cvpr cvpr2013 cvpr2013-125 knowledge-graph by maker-knowledge-mining
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Author: Yi-Chen Chen, Vishal M. Patel, Jaishanker K. Pillai, Rama Chellappa, P. Jonathon Phillips
Abstract: We propose a novel dictionary-based learning method for ambiguously labeled multiclass classification, where each training sample has multiple labels and only one of them is the correct label. The dictionary learning problem is solved using an iterative alternating algorithm. At each iteration of the algorithm, two alternating steps are performed: a confidence update and a dictionary update. The confidence of each sample is defined as the probability distribution on its ambiguous labels. The dictionaries are updated using either soft (EM-based) or hard decision rules. Extensive evaluations on existing datasets demonstrate that the proposed method performs significantly better than state-of-the-art ambiguously labeled learning approaches.
Reference: text
sentIndex sentText sentNum sentScore
1 Jonathon Phillips † Abstract We propose a novel dictionary-based learning method for ambiguously labeled multiclass classification, where each training sample has multiple labels and only one of them is the correct label. [sent-4, score-0.734]
2 The dictionary learning problem is solved using an iterative alternating algorithm. [sent-5, score-0.426]
3 At each iteration of the algorithm, two alternating steps are performed: a confidence update and a dictionary update. [sent-6, score-0.597]
4 The confidence of each sample is defined as the probability distribution on its ambiguous labels. [sent-7, score-0.258]
5 The dictionaries are updated using either soft (EM-based) or hard decision rules. [sent-8, score-0.336]
6 Extensive evaluations on existing datasets demonstrate that the proposed method performs significantly better than state-of-the-art ambiguously labeled learning approaches. [sent-9, score-0.629]
7 Introduction In many practical image and video applications, one has access only to ambiguously labeled data. [sent-11, score-0.59]
8 The problem of learning identities where each example is associated with multiple labels, when only one of which is correct is often known as ambiguously labeled learning. [sent-13, score-0.629]
9 A semi− supervised dictionary− based learning method was proposed in [18] under the formulation where there are either labeled samples or totally unlabeled sam− ples available for training. [sent-16, score-0.291]
10 The method iteratively estimates the confidence of unlabeled samples belonging to each class ∗Yi− Chen Chen, Vishal M. [sent-17, score-0.348]
11 We say a signal x is sparse in dictionary D if it can be approxi− mated by x = Dt, where t is a sparse vector and D is a dic− tionary that contains atoms as its columns. [sent-32, score-0.472]
12 The dictionary D can be analytic such as a redundant Gabor dictionary or it can be trained directly from data. [sent-33, score-0.714]
13 It has been observed that learning a dictionary directly from training data rather than using a predetermined dictionary usually leads to bet− ter representation. [sent-34, score-0.753]
14 Thus, learned dictionaries generally have superior results in many practical image processing applica− tions such as restoration and classification. [sent-35, score-0.195]
15 This has moti− vated researchers to develop dictionary learning algorithms for supervised [15], [11], [17], [14], [16], semi− supervised [18] and unsupervised [20], [4], [9] learning. [sent-36, score-0.396]
16 In this paper, we consider a dictionary learning problem where each train− ing sample is provided with a set of possible labels and only one label among them is the true one. [sent-37, score-0.589]
17 We develop dictio− nary learning algorithms that process ambiguously labeled data. [sent-38, score-0.659]
18 faces), the algorithm consists of two main steps: confidence update and dictionary update. [sent-43, score-0.567]
19 The con− fidence for each sample is defined as the probability distri− bution on its ambiguous labels. [sent-44, score-0.191]
20 In the confidence update phase, the confidence is updated for each sample according to its residuals when the sample is projected onto different class dictionaries. [sent-45, score-0.557]
