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Manifold Learning in Quotient Spaces

Eloi Mehr, Andre Lieutier, Fernando Sanchez Bermudez, Vincent Guitteny, Nicolas Thome, Matthieu Cord
2018 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition  
To get rid of undesirable input variability our model learns a manifold in a quotient space of the input space. Typically, we propose to quotient the space of 3D models by the action of rotations.  ...  Thus, our quotient autoencoder allows to directly learn in the space of interest, ignoring side information.  ...  Manifold learning aims at representing the data by learning an embedding in a low-dimensional space.  ... 
doi:10.1109/cvpr.2018.00955 dblp:conf/cvpr/MehrLBGTC18 fatcat:5ib5i2txtfdfxkljuvjmrlnv64

Fixed-rank matrix factorizations and Riemannian low-rank optimization [article]

B. Mishra, G. Meyer, S. Bonnabel, R. Sepulchre
2013 arXiv   pre-print
We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent  ...  We adopt the geometric framework of optimization on Riemannian quotient manifolds.  ...  The set W is the total space of the quotient manifold W. A popular example of quotient manifold is the Grassmann manifold Gr(r, d), that is, the set of r-dimensional subspaces in R d×r .  ... 
arXiv:1209.0430v2 fatcat:6riq4ml5mres5owb3x636mi3ee

Linear Regression under Fixed-Rank Constraints: A Riemannian Approach

Gilles Meyer, Silvere Bonnabel, Rodolphe Sepulchre
2011 International Conference on Machine Learning  
In this paper, we tackle the problem of learning a linear regression model whose parameter is a fixed-rank matrix.  ...  machine learning algorithms.  ...  In a nutshell, a quotient manifold is a set of equivalence classes.  ... 
dblp:conf/icml/MeyerBS11 fatcat:bwov435ewnfujcxsp6a6z4h64m

A Manifold Approach to Learning Mutually Orthogonal Subspaces [article]

Stephen Giguere, Francisco Garcia, Sridhar Mahadevan
2017 arXiv   pre-print
Each point on the manifold defines a partitioning of the input space into mutually orthogonal subspaces, where the number of partitions and their sizes are defined by the user.  ...  Although many machine learning algorithms involve learning subspaces with particular characteristics, optimizing a parameter matrix that is constrained to represent a subspace can be challenging.  ...  Quotient manifolds inherit a great deal of structure from their parent manifold. In particular, constraints that define the parent manifold also apply to the quotient.  ... 
arXiv:1703.02992v1 fatcat:iuoxkoni7vahxf64slaxjdl2ne

Equivariant Manifold Flows [article]

Isay Katsman, Aaron Lou, Derek Lim, Qingxuan Jiang, Ser-Nam Lim, Christopher De Sa
2022 arXiv   pre-print
In this paper, we lay the theoretical foundations for learning symmetry-invariant distributions on arbitrary manifolds via equivariant manifold flows.  ...  Tractably modelling distributions over manifolds has long been an important goal in the natural sciences.  ...  In addition, we thank the National Science Foundation for awarding Prof.  ... 
arXiv:2107.08596v2 fatcat:xgs3a4n3xzailmbx4o6alg4q34

Ultrahyperbolic Neural Networks

Marc Law
2021 Neural Information Processing Systems  
Riemannian space forms, such as the Euclidean space, sphere and hyperbolic space, are popular and powerful representation spaces in machine learning.  ...  This paper introduces a method to learn parametric models in ultrahyperbolic space.  ...  In this paper, we propose a parametric model that learns representations lying on the quotient manifold P p,q r .  ... 
dblp:conf/nips/Law21 fatcat:tz66v5qas5dfrovorqvjhfhbbu

Advances in matrix manifolds for computer vision

Yui Man Lui
2012 Image and Vision Computing  
The increased popularity of matrix manifolds is due partly to the need to characterize image features in non-Euclidean spaces.  ...  Matrix manifolds provide rigorous formulations allowing patterns to be naturally expressed and classified in a particular parameter space.  ...  Whereas discriminant analysis learns a projection that maximizes the trace quotient, multivariate regression learns a projection by fitting the class labels in a regression space.  ... 
doi:10.1016/j.imavis.2011.08.002 fatcat:wdzjikdbpfgobgopkvc4rmqkqy

