iccv iccv2013 iccv2013-295 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Stefanos Zafeiriou, Irene Kotsia
Abstract: Kernels have been a common tool of machine learning and computer vision applications for modeling nonlinearities and/or the design of robust1 similarity measures between objects. Arguably, the class of positive semidefinite (psd) kernels, widely known as Mercer’s Kernels, constitutes one of the most well-studied cases. For every psd kernel there exists an associated feature map to an arbitrary dimensional Hilbert space H, the so-called feature space. Tdihme mnsaiionn reason ebreth sipnadc ep s Hd ,ke threne slos’-c c aplolpedul aferiattyu rise the fact that classification/regression techniques (such as Support Vector Machines (SVMs)) and component analysis algorithms (such as Kernel Principal Component Analysis (KPCA)) can be devised in H, without an explicit defisnisiti (oKnP of t)h)e c feature map, only by using athne xkperlniceitl (dtehfeso-called kernel trick). Recently, due to the development of very efficient solutions for large scale linear SVMs and for incremental linear component analysis, the research to- wards finding feature map approximations for classes of kernels has attracted significant interest. In this paper, we attempt the derivation of explicit feature maps of a recently proposed class of kernels, the so-called one-shot similarity kernels. We show that for this class of kernels either there exists an explicit representation in feature space or the kernel can be expressed in such a form that allows for exact incremental learning. We theoretically explore the properties of these kernels and show how these kernels can be used for the development of robust visual tracking, recognition and deformable fitting algorithms. 1Robustness may refer to either the presence of outliers and noise the robustness to a class of transformations (e.g., translation). or to ∗ Irene Kotsia ,†,? ∗Electronics Laboratory, Department of Physics, University of Patras, Greece ?School of Science and Technology, Middlesex University, London i .kot s i @mdx . ac .uk a
Reference: text
sentIndex sentText sentNum sentScore
1 uk Abstract Kernels have been a common tool of machine learning and computer vision applications for modeling nonlinearities and/or the design of robust1 similarity measures between objects. [sent-4, score-0.184]
2 For every psd kernel there exists an associated feature map to an arbitrary dimensional Hilbert space H, the so-called feature space. [sent-6, score-0.484]
3 Recently, due to the development of very efficient solutions for large scale linear SVMs and for incremental linear component analysis, the research to- wards finding feature map approximations for classes of kernels has attracted significant interest. [sent-8, score-0.544]
4 In this paper, we attempt the derivation of explicit feature maps of a recently proposed class of kernels, the so-called one-shot similarity kernels. [sent-9, score-0.29]
5 We show that for this class of kernels either there exists an explicit representation in feature space or the kernel can be expressed in such a form that allows for exact incremental learning. [sent-10, score-0.975]
6 We theoretically explore the properties of these kernels and show how these kernels can be used for the development of robust visual tracking, recognition and deformable fitting algorithms. [sent-11, score-0.483]
7 Introduction In kernel learning2 [26], for each positive semi definite (psd) kernel k there exists an associated feature map φ to an arbitrary dimensional (in some cases infinite) feature space H. [sent-21, score-0.794]
8 In that case, the kernel learning problem, even though being n tohnatli cneasaer ,in th tehe k original space, pbroecbolemmes, elvineenar t hionu utgheh new space H. [sent-22, score-0.397]
9 The explicit form of φ is not required to perfnoewrm s palal computations, as tth feo rsmo-c oafl φled is k neornte rel qtruiicrke d(i t. [sent-23, score-0.145]
10 Recently, the following reverse problem has attracted a lot of attention: Given a kernel k, one should find an efficient and effective approximation of φ that successfully replaces the kernel [27, 18, 2, 15, 16, 2]. [sent-26, score-0.794]
11 The second concerned the unavailability of both exact (or effective) and efficient incremental versions of Principal Component Analysis (PCA) algorithms with kernels [4, 11, 7] (there exist only for the linear case [14, 23]). [sent-29, score-0.462]
12 Indeed, the most well known incremental Kernel PCA (KPCA) algorithms presented in [11, 7, 4] use only approximations. [sent-30, score-0.25]
13 In [4], the authors kernelized an exact algorithm for incremental PCA [14, 23], but, in order to maintain a constant update speed, they constructed a reduced set of expansions of the kernel principal components and of the mean, using pre-images. [sent-32, score-0.749]
14 2As kernel learning we refer to the general framework of classification, regression and component analysis with kernels. [sent-34, score-0.446]
15 22339922 As mentioned above, a kernel learning problem becomes linear in the feature space H. [sent-35, score-0.397]
