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Efficient Random Walks on Riemannian Manifolds [article]

Michael Herrmann, Simon Schwarz, Anja Sturm, Max Wardetzky
<span title="2022-02-02">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
According to a version of Donsker's theorem, geodesic random walks on Riemannian manifolds converge to the respective Brownian motion.  ...  As a result we obtain an efficient algorithm for sampling Brownian motion on compact Riemannian manifolds.  ...  Using retractions yields Algorithm 1 for simulating random walks on a Riemannian manifold.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2202.00959v1">arXiv:2202.00959v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gkk5bz3chnetxkmzbwhfn3xqcu">fatcat:gkk5bz3chnetxkmzbwhfn3xqcu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220204082235/https://arxiv.org/pdf/2202.00959v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ba/8d/ba8d40697cb9dd828350eef5f50ffb3f1cbc4273.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2202.00959v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Monte Carlo Tracking on the Riemannian Manifold of Multivariate Normal Distributions

Hichem Snoussi, Cedric Richard
<span title="">2009</span> <i title="IEEE"> 2009 IEEE 13th Digital Signal Processing Workshop and 5th IEEE Signal Processing Education Workshop </i> &nbsp;
In this contribution, a general scheme of particle filtering on Riemannian manifolds is proposed.  ...  In addition to the nonlinear dynamics, the system state is constrained to lie on a Riemannian manifold M, which dimension is much lower than the whole embedding space dimension.  ...  In order to describe the temporal correlation of the noise covariance trajectory on the Riemannian manifold of positive definite matrices S + , we define the Generalized Gaussian random walk Σ t ∼ GN (  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/dsp.2009.4785935">doi:10.1109/dsp.2009.4785935</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zabcd4suwnhrfbdof3parjzj4a">fatcat:zabcd4suwnhrfbdof3parjzj4a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810225544/http://www.cedric-richard.fr/Articles/snoussi2009monte.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3c/39/3c39f53c98ab7e541e04fc6db519ab46d9b7a8ec.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/dsp.2009.4785935"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Sampling and Optimization on Convex Sets in Riemannian Manifolds of Non-Negative Curvature [article]

Navin Goyal, Abhishek Shetty
<span title="2019-07-24">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We first present a random walk based sampling algorithm and then combine it with simulated annealing for solving convex optimization problems.  ...  The Euclidean space notion of convex sets (and functions) generalizes to Riemannian manifolds in a natural sense and is called geodesic convexity.  ...  In the Euclidean case, most sampling algorithms are based on geometric random walks.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1907.10531v1">arXiv:1907.10531v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hh765dhqgfazjdi6biq6t225gu">fatcat:hh765dhqgfazjdi6biq6t225gu</a> </span>
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Mosaicing of Confocal Microscopic In Vivo Soft Tissue Video Sequences [chapter]

Tom Vercauteren, Aymeric Perchant, Xavier Pennec, Nicholas Ayache
<span title="">2005</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
A robust estimator based on statistics for Riemannian manifolds is developed to find a globally consistent mapping of the input frames to a common coordinate system.  ...  Results on 50 images of a live mouse colon demonstrate the effectiveness of the proposed method.  ...  A fully automatic robust approach based on Riemannian statistics and efficient scattered data fitting techniques was proposed.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11566465_93">doi:10.1007/11566465_93</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/sqduenctprg4locr33hv7il4cq">fatcat:sqduenctprg4locr33hv7il4cq</a> </span>
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Representation Policy Iteration [article]

Sridhar Mahadevan
<span title="2012-07-04">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The key innovation is a coordinate-free representation of value functions, using the theory of smooth functions on a Riemannian manifold.  ...  Hodge theory yields a constructive method for generating basis functions for approximating value functions based on the eigenfunctions of the self-adjoint (Laplace-Beltrami) operator on manifolds.  ...  Reversible random walks greatly simplify spectral analysis since such random walks are similar to a symmetric operator on the state space.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1207.1408v1">arXiv:1207.1408v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/63hbb2ncpvg4xmwr2s2l4lwahq">fatcat:63hbb2ncpvg4xmwr2s2l4lwahq</a> </span>
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A General Metric for Riemannian Manifold Hamiltonian Monte Carlo [chapter]

