iccv iccv2013 iccv2013-177 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Pengfei Zhu, Lei Zhang, Wangmeng Zuo, David Zhang
Abstract: Most of the current metric learning methods are proposed for point-to-point distance (PPD) based classification. In many computer vision tasks, however, we need to measure the point-to-set distance (PSD) and even set-to-set distance (SSD) for classification. In this paper, we extend the PPD based Mahalanobis distance metric learning to PSD and SSD based ones, namely point-to-set distance metric learning (PSDML) and set-to-set distance metric learning (SSDML), and solve them under a unified optimization framework. First, we generate positive and negative sample pairs by computing the PSD and SSD between training samples. Then, we characterize each sample pair by its covariance matrix, and propose a covariance kernel based discriminative function. Finally, we tackle the PSDML and SSDMLproblems by using standard support vector machine solvers, making the metric learning very efficient for multiclass visual classification tasks. Experiments on gender classification, digit recognition, object categorization and face recognition show that the proposed metric learning methods can effectively enhance the performance of PSD and SSD based classification.
Reference: text
sentIndex sentText sentNum sentScore
1 hk Abstract Most of the current metric learning methods are proposed for point-to-point distance (PPD) based classification. [sent-4, score-0.361]
2 In many computer vision tasks, however, we need to measure the point-to-set distance (PSD) and even set-to-set distance (SSD) for classification. [sent-5, score-0.234]
3 In this paper, we extend the PPD based Mahalanobis distance metric learning to PSD and SSD based ones, namely point-to-set distance metric learning (PSDML) and set-to-set distance metric learning (SSDML), and solve them under a unified optimization framework. [sent-6, score-1.083]
4 First, we generate positive and negative sample pairs by computing the PSD and SSD between training samples. [sent-7, score-0.171]
5 Then, we characterize each sample pair by its covariance matrix, and propose a covariance kernel based discriminative function. [sent-8, score-0.256]
6 Finally, we tackle the PSDML and SSDMLproblems by using standard support vector machine solvers, making the metric learning very efficient for multiclass visual classification tasks. [sent-9, score-0.287]
7 Experiments on gender classification, digit recognition, object categorization and face recognition show that the proposed metric learning methods can effectively enhance the performance of PSD and SSD based classification. [sent-10, score-0.265]
8 Introduction How to select a proper distance metric is a key problem in pattern classification, while the optimal distance metric for a specific pattern classification task depends on the underlying data structure and distributions. [sent-12, score-0.697]
9 In recent years, it has been increasingly popular to learn a desired distance metric from the given training samples in many visual classification tasks, such as face/action/kinship verification [14], visual tracking [18], and image retrieval [1]. [sent-13, score-0.429]
10 Metric learning methods can be categorized into unsupervised [9], semi-supervised [3] and supervised ones [14, 18, 1], according to the availability of the class labels of training samples. [sent-14, score-0.108]
11 In general, metric learning aims to learn a valid distance ∗Corresponding author metric, measured by which the samples from the positive sample pair (i. [sent-15, score-0.557]
12 , samples with the same class label or similar samples) could be as close as possible, while the samples from the negative sample pair (i. [sent-17, score-0.34]
13 , samples with the different class labels or dissimilar samples) could be as far as possible. [sent-19, score-0.111]
14 In some cases, only positive pairs are used in metric learning [14]. [sent-21, score-0.312]
