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Low-Rank Representation over the Manifold of Curves [article]

Stephen Tierney, Junbin Gao, Yi Guo, Zhengwu Zhang
2016 arXiv   pre-print
In this paper we propose a method to analyse subspace structure of the functional data by using the state of the art Low-Rank Representation (LRR).  ...  The naive treatment of functional data as traditional multivariate data can lead to poor performance since the algorithms are ignoring the correlation in the curvature of each function.  ...  Acknowledgments Funding information hidden for the review process.  ... 
arXiv:1601.00732v2 fatcat:anrxcobtkzd6lcn5ykheqlppii

Structural Similarity and Distance in Learning [article]

Joseph Wang, Venkatesh Saligrama, David A. Castañón
2011 arXiv   pre-print
To learn structural information, low-dimensional structure of the data is captured by solving a non-linear, low-rank representation problem.  ...  We show that this low-rank representation can be kernelized, has a closed-form solution, allows for separation of independent manifolds, and is robust to noise.  ...  Our goal is to discover these manifolds using by using techniques to learn low-rank representations of the data.  ... 
arXiv:1110.5847v1 fatcat:kuf2yz7tibavnbtsf2lgujhx34

Tractable Clustering of Data on the Curve Manifold [article]

Stephen Tierney, Junbin Gao, Yi Guo, Zheng Zhang
2017 arXiv   pre-print
In this paper we propose a tractable method to cluster functional data or curves by adapting the Euclidean Low-Rank Representation (LRR) to the curve manifold.  ...  The naive treatment of functional data as traditional multivariate data can lead to poor performance since the algorithms are ignoring the correlation in the curvature of each function.  ...  The third author is partly supported by NSFC funded project (No. 41371362) and some of the work was carried out when he was with CSIRO. The fourth author is supported by SAMSI under grant DMS-1127914.  ... 
arXiv:1704.03963v1 fatcat:jyzs2w7cejb65oublpwtgrulpa

Page 9694 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
The authors define the notion of a low height representation of G and prove that, if G — SL(V) is a low height representation and E is a principal G-bundle on X which is semistable with respect to M, then  ...  Under mild restrictions on the prime p, the authors also prove a converse result, namely that if G is almost simple and G — SL(V) is a representation which is not of low height, then there exist a curve  ... 

Dimension induced clustering

Aristides Gionis, Alexander Hinneburg, Spiros Papadimitriou, Panayiotis Tsaparas
2005 Proceeding of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining - KDD '05  
We demonstrate the effectiveness of our algorithms for discovering low-dimensional m-flats embedded in high dimensional spaces, and for detecting low-rank submatrices.  ...  It is commonly assumed that high-dimensional datasets contain points most of which are located in low-dimensional manifolds.  ...  and local density can be used to detect low dimensional m-flats and low-rank sub-matrices.  ... 
doi:10.1145/1081870.1081880 dblp:conf/kdd/GionisHPT05 fatcat:bqtjs7ghgfebxi32li47zlcx7q

Geometry of Deep Generative Models for Disentangled Representations [article]

Ankita Shukla, Shagun Uppal, Sarthak Bhagat, Saket Anand, Pavan Turaga
2019 arXiv   pre-print
Deep generative models like variational autoencoders approximate the intrinsic geometry of high dimensional data manifolds by learning low-dimensional latent-space variables and an embedding function.  ...  We use several metrics to compare the properties of latent spaces of disentangled representation models in terms of class separability and curvature of the latent-space.  ...  network that maps the data manifold to low dimensional latent space, and a decoder network that learns to map these representations back to the data manifold.  ... 
arXiv:1902.06964v1 fatcat:ttyd7v3hqvgq7o35ydillvx5aa

Cover-based bounds on the numerical rank of Gaussian kernels

Amit Bermanis, Guy Wolf, Amir Averbuch
2014 Applied and Computational Harmonic Analysis  
This approach works under the assumption that, due to the low-dimensionality of the underlying manifold, the kernel has a low numerical rank.  ...  The cover-oriented methodology is also used to provide a relation between the geodesic length of a curve and the numerical rank of Gaussian kernel of datasets that are sampled from it.  ...  The second author was also supported by the Eshkol Fellowship from the Israeli Ministry of Science & Technology.  ... 
doi:10.1016/j.acha.2013.05.004 fatcat:27qevy32yjbate5jgmcouohgae

Combining Attention Model with Hierarchical Graph Representation for Region-Based Image Retrieval

S.-H. FENG, D. XU, B. LI
2008 IEICE transactions on information and systems  
the regions' significance. (2) A hierarchical graph representation which combines region-level with image-level similarities is utilized for the manifold-ranking method .  ...  Experimental results demonstrate that the proposed approach shows the satisfactory retrieval performance compared to the global-based and the block-based manifold-ranking methods.  ...  The results are averaged over the 5,000 queries. The precision versus scope curve is used to evaluate the performance of vari- ous methods.  ... 
doi:10.1093/ietisy/e91-d.8.2203 fatcat:tciwxcprhnbfvdrdwurixjz5gi