Michael Betancourt
<span title="">2013</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Current RMHMC implementations, however, rely on a Riemannian metric that limits their application to analytically-convenient models.  ...  Markov Chain Monte Carlo (MCMC) is an invaluable means of inference with complicated models, and Hamiltonian Monte Carlo, in particular Riemannian Manifold Hamiltonian Monte Carlo (RMHMC), has demonstrated  ...  Because the resulting Hamiltonian trajectories are related to geodesics on a Riemannian manifold with metric Σ(q), this choice is known as Riemannian Manifold Hamiltonian Monte Carlo (RMHMC) [4] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-40020-9_35">doi:10.1007/978-3-642-40020-9_35</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/thjrizujonenhlmmgdgld3nql4">fatcat:thjrizujonenhlmmgdgld3nql4</a> </span>
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Curvature of Hypergraphs via Multi-Marginal Optimal Transport [article]

Shahab Asoodeh, Tingran Gao, James Evans
<span title="2018-03-22">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
is related to the scalar curvature when the hypergraph arises naturally from a Riemannian manifold.  ...  We introduce a novel definition of curvature for hypergraphs, a natural generalization of graphs, by introducing a multi-marginal optimal transport problem for a naturally defined random walk on the hypergraph  ...  In a d-dimensional Riemannian manifold (M, d M ), consider the random walk (c.f.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1803.08584v1">arXiv:1803.08584v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hpolid3bebatboi47whtm7nhae">fatcat:hpolid3bebatboi47whtm7nhae</a> </span>
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Internal and Contextual Attention Network for Cold-start Multi-channel Matching in Recommendation

Ruobing Xie, Zhijie Qiu, Jun Rao, Yi Liu, Bo Zhang, Leyu Lin
<span title="">2020</span> <i title="International Joint Conferences on Artificial Intelligence Organization"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/vfwwmrihanevtjbbkti2kc3nke" style="color: black;">Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence</a> </i> &nbsp;
In experiments, we conduct both offline and online evaluations with case studies on a real-world integrated recommendation system.  ...  Hence, most large-scale recommendation systems consist of two modules: a multi-channel matching module to efficiently retrieve a small subset of candidates, and a ranking module for precise personalized  ...  We will construct Brownian motion out of random walks on a manifold. We first fix a small time step τ > 0.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.24963/ijcai.2020/375">doi:10.24963/ijcai.2020/375</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/ijcai/ReyMP20.html">dblp:conf/ijcai/ReyMP20</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bvsodjpkvnar3dgdqbcs7lrq7e">fatcat:bvsodjpkvnar3dgdqbcs7lrq7e</a> </span>
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Wrapped Gaussian Process Regression on Riemannian Manifolds

Anton Mallasto, Aasa Feragen
<span title="">2018</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ilwxppn4d5hizekyd3ndvy2mii" style="color: black;">2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition</a> </i> &nbsp;
We tackle the problem by defining wrapped Gaussian processes (WGPs) on Riemannian manifolds, using the probabilistic setting to generalize GP regression to the context of manifold-valued targets.  ...  The method is validated empirically on diffusion weighted imaging (DWI) data, directional data on the sphere and in the Kendall shape space, endorsing WGP regression as an efficient and flexible tool for  ...  Probabilistic notions Let X be a random point on a Riemannian manifold M , the set E[X] := p | p ∈ arg min q∈M (E[d(q, X) 2 ]) . (7) is called the Fréchet means of X.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/cvpr.2018.00585">doi:10.1109/cvpr.2018.00585</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/cvpr/MallastoF18.html">dblp:conf/cvpr/MallastoF18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bhkwtedat5emzo6rydu52lcbmq">fatcat:bhkwtedat5emzo6rydu52lcbmq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200318233146/http://openaccess.thecvf.com/content_cvpr_2018/papers/Mallasto_Wrapped_Gaussian_Process_CVPR_2018_paper.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/4f/c2/4fc2e9783d594dab6bb1c044ffae7ca5caf97360.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/cvpr.2018.00585"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Incomplete Reparameterizations and Equivalent Metrics [article]