16 i cHieennct aen, wd hleownl-odwim-ednimsieonnsailo approximations exist for φ we can take full advantage of efficient packages for regression and classification [9] but also of exact and low cost incremental PCA algorithms [14, 23]. [sent-37, score-0.413]
17 However, it was recently shown that for some particular classes of kernels such approximations do exist. [sent-39, score-0.277]
18 The main lines of research towards efficient approximation of features map include (a) exploiting particular kernel properties to find the approximation (e. [sent-40, score-0.425]
19 , in [27, 18] the authors exploited various properties to propose efficient and effective approximations of large families of additive kernels); (b) the application of random sampling on Fourier features [15, 16, 2] (e. [sent-42, score-0.148]
20 , in [22] methodologies have been proposed for encoding stationary kernels by randomly sampling their Fourier features); (c) the application of the so-called Nystrom method, which is a data-dependent methodology that requires training [29, 28, 21]. [sent-44, score-0.204]
21 In this paper, we study a recently proposed kernel, the so-called one-shot similarity kernel, which was shown to be particularly useful for the recently introduced similarity problems (face and action similarity [32, 30, 12]). [sent-46, score-0.616]
22 In particular, we show that (1) a special form of the kernel has a closed form feature map and (2) the general kernel can be written in a form which allows for efficient incremental solutions. [sent-47, score-1.246]
23 Hence, the proposed form of the one-shot similarity kernel makes it suitable for incremental PCA, which is particular useful for visual tracking [23]. [sent-48, score-0.961]
24 Summarizing, the contributions of this paper are: • • We study the recently proposed class of one-shot similarity u kdeyrn tehles raencde snhtlyow p rthopato sthedere c aexssist o fcl oonsee-ds hfootrm si solutions that can be acquired after simple data normalization. [sent-49, score-0.216]
25 For this case we show that (1) the use of oneshot similarity kernel with SVMs can be re-interpreted as a margin maximization and (2) the one-shot similarity kernels can be used with the recently introduced SVM packages which can train linear SVMs in linear time (i. [sent-50, score-1.104]
26 We show that the proposed one-shot similarity kernel can s bheo fwor thmautl athteed p rino a sfoedrm o nwe-hsichhot a slilmowilas fitoyr kinercnree-l mental Principal Component Analysis. [sent-53, score-0.581]
27 • We apply the one-shot similarity kernel to object tracking waphpelrye t shtea toen-eo-f-sthhoet-a sirmt rielasruiltyts are ealch toie ovebjde. [sent-54, score-0.669]
28 The One-Shot Similarity Kernel In this section, we will define the one-shot and multipleshot similarity kernels, having as an example the recently introduced face similarity problem in the wild [30, 12, 3 1, 32]. [sent-56, score-0.576]
29 Face similarity is conceptually different to the standard face recognition, in which the algorithm, given a test facial image, should find the most similar face (or the kmost similar faces) from a pre-defined dataset (corresponding to the same identity). [sent-57, score-0.465]
30 Indeed, face similarity tries to determine whether two given facial images belong to the same face or not. [sent-58, score-0.465]
31 Furthermore, there is a subtle, yet crucial difference between the face similarity and verification problems [32, 8, 34, 35]. [sent-59, score-0.341]
32 In face verification the identity being claimed is known, hence person specific models can be learned and used. [sent-60, score-0.185]
33 This is not the case with face similarity, as such models can not be used or trained (the interested reader can refer to [8, 32] and in the references within for more details regarding the face similarity problem). [sent-61, score-0.541]
34 In order to construct the one-shot similarity kernel, background samples are required. [sent-62, score-0.291]
35 For example, in face recognition, as background samples we can consider a set of facial images that do not belong to the list of faces of the system (very similar to the so-called world model in the face verification problem). [sent-68, score-0.473]
36 Their oneshoLt similarity score ois computed by considering the set of background samples A with cardinality NA, which contains samples nnodt belonging wtoit hthcea same tcylaNss as neither x nor y but otherwise not-labeled [32]. [sent-71, score-0.353]
37 The one-shot similarity score in the Fisher’s Linear Discriminant Analysis (FLDA) framework can be described as follows. [sent-72, score-0.184]
38 the similarity between x and y via the background samples? [sent-77, score-0.229]
39 (3) 22339933 The above kernel in (3), as proven in [30], is a psd kernel (for further details regarding the one-shot similarity kernel the interested reader may refer to [30, 12, 31, 32]). [sent-80, score-1.581]
40 (4) The kernel k can thus take the following functional form: k(x, y) = f(x)Tg(y) (5) where f(u) =⎡⎢⎣−√12√? [sent-85, score-0.448]
