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The Manifold Tangent Classifier

Salah Rifai, Yann N. Dauphin, Pascal Vincent, Yoshua Bengio, Xavier Muller
2011 Neural Information Processing Systems  
We combine three important ideas present in previous work for building classifiers: the semi-supervised hypothesis (the input distribution contains information about the classifier), the unsupervised manifold  ...  insensitive to local directions changes along the manifold.  ...  Acknowledgments The authors would like to acknowledge the support of the following agencies for research funding and computing support: NSERC, FQRNT, Calcul Québec and CIFAR.  ... 
dblp:conf/nips/RifaiDVBM11 fatcat:4q3jlpzi35erhkxue7dfkk43ta

Global Versus Localized Generative Adversarial Nets

Guo-Jun Qi, Liheng Zhang, Hao Hu, Marzieh Edraki, Jingdong Wang, Xian-Sheng Hua
2018 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition  
Furthermore, it can prevent the manifold from being locally collapsed to a dimensionally deficient tangent subspace by imposing an orthonormality prior between tangents.  ...  We will also demonstrate the LGAN can be applied to train a robust classifier that prefers locally consistent classification decisions on the manifold, and the resultant regularizer is closely related  ...  Acknowledgement The research was partly supported by NSF grant #1704309 and IARPA grant #D17PC00345.  ... 
doi:10.1109/cvpr.2018.00164 dblp:conf/cvpr/QiZHEWH18 fatcat:5axtaxoehjckjciyl2nmqklcwa

Human Detection via Classification on Riemannian Manifolds

Oncel Tuzel, Fatih Porikli, Peter Meer
2007 2007 IEEE Conference on Computer Vision and Pattern Recognition  
We present a novel approach for classifying points lying on a Riemannian manifold by incorporating the a priori information about the geometry of the space.  ...  The space of d-dimensional nonsingular covariance matrices can be represented as a connected Riemannian manifold.  ...  The tangent space is a vector space and we learn the classifiers on this space. The classifiers can be trained on the tangent space at any point on the manifold.  ... 
doi:10.1109/cvpr.2007.383197 dblp:conf/cvpr/TuzelPM07 fatcat:hdzasvokergpbjujvnth34as5e

Global versus Localized Generative Adversarial Nets [article]

Guo-Jun Qi, Liheng Zhang, Hao Hu, Marzieh Edraki, Jingdong Wang and Xian-Sheng Hua
2018 arXiv   pre-print
Furthermore, it can prevent the manifold from being locally collapsed to a dimensionally deficient tangent subspace by imposing an orthonormality prior between tangents.  ...  We will also demonstrate the LGAN can be applied to train a robust classifier that prefers locally consistent classification decisions on the manifold, and the resultant regularizer is closely related  ...  Experimental Results for Conv-Large We compare the LGAN using Conv-Large discriminator with state-of-the-art semi-supervised baselines. The results are reported in Table 4 .  ... 
arXiv:1711.06020v2 fatcat:olr7mkzorja4tigie2dakcf7wq

Local Subspace Classifier with Transform-Invariance for Image Classification

S. HOTTA
2008 IEICE transactions on information and systems  
After that, the input sample is classified into the class to which the nearest manifold belongs.  ...  The proposed method classifies the in- put sample to the class to which the nearest manifold be- longs.  ... 
doi:10.1093/ietisy/e91-d.6.1756 fatcat:t6toltpvcfgv5dnqmpesf3oa6m

Learning on Manifolds [chapter]

Fatih Porikli
2010 Lecture Notes in Computer Science  
to search, cluster, classify, and recognize given observations on smooth manifolds without flattening, charting, or dimensionality reducing them.  ...  and recognition problems are demonstrated after reviewing some of the fundamental preliminaries. ★ Throughout this paper, learning on manifolds refers to the family of supervised and unsupervised methods  ...  , thus, the classifiers can be trained on the tangent space at any point on the manifold.  ... 
doi:10.1007/978-3-642-14980-1_2 fatcat:kszllhejt5gmhac25653zdfjae

Manifold Matching for High-Dimensional Pattern Recognition [chapter]

Seiji Hotta
2008 Pattern Recognition Techniques, Technology and Applications  
Combination of manifold matching and tangent distance Let us start with a brief review of tangent distance before introducing the way of combining manifold matching and tangent distance.  ...  The manifold S q is approximated linearly by its tangent hyperplane at a point q: where t q i is the ith d-dimensional tangent vector (TV) that spans the r-dimensional tangent hyperplane (i.e., the number  ...  Manifold Matching for High-Dimensional Pattern Recognition, Pattern Recognition Techniques, Technology and Applications, Peng-Yeng Yin (Ed.), ISBN: 978-953-7619-24-4, InTech, Available from: http://www.intechopen.com  ... 
doi:10.5772/6247 fatcat:cqjzuf35tzfhpmckazxnskhjce

Color Texture Discrimination Using the Principal Geodesic Distance on a Multivariate Generalized Gaussian Manifold [chapter]