Michael Betancourt
<span title="2019-10-09">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
I then consider how these changing interactions manifest in the context of Markov chain Monte Carlo algorithms defined on Riemannian manifolds.  ...  Random Walk Metropolis-Hastings Repeatedly sampling a random direction and then following the corresponding geodesic for some finite time generates a second-order Markov process on the base manifold (  ...  In the global coordinates of a Euclidean manifold this construction reduces to the usual random walk Metropolis algorithm.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1910.09407v1">arXiv:1910.09407v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4cxaldgd5jbipoxdq62fjoffwi">fatcat:4cxaldgd5jbipoxdq62fjoffwi</a> </span>
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Latent Space Oddity: on the Curvature of Deep Generative Models [article]

Georgios Arvanitidis, Lars Kai Hansen, Søren Hauberg
<span title="2018-01-31">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Results are demonstrated on convolutional and fully connected variational autoencoders, but the formalism easily generalize to other deep generative models.  ...  Under mild conditions, we show that this distortion can be characterized by a stochastic Riemannian metric, and demonstrate that distances and interpolants are significantly improved under this metric.  ...  based on Euclidean random walks.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1710.11379v2">arXiv:1710.11379v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3d4xh6qwejekfgignwkknjboga">fatcat:3d4xh6qwejekfgignwkknjboga</a> </span>
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Data Augmentation with Variational Autoencoders and Manifold Sampling [article]

Clément Chadebec, Stéphanie Allassonnière
<span title="2021-09-28">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We propose a new efficient way to sample from a Variational Autoencoder in the challenging low sample size setting.  ...  Such results were also observed on 3 standard data sets and with other classifiers. A code is available at https://github.com/clementchadebec/Data_Augmentation_with_VAE-DALI.  ...  Riemannian Random Walk A natural way to explore the latent space of a VAE consists in using a random walk like algorithm which moves from one location to another with a certain probability.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2103.13751v3">arXiv:2103.13751v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kd3fsdpi55fs7ja6bufp6f527m">fatcat:kd3fsdpi55fs7ja6bufp6f527m</a> </span>
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Diffusion Curvature for Estimating Local Curvature in High Dimensional Data [article]

Dhananjay Bhaskar, Kincaid MacDonald, Oluwadamilola Fasina, Dawson Thomas, Bastian Rieck, Ian Adelstein, Smita Krishnaswamy
<span title="2022-06-08">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point  ...  We show that this laziness directly relates to volume comparison results from Riemannian geometry.  ...  In the Riemannian setting, if one defines the random walk to be dm r x (y) = dvol(y) /volB(x,r) then Ollivier demonstrates that k (x, y) = r 2 Ric(v,v) /2d+2 + O(r 3 + d(x, y)r 2 ) where v is a unit tangent  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2206.03977v1">arXiv:2206.03977v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ulva4dtedffnpn6vqccll33b5u">fatcat:ulva4dtedffnpn6vqccll33b5u</a> </span>
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Hamiltonian Monte Carlo for Hierarchical Models [chapter]

Michael Betancourt, Mark Girolami
<span title="2015-05-14">2015</span> <i title="Chapman and Hall/CRC"> Current Trends in Bayesian Methodology with Applications </i> &nbsp;
By capturing these relationships, however, hierarchical models also introduce distinctive pathologies that quickly limit the efficiency of most common methods of inference.  ...  indebted to Simon Byrne, Bob Carpenter, Michael Epstein, Andrew Gelman, Yair Ghitza, Daniel Lee, Peter Li, Sam Livingstone, and Anne-Marie Lyne for many fruitful discussions as well as invaluable comments on  ...  With the Hamiltonian now generating dynamics on a Riemannian manifold with metric Σ, we follow the convention established above and denote the resulting algorithm as Riemannian Hamiltonian Monte Carlo  ... 
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Eigenvalues of Graphs [chapter]

Fan R. K. Chung
<span title="">1995</span> <i title="Birkhäuser Basel"> Proceedings of the International Congress of Mathematicians </i> &nbsp;
Eigenvalue bounds for special families of graphs, such as the convex subgraphs of homogeneous graphs, with applications to random walks and efficient approximation algorithms.  ...  One way to use the above theorem is to bound the heat kernel of a graph by the (continuous) heat kernel of the Riemannian manifolds, for certain graphs that we call convex subgraphs.  ... 
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