41 r dA p nriocdeu property toofr sthe kernel is: k(x, y) = f(x)Tg(y) = f(y)Tg(x) = g(y)Tf(x). [sent-104, score-0.397]
42 (8) In the next section we are going to exploit this property to formulate an exact and incremental Principal Component Analysis (iPCA). [sent-105, score-0.294]
43 A Special Case of the one-shot similarity kernel Assuming that all data are normalized such that | |x||2S = ˜x T x˜ = ˜y T y˜ = 1 (i. [sent-118, score-0.581]
44 (9) Hence, the closed feature map that can be used has the following closed form: φ(x) = S−12x. [sent-121, score-0.152]
45 (10) We will now study the interpretation of the application of this kernel within the SVMs framework. [sent-122, score-0.397]
46 Typically, w and b are found by solving the Wolf dual problem where in the case of the kernel (3) can be written as: max0≤αi≤CαT1 −12αTKsα, s. [sent-131, score-0.397]
47 (12) Thus, when the one-shot similarity kernel is used with SVMs, it attempts to maximize a squared Mahalanobis type distance margin, which is inversely proportional to wTSw. [sent-142, score-0.581]
48 Hence, the one-shot similarity kernel can be interpreted as a type of margin being maximized within the linear SVM framework. [sent-143, score-0.611]
49 In case the matrix S is singular or in non-linear case where the one-shot similarity kernel is used in a features space (i. [sent-144, score-0.581]
50 , a kernel is used in the SVM problem (12)), solutions can be provided by using the tools in ([36, 13]). [sent-146, score-0.397]
51 Since the kernel has a closed form it can be directly used with the recently proposed linear SVMs which can be trained in linear time with regards to the number of training samples and solve the following reformulated optimization 22339944 problem 3: mwi,ξn 21wTw + Cξ s. [sent-147, score-0.609]
52 It is worth noting here that the functional form of the similarity kernel in (3) does not allow the use of fast cutting plane algorithm for solving (13), as proposed in [10]. [sent-152, score-0.725]
53 Exact Incremental Component Analysis using the one-shot similarity kernel In this section, we will show how the property in (8) can be harnessed in order to define a special version of KPCA. [sent-154, score-0.581]
54 The proposed KPCA, contrary to the general incremental KPCA approaches [4], does not require the computation of pre-images. [sent-155, score-0.25]
55 have explicit mappings for Xg and Xf makes feasible the computation of incremental PCA without the use of pre-images. [sent-178, score-0.378]
56 4Centering in the feature space is straightforward by centering the kernel matrix [25]. [sent-190, score-0.397]
57 Additionally, using the kernel properties (8) the following properties hold: UφTφ(x) = UfTg(x) = UgTf(x) (17) and also Uf and Ug are mutually orthogonal UfTUg = Λ −−1212V T X φTfTX φgV Λ −−112 = UφTUφ = I. [sent-192, score-0.453]
58 (18) We proceed with showing that by using the explicit definition of Uf and Ug we can define an incremental KPCA without the need of pre-images. [sent-193, score-0.324]
59 ect approach to KPCA, the storage requirements for the incremental update is of fixed complexity (e. [sent-264, score-0.25]
60 ,T wheh complexity of the update is also fixed for our kernel (e. [sent-268, score-0.397]
61 One of the main applications of iPCA is object tracking [23], in which the object subspace is adaptively learned and online updated. [sent-275, score-0.136]
62 In this paper we combine the proposed kernel with the tracking framework proposed in [23], but instead of PCA we use the KPCA with the proposed kernel. [sent-276, score-0.485]
63 The particle chosen is the one corresponding to an image which can be best reconstructed within a subspace of choice (in our case, our kernel subspace). [sent-279, score-0.478]
64 In the tracking framework the set of background samples A needed can be images of objects ootfh bearc cthkganro tuhned one we sw Aish n teoe sample. [sent-281, score-0.195]
65 Experimental Results For our experiments, we evaluated the proposed kernel in a number of applications including face recognition, object tracking and deformable model fitting. [sent-285, score-0.661]
66 For face recognition we used a similar framework to [32]. [sent-286, score-0.119]
67 We show that by exploiting the functional form of the one-shot similarity kernel state-of-the-art tracking results can be produced. [sent-318, score-0.762]
68 Finally, we combined the oneshot similarity kernel- SVMs with the recently introduced discriminative fitting algorithms, such as the Constrained Local Models (CLMs) [24] and we report state-of-the-art fitting results. [sent-319, score-0.407]
69 Face Recognition The usefulness of the one-shot similarity kernel for face verification has been shown in [32] , using the LFW database [8], hence we do not repeat these experiments. [sent-322, score-0.766]
70 We did perform multi-identity face classification in the LFW image set to show the gain in computational time by using the proposed formulation of the one-shot similarity kernel. [sent-323, score-0.303]