Geert Verdoolaege, Aqsa Shabbir
2015 Lecture Notes in Computer Science  
It is shown to perform significantly better than several other classifiers.  ...  Then, the similarity of a texture to a class is defined in terms of the Rao geodesic distance on the manifold from the texture's distribution to its projection on the principal geodesic of that class.  ...  The resulting tangent vectors, which are the eigenvectors of the covariance matrix in the tangent space, uniquely define a set of geodesic subspaces of the manifold.  ... 
doi:10.1007/978-3-319-25040-3_41 fatcat:bgaovkefdffqncjcfuzphrasbu

Distance Learner: Incorporating Manifold Prior to Model Training [article]

Aditya Chetan, Nipun Kwatra
2022 arXiv   pre-print
The manifold hypothesis (real world data concentrates near low-dimensional manifolds) is suggested as the principle behind the effectiveness of machine learning algorithms in very high dimensional problems  ...  We also evaluate our method on the task of adversarial robustness, and find that it not only outperforms standard classifier by a large margin, but also performs at par with classifiers trained via state-of-the-art  ...  Other methods such as Manifold Tangent Classifier [Rifai et al., 2011] which encourage network invariance along learnt tangent spaces have also been proposed.  ... 
arXiv:2207.06888v1 fatcat:7z7afm7sbzaxpj6gqcsr5h5xgq

K-tangent spaces on Riemannian manifolds for improved pedestrian detection

Andres Sanin, Conrad Sanderson, Mehrtash T. Harandi, Brian C. Lovell
2012 2012 19th IEEE International Conference on Image Processing  
This is often done through representing the descriptors as points on Riemannian manifolds, with the discrimination accomplished on a tangent space.  ...  However, such treatment is restrictive as distances between arbitrary points on the tangent space do not represent true geodesic distances, and hence do not represent the manifold structure accurately.  ...  Acknowledgements: NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy, as well as the Australian Research Council through  ... 
doi:10.1109/icip.2012.6466899 dblp:conf/icip/SaninSHL12 fatcat:kfvyz5ihcrap5djtuyr5o4sice

Semi-supervised Learning with GANs: Manifold Invariance with Improved Inference [article]

Abhishek Kumar, Prasanna Sattigeri, P. Thomas Fletcher
2017 arXiv   pre-print
Motivated by the ability of the GANs generator to capture the data manifold well, we propose to estimate the tangent space to the data manifold using GANs and employ it to inject invariances into the classifier  ...  Most of these methods use a shared discriminator/classifier which discriminates real examples from fake while also predicting the class label.  ...  We propose to use the generator to obtain the tangents to the image manifold and use these to inject invariances into the classifier [36] .  ... 
arXiv:1705.08850v2 fatcat:bhvyhsbujfenvprmi6ttvpy4vu

Spherical Embedding and Classification [chapter]

Richard C. Wilson, Edwin R. Hancock
2010 Lecture Notes in Computer Science  
We use the Lie group representation of the hypersphere and its associated Lie algebra to define the exponential map between the manifold and its local tangent space.  ...  We also define the nearest mean classifier on the manifold and give results for the embedding accuracy, the nearest mean classifier and the nearest-neighbor classifier on a variety of indefinite datasets  ...  The exponential map is a map from points on the manifold to points on a tangent space of the manifold. As the tangent space is flat (i.e.  ... 
doi:10.1007/978-3-642-14980-1_58 fatcat:qrx7jlegh5gzdcf4inpud4sbbi

Page 901 of Mathematical Reviews Vol. 36, Issue 4 [page]

1968 Mathematical Reviews  
Stasheff, Topology 2 (1963), 239-246 ; MR 27 #4235], the authors prove that if r: M — BO, is the classifying map for the tangent bundle of a differentiable n-manifold and if the map M— BO, BF, can be factored  ...  Consider the H*(B,)-structures on H*(B,) and H*(B) induced by the map of classifying spaces B, > B, and the classifying map B—> Bg.  ... 

Pedestrian Detection via Classification on Riemannian Manifolds

O. Tuzel, F. Porikli, P. Meer
2008 IEEE Transactions on Pattern Analysis and Machine Intelligence  
The main contribution of the paper is a novel approach for classifying points lying on a connected Riemannian manifold using the geometry of the space.  ...  Since the descriptors do not form a vector space, well-known machine learning techniques are not well suited to learn the classifiers.  ...  , and Dariu Gavrila for providing the data sets and the results of their experiments.  ... 
doi:10.1109/tpami.2008.75 pmid:18703826 fatcat:7eslwqitefbo3ewx5jfx4ksvpm

Transductive Few-Shot Classification on the Oblique Manifold [article]

Guodong Qi, Huimin Yu, Zhaohui Lu, Shuzhao Li
2021 arXiv   pre-print
Furthermore, we design an Oblique Distance-based Classifier (ODC) that achieves classification in the tangent spaces which better approximate OM locally by learnable tangency points.  ...  In this work, we perform the feature extraction in the Euclidean space and the geodesic distance metric on the Oblique Manifold (OM).  ...  For clarity and visibility, we take 2-way 2-shot classification as an example, plot partially, and push the tangent spaces away from the manifold in the classifier.  ... 
arXiv:2108.04009v1 fatcat:q2qdzvob4rennckiazr7cfkm5m
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