21 Then, the dictionary is updated using a fixed confidence. [sent-46, score-0.401]
22 (b) An illustration of how common label samples are collected to learn intermediate dictionaries, which are used to update the confidence for sample xi. [sent-70, score-0.429]
23 We propose a dictionary− based learning method when ambiguously labeled data are provided for training. [sent-72, score-0.629]
24 We show that our dictionary learning with soft decision rule is an EM− based dictionary learning method. [sent-76, score-0.889]
25 We propose a weighted K− SVD [1] algorithm to weigh the importance of samples according to their confidences during the learning process. [sent-78, score-0.155]
26 The true label zi of the ith training sample is in the multi− label set Li. [sent-95, score-0.162]
27 For each feature vector xi and for each class j, we define a latent variable pi,j, which represents the confidence of xi belong− ing to the jth class. [sent-97, score-0.383]
28 Define Cj to be the collection of samples in class j represented as a matrix and C = [C1, C2, · · · , CK] be the concatenation of all samples from different, c·l·a·ss ,Ces. [sent-107, score-0.22]
29 Similarly, let Dj be the dictionary that is learned from the data in Cj and D = [D1, D2 , · · · , DK] be the concatenation of all dictionaries. [sent-108, score-0.357]
30 Given this ambiguously labeled data, how can one learn dictionaries to represent each class? [sent-110, score-0.785]
31 We solve the dictionary learning problem using an iter− ative alternating algorithm. [sent-111, score-0.426]
32 At each iteration, two major steps are performed: confidence update and dictionary up− date. [sent-112, score-0.567]
33 Confidence Update: We use the notation D(t) , P(t) to de− note the dictionary matrix and confidence matrix respec− tively, in the tth iteration. [sent-118, score-0.494]
34 Keeping the dictionary D(t) fixed, the confidence of a feature vector belonging to classes outside its label set is fixed at 0 and is not updated. [sent-119, score-0.593]
35 To update the confidence of a sample belonging to classes in its label set, we first make the observation that a sample 1We refer to class matrices and clusters interchangeably. [sent-120, score-0.475]
36 333555224 which is well represented by the dictionary of class j, should have high confidence. [sent-121, score-0.415]
37 In other words, the confidence of a sample xi belonging to a class j should be inversely proportional to the reconstruction error that results when xi is projected onto Dj . [sent-122, score-0.485]
38 This can be done by updating the confidence matrix P(t) as follows xi wher βj(t)apn(it,dj)σ=(jt)? [sent-123, score-0.214]
39 3, we derive (2) under the assumption that the likelihood of each sample xi is a mixture of Gaussian densities, is the weight associated with the density of label j. [sent-134, score-0.185]
40 and β(jt) Cluster Update:2 Once the confidence matrix P(t) is up− dated, we use it to update the class matrix C(t+1) . [sent-135, score-0.268]
41 Given a class matrix we seek a dictionary that provides C(jt+1), D(jt+1) the sparsest representation for each example feature in this matrix, by solving the following optimization problem (D(jt+1),Γj(t+1)) = arDg,mΓin? [sent-139, score-0.415]
42 The K− SVD algorithm alternates between sparse− coding and dictionary update steps. [sent-152, score-0.43]
43 dictionary is updated atom− by− atom in an efficient way. [sent-155, score-0.401]
44 The entire approach for learning dictionaries from ambiguously labeled data using hard decisions is summarized in Algo− rithm 1. [sent-156, score-0.878]
45 The Dictionary proach Learning Soft Decision ap- The dictionary learning soft decision (DLSD) approach learns dictionaries that are used to update the confidence for each sample xi, based on the weighted distribution of other samples that share the same candidate label belonging to Li. [sent-159, score-1.194]
46 The weighted distribution of other samples sharing a × given candidate label c is computed through the normaliza− tion of all pl,c’s with l i. [sent-160, score-0.17]
47 = Confidence Update: In this step, given the intermediate dictionary D(t),i learned from the previous iteration for each sample xi, we calculate the residuals using for all jl in Li as e(itj)l,i = ? [sent-162, score-0.542]
