cvpr cvpr2013 cvpr2013-367 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Rui Caseiro, Pedro Martins, João F. Henriques, Fátima Silva Leite, Jorge Batista
Abstract: In the past few years there has been a growing interest on geometric frameworks to learn supervised classification models on Riemannian manifolds [31, 27]. A popular framework, valid over any Riemannian manifold, was proposed in [31] for binary classification. Once moving from binary to multi-class classification thisparadigm is not valid anymore, due to the spread of multiple positive classes on the manifold [27]. It is then natural to ask whether the multi-class paradigm could be extended to operate on a large class of Riemannian manifolds. We propose a mathematically well-founded classification paradigm that allows to extend the work in [31] to multi-class models, taking into account the structure of the space. The idea is to project all the data from the manifold onto an affine tangent space at a particular point. To mitigate the distortion induced by local diffeomorphisms, we introduce for the first time in the computer vision community a well-founded mathematical concept, so-called Rolling map [21, 16]. The novelty in this alternate school of thought is that the manifold will be firstly rolled (without slipping or twisting) as a rigid body, then the given data is unwrapped onto the affine tangent space, where the classification is performed.
Reference: text
sentIndex sentText sentNum sentScore
1 A popular framework, valid over any Riemannian manifold, was proposed in [31] for binary classification. [sent-3, score-0.066]
2 Once moving from binary to multi-class classification thisparadigm is not valid anymore, due to the spread of multiple positive classes on the manifold [27]. [sent-4, score-0.455]
3 It is then natural to ask whether the multi-class paradigm could be extended to operate on a large class of Riemannian manifolds. [sent-5, score-0.176]
4 We propose a mathematically well-founded classification paradigm that allows to extend the work in [31] to multi-class models, taking into account the structure of the space. [sent-6, score-0.216]
5 The idea is to project all the data from the manifold onto an affine tangent space at a particular point. [sent-7, score-0.672]
6 To mitigate the distortion induced by local diffeomorphisms, we introduce for the first time in the computer vision community a well-founded mathematical concept, so-called Rolling map [21, 16]. [sent-8, score-0.104]
7 The novelty in this alternate school of thought is that the manifold will be firstly rolled (without slipping or twisting) as a rigid body, then the given data is unwrapped onto the affine tangent space, where the classification is performed. [sent-9, score-1.046]
8 Introduction Applications in computer vision often involve the study of real world problems where the nonlinear constraints lead to data that lies on curved spaces [19, 28, 3]. [sent-11, score-0.069]
9 When treating cases that cannot be solved within the standard Euclidean tools, it is usual to resort to some local linear approximations or to use ad hoc solutions. [sent-12, score-0.027]
10 Those solutions are not always valid, which poses a challenge for several computer vision applications where data often lies in complex manifolds, namely in Riemannian manifolds i. [sent-13, score-0.256]
11 Prior Work : Recently, the development of geometric frameworks to learn supervised classification models on Riemannian manifolds has been addressed in the computer vision community [3 1, 27]. [sent-23, score-0.395]
12 This classifier is an additive model, where a set of weak learners are built by regression over the mappings of the data points on appropriate tangent planes (at the Karcher mean of the positive training points) and combined through boosting. [sent-26, score-0.176]
13 The consideration of the negative samples in the mean computation would bias the result, since they are assumed to be spread on the manifold [3 1, 27]. [sent-27, score-0.343]
14 This framework was tested to detect pedestrians in images using as descriptor a region covariance matrix [30] (Sym+ - symmetric positive definite matrices), but the algorithm is valid over any Riemannian manifold and can be combined with several different boosting (classification) methods. [sent-28, score-0.456]
15 Learning problems on Riemannian manifolds are generally solved by flattening the manifold via local diffeomorphisms [5], i. [sent-30, score-0.758]
16 the manifold is locally embedded into an Euclidean space. [sent-32, score-0.294]
17 However, embedding the manifold using those local diffeomorphisms leads to some problems. [sent-33, score-0.446]
18 The exponential map is onto but only one-to-one in a neighborhood of a point. [sent-34, score-0.061]
19 444111 whole manifold look like an Euclidean space. [sent-38, score-0.294]
20 [27] : once we try to change the paradigm from binary to multi-class classification the Tuzel’s framework [3 1] is not valid anymore due to the spread of multiple positive classes on the manifold. [sent-40, score-0.31]
21 From this perspective, it is natural to see efforts for solve this bottleneck. [sent-41, score-0.027]
22 Tuzel [3 1] endowed the Sym+ manifold with the well-known AffineInvariant metric, however a thorough analysis of this space opens a new perspective. [sent-42, score-0.393]