15 In [27], metric learning is formulated as a kernel classification model and the relations with LMNN and ITML are discussed. [sent-22, score-0.287]
16 Metric learning algorithms have also been developed for multi-task learning [24], multiple instance learning [15] and nonlinear metrics [19]. [sent-23, score-0.147]
17 Currently, almost all the metric learning methods focus on the learning of a point-to-point distance (PPD) metric in couple with the nearest neighbor classifier (NNC). [sent-24, score-0.691]
18 , face recognition), however, we need to measure the distance between an image (i. [sent-27, score-0.117]
19 In video based recognition tasks [29] or multi-view object recognition [20], we even need to measure the distance between two image sets. [sent-32, score-0.117]
20 Therefore, it is highly desired to design effective point-to-set distance (PSD) and set-to-set distance (SSD) metric learning methods. [sent-33, score-0.5]
21 Unfortunately, many PPD metric learning methods cannot be readily applied to PSD and SSD based classification. [sent-34, score-0.244]
22 A set is often modeled as a hull, a convex hull (CH), or an affine hull (AH), and PSD can then be defined as the distance from a point to this hull. [sent-35, score-0.863]
23 Correspondingly, the nearest subspace classifier (NSC), nearest convex hull classifier (NCH) [26], and nearest convex affine classifier (NAH) [26] are proposed for PSD based classification. [sent-36, score-0.747]
24 In [6], a set is modeled as a bounding hyperdisk (the set formed by intersecting their affine hull and their smallest bounding hypersphere), and a nearest hyperdisk classifier (NHD) is pro2664 posed for classification [6]. [sent-37, score-0.678]
25 Given a query sample, those PSD based classifiers (NSC, NCH, NAH and NHD) compute its distance to each class, i. [sent-38, score-0.201]
26 , the PSD between the query samples and the set of templates of this class, and classify it to the class with the minimal point-to-set distance. [sent-40, score-0.218]
27 In [30], an image to class distance is learned in a multi-task way by considering each class as one task. [sent-41, score-0.235]
28 In [36], an image to class distance is defined by minimizing the distance over all possible object configurations and all possible object matchings, and then the distance function parameters are learned. [sent-42, score-0.41]
29 The work in [30] and [36] both focus on a special image to class distance rather than a general point to set distance. [sent-43, score-0.176]
30 In [5], by modeling each set as a CH/AH, the CH/AH based image set distance (CHISD/AHISD) is defined. [sent-45, score-0.117]
31 In [16], sparsity is imposed on the AH model and a sparse approximation nearest points (SANP) method is proposed for image set classification. [sent-46, score-0.079]
32 In [35], a regularized affine hull (RAH) is proposed to model a set, and the SSD is defined between two RAHs. [sent-47, score-0.414]
33 In [34], each set is represented by a linear subspace and the angles between two subspaces are utilized to measure the similarity of two sets. [sent-48, score-0.029]
34 In [29], an image set is modeled as a manifold and a manifold-to-manifold distance (MMD) is proposed. [sent-50, score-0.144]
35 After calculating the distance from the query set to each template set, those SSD based classifiers classify the query set to the class with the minimal set-to-set distance. [sent-51, score-0.401]
36 To introduce discriminative information to SSD, projection matrix is learned in a large margin manner, e. [sent-52, score-0.051]
37 , discriminative canonical correlation (DCC) [20] and manifold discriminant analysis (MDA) [28]. [sent-54, score-0.029]
38 In [32], a set based discriminative ranking model is proposed by iterating between SSD finding and discriminative feature space projection. [sent-55, score-0.058]
39 PSD (left) and SSD (right) Metric learning Inspired by the success of metric learning in PPD based classification, the performance of PSD and SSD based classification can also be boosted by metric learning. [sent-57, score-0.531]