Multiple Atlas Construction From A Heterogeneous Brain MR Image Collection

Yuchen Xie, Jeffrey Ho, Baba C. Vemuri
2013 IEEE Transactions on Medical Imaging  
The geometric notion that underlies our approach is the idea of manifold learning in a quotient space, the quotient space of the image space by the rotations.  ...  We present an extension of the existing manifold learning approach to quotient spaces by using invariant metrics, and utilizing the manifold structure for partitioning the images into more homogeneous  ...  Second, the proposed method explicitly models the intrinsic manifold structure in the quotient space using manifold learning ( -NN graph) [13] , [14] , [22] .  ... 
doi:10.1109/tmi.2013.2239654 pmid:23335665 pmcid:PMC3595350 fatcat:dojcxcmi5fcwtfne7ps4nfv72y

A Differential Topological View of Challenges in Learning with Feedforward Neural Networks [article]

Hao Shen
2018 arXiv   pre-print
Finally, in the setting of deep representation learning, we further apply the quotient topology to investigate the architecture of DNNs, which enables to capture nuisance factors in data with respect to  ...  a specific learning task.  ...  Expressiveness of DNNs: Width vs depth Most data studied in machine learning often share some low-dimensional structure. In this work, we endow the input space X with a smooth manifold structure.  ... 
arXiv:1811.10304v1 fatcat:wvdwdp76hrgmfmtrjbc5nmlgvq

Trace Quotient Problems Revisited [chapter]

Shuicheng Yan, Xiaoou Tang
2006 Lecture Notes in Computer Science  
First, considering that the feasible solutions are constrained on a Grassmann manifold, we present a necessary condition for the optimal solution of the trace quotient problem, which then naturally elicits  ...  In this paper, we present a direct solution to the former formulation.  ...  Quotient Trace Problem A large family of algorithms for subspace learning [6] ends with solving a trace quotient problem as in (1).  ... 
doi:10.1007/11744047_18 fatcat:v5k3e7qxvfadnhicpux6kivu7a

Groups not acting on manifolds [article]

David Fisher, Lior Silberman
2008 arXiv   pre-print
In this article we collect a series of observations that constrain actions of many groups on compact manifolds.  ...  In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the  ...  By a non-positively curved Hilbert manifold, we mean a complete geodesic CAT(0) metric space all of whose tangent cones are (isometric to) Hilbert spaces.  ... 
arXiv:0801.0875v3 fatcat:k4pnjdqajzfqnhs66i3kcnqtde

Groups Not Acting on Manifolds

D. Fisher, L. Silberman
2010 International mathematics research notices  
In this article we collect a series of observations that constrain actions of many groups on compact manifolds.  ...  In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the  ...  By a non-positively curved Hilbert manifold, we mean a complete geodesic CAT(0) metric space all of whose tangent cones are (isometric to) Hilbert spaces.  ... 
doi:10.1093/imrn/rnn060 fatcat:qqaks7iievagxddhlcuj6ymhfy

QGOpt: Riemannian optimization for quantum technologies

Ilia Luchnikov, Alexander Ryzhov, Sergey Filippov, Henni Ouerdane
2021 SciPost Physics  
., orthogonality of isometric and unitary matrices, CPTP property of quantum channels, and conditions on density matrices, can be seen as quotient or embedded Riemannian manifolds.  ...  In the present work, we introduce QGOpt, the library for constrained optimization in quantum technology.  ...  The projection method of quotient manifold performs the projection on the horizontal space.  ... 
doi:10.21468/scipostphys.10.3.079 fatcat:kmrbtvuy3bhcthkijj7gj3hc4a

Low-Rank Riemannian Optimization on Positive Semidefinite Stochastic Matrices with Applications to Graph Clustering

Ahmed Douik, Babak Hassibi
2018 International Conference on Machine Learning  
In particular, the paper explicitly derives the tangent space, Riemannian gradients and retraction operator that allow the design of efficient optimization methods on the proposed manifolds.  ...  The reparameterization of the problem allows its formulation as an optimization over either an embedded or quotient Riemannian manifold whose geometries are investigated.  ...  Acknowledgements The authors would like to thank Ramya Korlakai Vinayak for collecting and providing the real-world data used in this manuscript.  ... 
dblp:conf/icml/DouikH18 fatcat:txevwnt5tbdyvhjhm2ac6vgtpi

QGOpt: Riemannian optimization for quantum technologies [article]

I. A. Luchnikov, A. Ryzhov, S. N. Filippov, H. Ouerdane
2021 arXiv   pre-print
., orthogonality of isometric and unitary matrices, CPTP property of quantum channels, and conditions on density matrices, can be seen as quotient or embedded Riemannian manifolds.  ...  In the present work, we introduce QGOpt, the library for constrained optimization in quantum technology.  ...  The projection method of quotient manifold performs the projection on the horizontal space.  ... 
arXiv:2011.01894v3 fatcat:ruyx3tlmabefjj2bdw3euah22i
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