Clifford structures on Riemannian manifolds

Andrei Moroianu, Uwe Semmelmann
2011 Advances in Mathematics  
We give the complete classification of manifolds carrying parallel even Clifford structures: K\"ahler, quaternion-K\"ahler and Riemannian products of quaternion-K\"ahler manifolds, several classes of 8  ...  We introduce the notion of even Clifford structures on Riemannian manifolds, a framework generalizing almost Hermitian and quaternion-Hermitian geometries.  ...  We also thank the anonymous referee for his suggestions which helped us to improve the exposition.  ... 
doi:10.1016/j.aim.2011.06.006 fatcat:5r3czmlqizd7tb6k7zuuxipfwi

SAR Target Recognition via Local Sparse Representation of Multi-Manifold Regularized Low-Rank Approximation

2018 Remote Sensing  
We first propose a low-dimensional representation model via incorporating multi-manifold regularization term into the low-rank matrix factorization framework.  ...  By alternately optimizing the matrix factorization and manifold selection, the feature representation model can not only acquire the optimal low-rank approximation of original samples, but also capture  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/rs10020211 fatcat:fsqqjofnhfhjzblrgwggutqcr4

Coverings of dehn fillings of surface bundles

John Hempel
1986 Topology and its Applications  
Acknowledgement I would like to thank Marc Baker for pointing out an error, relative to the arguments of Section 4, in an earlier version of this paper.  ...  This paper concerns the study of 3-manifolds which are obtained by Dehn filling on a surface bundle over S' (every closed 3-manifold is so representable) and directed toward the question: (or of any  ...  In this paper we approach this problem in light of the result [14] that every closed, orientable 3-manifold contains a knot (embedded simple closed curve) whose complement fibers over S'.  ... 
doi:10.1016/0166-8641(86)90058-1 fatcat:nuxhzqi2b5hchotd2uso6fih4m

Low Rank Representation on Riemannian Manifold of Square Root Densities [article]

Yifan Fu and Junbin Gao and Xia Hong and David Tien
2015 arXiv   pre-print
In this paper, we present a novel low rank representation (LRR) algorithm for data lying on the manifold of square root densities.  ...  distance of the manifold.  ...  Recently, a low rank representation on Grassmann manifolds has been proposed in [32] by mapping the Grassmann manifold onto the Euclidean space of symmetric matrices.  ... 
arXiv:1508.04198v1 fatcat:bqfyxuc5bvcu3dr6o7yzckhkc4

A New Method for Performance Analysis in Nonlinear Dimensionality Reduction [article]

Jiaxi Liang, Shojaeddin Chenouri, Christopher G. Small
2017 arXiv   pre-print
We find that the local rank correlation closely corresponds to our visual interpretation of the quality of the output.  ...  In addition, we demonstrate that the local rank correlation is useful in estimating the intrinsic dimensionality of the original data, and in selecting a suitable value of tuning parameters used in some  ...  As the dimensionality of the representation increases, while remaining below the correct dimension q of the manifold, the local rank correlation should increase.  ... 
arXiv:1711.06252v1 fatcat:ynq7fvrxgnea3l5quwyfhjxy3e

Horizontal Dimensionality Reduction and Iterated Frame Bundle Development [chapter]

Stefan Sommer
2013 Lecture Notes in Computer Science  
The paper gives examples of how low-dimensional horizontal components successfully approximate multimodal distributions.  ...  In contrast, distances do not split into orthogonal components and centered analysis distorts inter-point distances in the presence of curvature.  ...  The ability of the method to provide a low-dimensional representation and visualization with correctly estimated variance is illustrated on low-dimensional manifolds.  ... 
doi:10.1007/978-3-642-40020-9_7 fatcat:gik2hdshrzdq3bu5ienmohvs5m

Double Nuclear Norm Based Low Rank Representation on Grassmann Manifolds for Clustering

Xinglin Piao, Yongli Hu, Junbin Gao, Yanfeng Sun, Baocai Yin
2019 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)  
Inspired of Low Rank representation theory, researchers proposed a series of effective clustering methods for high-dimension data with non-linear metric.  ...  In this paper, we propose a new low rank model for high-dimension data clustering task on Grassmann manifold based on the Double Nuclear norm which is used to better approximate the rank minimization of  ...  Acknowledgement The research project is supported by National Natural  ... 
doi:10.1109/cvpr.2019.01235 dblp:conf/cvpr/PiaoHGSY19 fatcat:cxrnksbtonempfrhe6rmlgd6rq
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