48 We then use (2) to update the confidence replaced by e(itj)l,i. [sent-165, score-0.21]
49 =hes ie common ambiguous label samples are collected to learn the intermediate dictionaries The cell marked with ‘ ’ at the (i, j) entry indicates a non− zerop(it,j). [sent-177, score-0.427]
50 To learn the intermediate dictionaries for xi, exclusion of xi (corresponding to red cells) is necessary to enhance discriminative learning. [sent-181, score-0.302]
51 p(iNt)(i,jl),jl], where the weight wm reflects the relative amount of contri− bution from xim when learning the dictionary. [sent-199, score-0.182]
52 After Tc soft decision iterations, for each training sam− ple, we assign the label with the maximum confidence. [sent-223, score-0.151]
53 The labeled class matrices are used to learn the final dictionary D∗ = D(Tc) = via the K− SVD algorithm. [sent-224, score-0.532]
54 ,(8) where Zl is the random variable that corresponds to the true label zl of the observed sample xl. [sent-239, score-0.157]
55 Determining initial dictionaries The performance of both DLSD and DLHD will depend on the initial dictionaries as they determine how well the final dictionaries are learned through successive alternating iterations. [sent-317, score-0.615]
56 As a result, initializing our method with proper dictionaries is critical. [sent-318, score-0.195]
57 In this section, we propose an algo− rithm that uses both ambiguous labels and features to deter− mine the initial dictionaries. [sent-319, score-0.172]
58 At iteration t = 0, we build dictionaries for the sample xi, denoted by D(0),i = where the [Dj(10),i|D(j02),i|. [sent-328, score-0.249]
59 |Dj(|0L),ii|], intermediatedictionaryDj(k0),iislearnedfromsamplesother than xi with ambiguous label jk ∈ Li. [sent-331, score-0.238]
60 Each initial dictionary is then learned from the corresponding cluster using the K− SVD algorithm [1]. [sent-339, score-0.357]
61 Note that our method is very different from the approach oflearning dictionaries from partially labeled data [18]. [sent-341, score-0.312]
62 The work in [18] learns class discriminative dictionaries while our work learns class reconstructive dictionaries. [sent-342, score-0.365]
63 In addi− tion, from the formulation in [18] we see there are either labeled samples or totally unlabeled samples available for training. [sent-343, score-0.333]
64 In contrast, in our formulation, all samples are am− biguously labeled according to three controlled parameters. [sent-344, score-0.198]
65 In fact, formulations in [18] and [20] (for totally unlabeled samples) are special cases of the ambiguously labeled for− mulation presented in this paper. [sent-345, score-0.644]
66 Experiments To evaluate the performance of our dictionary method, we performed two sets of experiments defined in [5][6]: in− ductive experiments and transductive experiments. [sent-347, score-0.451]
67 We re− port the average test error rates (for inductive experiments) and the average labeling error rates (for transductive exper− iments), which were computed over 5 trials. [sent-348, score-0.411]
68 In an inductive experiment, samples are split in half into a training set and a test set. [sent-349, score-0.149]
69 Each sample in the training set is ambiguously labeled according to controlled parameters, while each sample in the test set is unlabeled. [sent-350, score-0.698]
70 In each trial, using the learned dictionaries from the training set, the test 333555557 error rate is calculated as the ratio of the number of test samples that are erroneously labeled, to the total number of test samples. [sent-351, score-0.313]
71 In a transductive experiment, all samples with ambiguous labels are used to train the dictionaries. [sent-352, score-0.293]
72 In each trial, the labeling error rate is calculated as the ratio of the × number of training samples that are erroneously labeled, to the total number of training samples. [sent-353, score-0.167]
73 Following the notations in [6], the controlled parameters are: p (proportion of ambiguously labeled samples), q (the number of extra labels for each ambiguously labeled sam− ple) and ? [sent-354, score-1.269]