40 1(a), the query image y (represented as a red dot) has the same class label as template set X1 (represented as a red hull) but it will be misclassified since it has a closer PSD to set X2. [sent-59, score-0.207]
41 If a proper metric learning method can be developed, it is possible that with the new distance metric, the PSD between y and X1 is smaller than that between y and X2, and consequently y can be correctly classified, as shown in the bottom part of Fig. [sent-60, score-0.391]
42 Similar anticipation goes to the metric learning of SSD based classification, as illustrated in Fig. [sent-62, score-0.244]
43 1(b), where the query set Y can be correctly classified with some proper SSD based distance metric. [sent-63, score-0.255]
44 With the above considerations, in this paper we propose two novel metric learning models, PSD metric learning (PSDML) and SSD metric learning (SSDML), to enhance the performance of PSD and SSD based classification. [sent-64, score-0.753]
45 One image (or image set) and one similarly labeled image set construct a positive pair, while one image (or image set) and one differently labeled set construct a negative pair. [sent-65, score-0.073]
46 Then the PSDML and SSDML problems are formulated as a sample pair classification problem. [sent-66, score-0.152]
47 Each sample pair is characterized by the covariance matrix of its two samples, and a covariance kernel is introduced. [sent-67, score-0.249]
48 A discriminative function is then proposed for sample pair classification, and finally the PSDML and SSDML can be solved by using an SVM model. [sent-68, score-0.138]
49 The proposed PSDML and SSDML methods can effectively improve the performance of PSD and SSD based classification, and are much more efficient than state-of-the- art metric learning methods. [sent-69, score-0.244]
50 The main abbreviations used in this paper are summarized in the following Table 1. [sent-70, score-0.051]
51 The main abbreviations used in this paper PPDpoint to point distance PSD point to set distance SSD set to set distance PSDML point to set distance metric learning SSDML set to set distance metric learning 2. [sent-72, score-1.124]
52 Set based distances Before distance metric learning, we need to first define how the distance is measured. [sent-73, score-0.429]
53 , a subspace spanned by all the available samples in the set. [sent-79, score-0.081]
54 ( ai =) =1 i {s required aenrde ai i=s required to be bounded: H(D) = {? [sent-95, score-0.142]
55 If τ1 <= 0 − ainndf τ2 >d τ0, H(D) is a reduced affine hull [5]. [sent-100, score-0.378]
56 If 2665 = 0 and τ2 = 1, H(D) is a convex hull [26]. [sent-101, score-0.341]
57 If τ1 = 0 and τ2 < 1, H(D) is a reduced convex hull [5]. [sent-102, score-0.341]
58 To rule out the meaningless points which are too far from the sample mean, the regularized affine hull (RAH) [35] is defined as follows to model an image set: τ1 H(D) = ? [sent-103, score-0.479]
59 ce (PSD) Given a sample x and a set of samples D, a point to set distance d(x, D) between x and D can be defined as follows: d(x, D) = ? [sent-114, score-0.234]
60 x − H(D) When H(D) is a hull, the solution of mina ? [sent-117, score-0.059]
61 P can be introduced to project the samples into a desired space. [sent-125, score-0.074]
62 22, and M = PTP, (5) When ˆa is obtained, we can form a sample pair (x, D aˆ). [sent-137, score-0.109]
63 (4) can be viewed as a Mahalanobis distance [10] between x and D aˆ, and the matrix M is always semi-positive definite. [sent-139, score-0.139]
64 In PSD based classification, the distance between the query sample y and the template set of each class X1, X2 , . [sent-140, score-0.359]
65 Suppose that the nearest subspace classifier (NSC) is used. [sent-144, score-0.115]
66 Given M, for class i, we have ai = Wiy, where Wi = ? [sent-145, score-0.119]
67 (7) The class with the minimal PSD is assigned to y: Label(y) = arg mini{dM (y, Xi)}. [sent-151, score-0.116]