71 As in [32, 33], we fused various features and measured the face recognition performance by varying the number of probes (from 5, 10, 20 and 50 subjects) and performing 20 random repetitions per experiment (for more details regarding the features the interested reader can refer to [32, 33]). [sent-327, score-0.238]
72 We used the one-vs-all linear SVM proposed in [6] with the original form of the one-shot similarity kernel in (3). [sent-328, score-0.623]
73 We also used the proposed form of the kernel (9) with a fast implementation of one-vs-all linear SVM in [10], for comparison reasons. [sent-329, score-0.439]
74 This implementation can be used only for the case of linear kernels (or, as in our case, only when φ(x) is known). [sent-330, score-0.168]
75 0624 shot similarity kernel and the closed form of the one-shot similarity kernel given in (9)) are denoted as OSK and FOSK, respectively. [sent-368, score-1.28]
76 As we can see, the proposed closed form solution of the one-shot similarity kernel in (9) produces similar results as the original form of the one-shot similarity kernel, but in at least one order of magnitude less time (O(n) over O(n2) where n are the training samples). [sent-369, score-0.925]
77 Similar gain in computational time was recently reported in [27] using approximations of various additive kernels. [sent-370, score-0.152]
78 Object Tracking We evaluated the performance of our subspace learning algorithms for the application of appearance-based face tracking. [sent-373, score-0.167]
79 The appearance-based approach to tracking has been one of the de facto choices for tracking objects in image sequences. [sent-374, score-0.176]
80 As discussed, the proposed kernel subspacebased tracking algorithm is closely related to the incremental visual tracker in [23] (abbreviated as IVT). [sent-375, score-0.819]
81 1 tracker proposed in [19] and the Multiple Instance Learning (MIL) tracker in [1]. [sent-379, score-0.168]
82 The Mean RMS is summarized in Table 2 (the proposed tracker is under the abbreviation OSK-IVT, while the oneshot similarity kernel using the online kernel learning technique with pre-images [4] is abbreviated as OSK-IVT-K). [sent-419, score-1.169]
83 As can be seen, by exploiting the functional form (8) of the proposed kernel we obtain an adaptive tracking algorithm with the same complexity of IVT while producing state-ofthe-art results. [sent-420, score-0.578]
84 Some indicative examples in which the proposed tracker (using the proposed kernel) outperforms the IVT kernel are shown in Fig. [sent-422, score-0.481]
85 Deformable Model Fitting The state-of-the-art algorithms for deformable model fitting are the recently introduced non-parametric CLMs using non-parametric density estimation. [sent-428, score-0.151]
86 We should note here that the functional form of the one-shot similarity kernel (3) cannot be combined with the linear cutting-plane SVMs. [sent-438, score-0.674]
87 For all one-shot similarity kernels the background samples for each of the points are selected as patches of other points (i. [sent-439, score-0.459]
88 In face deformable model fitting experiments, results are often reported in a curve of the proportion of the images vs the shape root mean square error (RMSE) between the predicted shape and the ground truth shape. [sent-448, score-0.238]
89 As we can see, the use of one-shot similarity kernels indeed increases the performance over standard SVMs. [sent-454, score-0.352]
90 Furthermore, the use the closed form of one-shot similarity kernel (9) with the cutting-plane SVMs boosts even further the results over the original one-shot similarity kernel. [sent-455, score-0.883]
91 Finally, in order to perform fair comparisons, we compared the actual time for solving the SVM optimization problem with the proposed kernel and either the cutting plane linear algorithm or the standard kernel SVM for the original version ofthe kernel. [sent-456, score-0.845]
92 Training with the proposed kernel and the original form required 3 hours and around 1day, respectively. [sent-458, score-0.439]
93 Conclusions In this paper we studied a recently introduced class of kernels, the one-shot similarity kernels. [sent-460, score-0.216]
94 We derived closed form feature maps and proved that they can be used for efficient exact incremental learning. [sent-461, score-0.412]
95 We successfully combined them with typical classification algorithms (SVMs) and incremental learning techniques (iPCA) and applied them in several problems (face recognition, object tracking and deformable model fitting), acquiring state-of-the-art results. [sent-462, score-0.395]
96 Labeled faces in the wild: A database forstudying face recognition in unconstrained environments. [sent-520, score-0.166]
97 Efficient online subspace learning with an indefinite kernel for visual tracking and recognition. [sent-579, score-0.533]
98 Using the nystrom method to speed up kernel machines. [sent-651, score-0.439]
99 Effective unconstrained face recognition by combining multiple descriptors and learned background statistics. [sent-671, score-0.164]
100 The discriminant elastic graph matching algorithm applied to frontal face verification. [sent-683, score-0.156]
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