74 (the degree of ambiguity −the maximum proba− bility of an extra label co− occurring with a true label, over all labels and inputs [6]). [sent-355, score-0.143]
75 Figures a3li(zae) da cnodl u(mb)n s−h voewcto average ×te1st) error rates (for inductive experiments) of the proposed dic− tionary method (DLHD and DLSD) versus p and ? [sent-365, score-0.266]
76 Both dictionary methods are compa− × rable to the Convex Learning from Partial Labels (CLPL) method (denoted as ‘mean’) [6]. [sent-368, score-0.357]
77 3(c) shows the aver− age labeling error rates (for transductive experiments) ver− sus q curves. [sent-370, score-0.243]
78 Figures 4(a) and (b) show the average labeling error rates versus p and q in transductive experi− ments. [sent-382, score-0.29]
79 Clearly, when either p or q is zero in transductive ex− periments, there exist no ambiguous labels and hence the × labeling errors are zero. [sent-384, score-0.261]
80 When 95% of samples are ambiguously labeled, the lowest average error labeling rate, 0. [sent-387, score-0.64]
81 It is observed that when 95% of samples are ambiguously labeled, DLSD achieves the lowest error labeling rate, of 14. [sent-400, score-0.64]
82 Discussions To explain the performance gain of our dictionary learn− ing approach, in Fig. [sent-404, score-0.391]
83 4, we show curves of two addi− tional baseline methods: ‘no dictionary learning (DL)’ and ‘equally− weighted K− SVD’ . [sent-405, score-0.431]
84 The ‘no DL’ method utilizes features and ambiguous labels only, without learning dictio− naries. [sent-406, score-0.157]
85 The ‘equally− weighted K− SVD’ method contrasts the DLSD method by simply us− ing equal weights among possible samples of each label for dictionary learning. [sent-409, score-0.561]
86 In other words, it ignores the weight matrix W in (7) and learns dictionaries by the standard K− SVD algorithm. [sent-410, score-0.222]
87 Performance of the proposed dictionary methods and other baselines [5], [6] on the LFW dataset. [sent-425, score-0.357]
88 (a) Average test error rates versus the proportion of ambiguously labeled samples (p ∈ [0, 0. [sent-426, score-0.818]
89 (b) Average test error rates versus the degree of ambiguity for each ambiguously labeled sample (p = 1, q = 1, ? [sent-428, score-0.791]
90 (c) Average labeling error rates versus the number of extra labels for each ambiguously labeled sample (p = 1, q ∈ [0, 1, . [sent-430, score-0.929]
91 The proposed dictionary methods are comparable to the CLPL method (‘mean’). [sent-434, score-0.357]
92 Performance of the proposed dictionary methods, two baseline methods (no dictionary learning −‘no DL’, and standard K− SVD ‘equally− weighted K− SVD’), CLPL (‘mean’) and ‘naive’ methods [5], [6] on transductive rates versus the proportion of ambiguously labeled samples (p ∈ experiments. [sent-436, score-1.663]
93 2(b), (c)) and noise, the learned dictionary atoms in our method are able to account for these variations to some de− gree. [sent-447, score-0.39]
94 5, we further show the initial (at t = 0) and updated (using DLSD at t = 20) confidence ma− trices corresponding to this experiment, where samples and labels are indexed vertically and horizontally, respectively. [sent-450, score-0.313]
95 Without any prior knowledge, ambiguously labeled samples have equally probable initial confidences. [sent-451, score-0.706]
96 Initial and updated confidence matrices on the TV series ‘LOST’ (12− class) dataset. [sent-458, score-0.21]
97 Conclusion We have extended the dictionary learning to the case of ambiguously labeled learning, where each example is sup− plied with multiple labels, only one of which is correct. [sent-462, score-0.986]
98 The proposed method iteratively estimates the confidence of samples belonging to each of the classes and uses it to refine the learned dictionaries. [sent-463, score-0.263]
99 Experiments using three publicly available datasets demonstrate the improved accuracy of the proposed method compared to state− of− the− art ambiguously labeled learning techniques. [sent-464, score-0.629]
100 The K− SVD: an algo− rithm for designing of overcomplete dictionaries for sparse represen− tation. [sent-473, score-0.249]
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