs.2016.07.004 fatcat:q2k4bbxrvndh3n7clgtl5crnx4

Discrete Surface Ricci Flow

M. Jin, J. Kim, F. Luo, X. Gu
2008 IEEE Transactions on Visualization and Computer Graphics  
This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary  ...  Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process.  ...  Smooth Surface Ricci Flow Suppose S is a smooth surface with a Riemannian metric g.  ... 
doi:10.1109/tvcg.2008.57 pmid:18599915 fatcat:z5zka2er35corovmdzazoxdpyy

Discrete Surface Ricci Flow [chapter]

Wei Zeng, Xianfeng David Gu
2013 SpringerBriefs in Mathematics  
This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary  ...  Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process.  ...  Smooth Surface Ricci Flow Suppose S is a smooth surface with a Riemannian metric g.  ... 
doi:10.1007/978-1-4614-8781-4_4 fatcat:begfg75rhzdw7lsz6mgneufm34

Only Bayes should learn a manifold (on the estimation of differential geometric structure from data) [article]

Søren Hauberg
2019 arXiv   pre-print
We investigate learning of the differential geometric structure of a data manifold embedded in a high-dimensional Euclidean space.  ...  Fully exploiting the recovered structure, however, requires the development of stochastic extensions to classic Riemannian geometry. We take early steps in that regard.  ...  SH was supported by a research grant (15334) from VILLUM FONDEN.  ... 
arXiv:1806.04994v3 fatcat:qhysomkjbjgnfdr6tqmfkjh47e

Applying Ricci Flow to High Dimensional Manifold Learning [article]

Yangyang Li, Ruqian Lu
2017 arXiv   pre-print
To overcome this drawback we propose a new algorithm called RF-ML to perform an operation on the manifold with help of Ricci flow before reducing the dimension of manifold.  ...  Traditional manifold learning algorithms often bear an assumption that the local neighborhood of any point on embedded manifold is roughly equal to the tangent space at that point without considering the  ...  Manifolds usually arise from data generated in some continuous process. The generated manifold is often embedded in some high-dimensional Euclidean space.  ... 
arXiv:1703.10675v4 fatcat:vzuxs7ddunac3izfnhgdnjvrrq

Nearest-neighbor search algorithms on non-Euclidean manifolds for computer vision applications

Pavan Turaga, Rama Chellappa
2010 Proceedings of the Seventh Indian Conference on Computer Vision, Graphics and Image Processing - ICVGIP '10  
Exact nearest neighbor methods that rely solely on the existence of a metric can be extended, albeit with a huge computational cost.  ...  We derive an approximate method of searching via approximate embeddings using the logarithmic map. We study the error incurred in such an embedding and show that it performs well in real experiments.  ...  Several methods have been proposed for embedding arbitrary spaces into a Euclidean or pseudo-Euclidean space [7, 44] .  ... 
doi:10.1145/1924559.1924597 dblp:conf/icvgip/TuragaC10 fatcat:n3osxgea45ayhpkgeujqonosnu
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