23 The space of Sym+ is a special Riemannian manifold since there is another metric, called Log-Euclidean [1], which allows to overcome the above limitations. [sent-43, score-0.328]
24 As showed in [1] the simple matrix exponential (exp) is a diffeomorphism from the Euclidean space of symmetric matrices to the Sym+ space. [sent-44, score-0.153]
25 The space of Sym+ endowed with a Log-Euclidean metric is in fact isomorphic (the algebraic structure of vector space is conserved) and isometric (distances are conserved) with the corresponding Euclidean space of symmetric matrices [1], i. [sent-45, score-0.404]
26 the LogEuclidean framework defines a mapping where the space of Sym+ is isomorphic, diffeomorphic and isometric to the associated space of symmetric matrices [1]. [sent-47, score-0.234]
27 This mapping is precisely the simple matrix logarithm (log), which can be seen as the logarithm map at the identity [1]. [sent-48, score-0.155]
28 By endowing the space of Sym+ with the Log-Euclidean metric, Tosato et al. [sent-49, score-0.104]
29 [27] proposed a mathematically wellfounded multi-class framework designed to operate on this particular manifold (Sym+). [sent-50, score-0.416]
30 All the data is projected onto a unique tangent space at the identity (simple matrix logarithm), where a typical multi-class LogitBoost algorithm is applied [7]. [sent-51, score-0.302]
31 [2] also embedded all the Sym+ manifold into an Euclidean space by endowing Sym+ with the Log-Euclidean metric to perform multiclass classification using linear SVM. [sent-53, score-0.481]
32 However, the Tosato/Carreira’s paradigm [27, 2] (embed all the manifold) is not generalizable in the sense that it cannot be applied to other Riemannian manifolds due to the specificity of the mapping/metric used. [sent-55, score-0.381]
33 It is then natural to ask whether the multi-class concept could be extended to operate on a large class of Riemannian manifolds. [sent-56, score-0.126]
34 Recently a new school of thought emerged [11, 12, 13, 5]. [sent-57, score-0.098]
35 This new paradigm suggests to embed the Riemannian manifold into a Reproducing Kernel Hilbert Space (RKHS) by using Mercer kernels on Riemannian manifolds. [sent-58, score-0.479]
36 [11, 12] proposed to use specific Grassmann kernels in order to embed the Grassmann manifold into a RKHS. [sent-60, score-0.384]
37 [13] used the Stein kernel to perform sparse coding and dictionary learning for symmetric positive definite matrices. [sent-62, score-0.131]
38 [5] proposed a novel kernel-based mean shift on general Riemannian man444222 ifolds, by using a general Riemannian kernel function, i. [sent-64, score-0.035]
39 However, the use of kernel-based algorithms for build classifiers on general Riemannian manifolds is not a good option. [sent-67, score-0.256]
40 Firstly, to our knowledge the heat kernel is the unique Mercer kernel suited to general Rieman- nian manifolds. [sent-68, score-0.135]
41 Secondly, the calculation of the heat kernel constitute a complex theoretical/technical problem and the computational burden is high. [sent-69, score-0.1]
42 Finally, by using Mercer kernels to implicitly project the data from the manifold we are restricted to use kernel-based classifiers. [sent-70, score-0.361]
43 The idea is to project all the data from the manifold onto an affine tangent space at a particular point (e. [sent-72, score-0.672]
44 To mitigate the distortion induced by local diffeomorphisms, we introduce for the first time in the computer vison community a well-founded mathematical concept, so-called Rolling map [16, 21]. [sent-75, score-0.139]
45 The novelty in this alternate school of thought is that the manifold will be firstly rolled (without slip and twist) as a rigid body, then the given data is unwrapped onto the affine tangent space, where the classification is performed. [sent-76, score-1.186]
46 For the sake of brevity the proof of concept will be done by testing with a multi-class LogitBoost algorithm [27, 7] on the Grassmann manifold [6, 25, 29, 28, 12, 16, 24]. [sent-77, score-0.339]
47 We remark that our paradigm is also valid with others Riemannian manifolds. [sent-78, score-0.161]
48 Rolling Maps on Riemannian Manifolds In the past few years there has been a growing interest in describing mathematically rolling motions, without slip and twist, of smooth manifolds (due to its analytic and geometric richness) [21, 15, 22, 16]. [sent-80, score-0.847]
49 The study of these kinematic problems proved to be relevant, in part because the knowledge on how to realize such virtual movements allows to solve complicated problems on certain manifolds, by reducing them to similar problems on much simpler manifolds. [sent-81, score-0.144]
50 For example, those rolling movements have been used with great success to compute interpolating curves and solve other optimal control problems on manifolds [15, 22, 16]. [sent-82, score-0.731]
51 The resulting curve is defined in explicit form, and has the advantage of being coordinate free [15, 16]. [sent-84, score-0.036]
52 Rollings motions are rigid motions in the embedding space, subject to some holonomic constraints (rolling con- Figure 1. [sent-85, score-0.263]
53 Rolling Map : M rolls upon M¯ = V ∼= TP0 M without slip or 1tw. [sent-86, score-0.175]
54 is Rt,o along a rolling curve α : [0, T] → M [22]. [sent-87, score-0.381]
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