68 Compared with the{ dnearest convex hull/affine hull classifier (NCH/NAH), which needs to solve c quadratic programming problems for the query sample y, NSC only needs to compute a set of linear projections of y with Wi, i= 1, 2, . [sent-152, score-0.521]
69 Set-to-set distance (SSD) Given two image sets D1 and D2, the set-to-set distance (SSD) between them can be defined as follows: d(D1,D2) = where ? [sent-159, score-0.234]
70 In [35], it has been shown that l2-norm regularized affine hull is much faster and can achieve comparable performance to convex/affine/sparse constraints. [sent-169, score-0.414]
71 Given a linear projection matrix P, the RNP model is: sm. [sent-170, score-0.022]
72 )2 (11) In SSD based classification, given a query image set Y , the SSD between it and each template set Xi, i= 1, 2, . [sent-186, score-0.118]
73 Y can then be classified by Label(Y ) = l(Xiˆ), where arg mini{dM (Y , Xi) }. [sent-190, score-0.058]
74 Distance metric learning With the definitions in Section 2, we can then design the metric learning algorithms for PSD and SSD based classification. [sent-192, score-0.488]
75 (7), the matrix M plays a critical role in the final distance dM (y, Xi). [sent-196, score-0.139]
76 It is expected that a good M can be learned from the training sample sets {X1, X2 , . [sent-197, score-0.065]
77 , Xc}, so that the PSD between a query sample y Xand the set Xl(y) can baet t rheedu PcSeDd, bwethwileee tnhe a Pq uSDer yb setawmepelne y and the other sets Xj ,j l(y), can be enlarged, where l(y) is the label of y. [sent-200, score-0.179]
78 , c, we propose the following metric learning = 2666 model: sm. [sent-203, score-0.244]
79 denotes the Frobenius norm, al(xi) and aj are cwoheeffriec ? [sent-212, score-0.066]
80 ts vector for Xl(xi) and Xj, b is the bias and ν is a positive constant. [sent-214, score-0.035]
81 ξiP and ξiNj are slack variables for positive and negative pairs. [sent-215, score-0.073]
82 dM (xi, Xl(xi) ) is the PSD distance from xi to the set it belongs to (i. [sent-216, score-0.257]
83 , the PSD of positive pairs), where l(xi) is the class label of xi, and dM (xi, Xj) ,j l(xi), is the PSD from xi to other classes (i. [sent-218, score-0.264]
84 (13) is a joint optimization problem of M and {al(xi) , aj }. [sent-223, score-0.066]
85 (13) by optimizing M and {al(xi) , aj } alternatively. [sent-225, score-0.066]
86 When M is foipxteimd, {al(xi) , aj }d are solved f}or a atellr nthateiv training samples. [sent-226, score-0.066]
87 fNixoetde ,t {haat here the}“ aleraev es-oolnveed-o fuot”r strategy aisi nuinsged s atom compute al(xi) . [sent-227, score-0.026]
88 That is, X¯l(xi) is the training sample set of class l(xi) but excluding sample xi. [sent-228, score-0.189]
89 Then the positive pairs are formed as (xi , X¯l(xi) aˆl(xi) ) and the negative pairs are formed as (xi , Xj,j? [sent-229, score-0.195]
90 We label the negative pair as “+1” and the positive pair is set as “-1”. [sent-232, score-0.191]
91 Let us denote by zi = (zi1, zi2) a generated sample pair. [sent-233, score-0.108]
92 The covariance matrix of the two samples in zi is Ci = (zi1 − zi2)(zi1 − zi2)T. [sent-234, score-0.176]
93 Suppose that we generated ns training sample pairs, and thus we have ns covariance F = matrices Ci, i = 1, 2, . [sent-235, score-0.18]
94 We label Ci as “+1” or “-1” based on the label of zi, and define the following kernel function to measure the similarity between Ci and Cj : k(Ci, Cj) = tr(CiCj) =< Ci, Cj > (14) where tr(·) is the trace operator of a matrix and < ·, · > means ttrhe( )inn iser th product oofp emraattroirc oefs. [sent-239, score-0.134]
95 Suppose that we have a query sample pair, denoted by z = (z1, z2). [sent-240, score-0.149]
96 We introduce the following discriminative function to judge whether z is positive or negative: f(C? [sent-242, score-0.064]
97 ) ≥iξ 1i − ξi,ξi≥ 0 (17) The Lagrange dual problem of the metric learning problem in Eq. [sent-261, score-0.244]
98 (18) can be easily solved by the support vector machine (SVM) solvers such as LIBSVM [7]. [sent-269, score-0.026]
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