cvpr cvpr2013 cvpr2013-66 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Yu-Tseh Chi, Mohsen Ali, Ajit Rajwade, Jeffrey Ho
Abstract: This paper proposes a dictionary learning framework that combines the proposed block/group (BGSC) or reconstructed block/group (R-BGSC) sparse coding schemes with the novel Intra-block Coherence Suppression Dictionary Learning (ICS-DL) algorithm. An important and distinguishing feature of the proposed framework is that all dictionary blocks are trained simultaneously with respect to each data group while the intra-block coherence being explicitly minimized as an important objective. We provide both empirical evidence and heuristic support for this feature that can be considered as a direct consequence of incorporating both the group structure for the input data and the block structure for the dictionary in the learning process. The optimization problems for both the dictionary learning and sparse coding can be solved efficiently using block-gradient descent, and the details of the optimization algorithms are presented. We evaluate the proposed methods using well-known datasets, and favorable comparisons with state-of-the-art dictionary learning methods demonstrate the viability and validity of the proposed framework.
Reference: text
sentIndex sentText sentNum sentScore
1 in Abstract This paper proposes a dictionary learning framework that combines the proposed block/group (BGSC) or reconstructed block/group (R-BGSC) sparse coding schemes with the novel Intra-block Coherence Suppression Dictionary Learning (ICS-DL) algorithm. [sent-8, score-0.749]
2 An important and distinguishing feature of the proposed framework is that all dictionary blocks are trained simultaneously with respect to each data group while the intra-block coherence being explicitly minimized as an important objective. [sent-9, score-1.123]
3 We provide both empirical evidence and heuristic support for this feature that can be considered as a direct consequence of incorporating both the group structure for the input data and the block structure for the dictionary in the learning process. [sent-10, score-0.982]
4 The optimization problems for both the dictionary learning and sparse coding can be solved efficiently using block-gradient descent, and the details of the optimization algorithms are presented. [sent-11, score-0.749]
5 We evaluate the proposed methods using well-known datasets, and favorable comparisons with state-of-the-art dictionary learning methods demonstrate the viability and validity of the proposed framework. [sent-12, score-0.631]
6 Introduction Sparse modeling and dictionary learning have emerged recently as an effective and popular paradigm for solving many important learning problems in computer vision. [sent-14, score-0.611]
7 λDgΨ(D) + λCΩ(C(g)), (1) where the X(g) are vectors/matrices generated from the data X, and Ψ, are regularizers on the learned dictionary D and sparse coefficients C(g) , respectively. [sent-19, score-0.804]
8 In dictionary learning, Ω(C) is usually based on various sparsity- Ω Figure1:Ilustraion fthepro sedBlock/GroupSarseCoding algorithm. [sent-20, score-0.57]
9 A group of data X(g) on the left is sparsely coded with respect to the dictionary D with block structure D[1] · · · D[b] . [sent-21, score-0.925]
10 promoting norms that depend on the extra structures placed on D, and it is the regularizer Ψ(D) that largely determines the nature of the dictionary D. [sent-22, score-0.584]
11 (1) provides the elegant theme, its variations are often composed of extra structures placed on the dictionary D ( [15, 10, 7, 13]), and less frequently, different ways of generating sparsely-coded data X(g) for training the dictionary. [sent-25, score-0.572]
12 For classification, a block structure is often imposed on D and hierarchical structures could be further specified using these blocks ([16, 10]), with the aim of endowing the learned dictionary certain predictive power. [sent-27, score-1.044]
13 To promote sparsity, the block structure on D is often accompanied by an appropriate block-based ? [sent-28, score-0.303]
14 On the other hand, for X(g) , a common approach is to generate a collection of groups of data vectors {xg1 , ··· , xgk} from X and to simultaneously sparse ctoordse { xthe ,d·a·ta· xvect}ors fr oinm e Xac han dda ttao group nXe(og)u [l1y] . [sent-34, score-0.282]
15 In a noiseless setting, our proposed problem of encoding sparse representations for a group of data samples X using 333777557 the minimum number of blocks from D can be cast as the following optimization program: P? [sent-38, score-0.473]
16 cator function ,p = 1, 2 and C[i] is the wi-thheer e bl Io(c·k) (ss aunb i-nmdaictriatxo) ro ffu nCc ithonat , corresponds t Co the i-th block of D as shown in Fig. [sent-45, score-0.267]
17 (3) We will call this program Block/Group Sparse Coding (BGSC) as it incorporates both the group structure in data and block structure in the dictionary. [sent-55, score-0.482]
18 In some applications of which the main concern is identifying contributing blocks rather than finding the sparse representation [7], the following optimization program is considered: P? [sent-56, score-0.374]
19 m Sharing of dictionary atoms for data in the same group had been shown to increase the discriminative power of the dictionary ([1]). [sent-85, score-1.386]
20 With the block structure added to dictionary D, our proposed BGSC and R-BGSC algorithms promote a group of data to share only few blocks of D for encoding. [sent-86, score-1.17]
21 Therefore, incorporating these SC algorithms in a dictionary learning framework, which iteratively updates coefficients of data and updates atoms of D, will result in training each block of D using only few groups of data. [sent-87, score-1.224]
22 This means that, for example, a badly written digit ’9’, which looks like a ’7’, when grouped together with other normally written ’9’s, will be encoded using atoms these ’9’s used. [sent-88, score-0.418]
23 The badly written ’9’ will, in turns, be used to train the atoms in D that represent ’9’s rather than those that represent ’7’s. [sent-89, score-0.31]
24 Another novelty of our framework is that we do not assign a class of signals to specific blocks of a dictionary, unlike other Sparse Representation based Classification (SRC) [6, 17] and [13]. [sent-90, score-0.266]
25 This would allow some blocks to store shared features between some different classes. [sent-91, score-0.232]
26 [13] trained a single dictionary block for each group of data. [sent-94, score-0.901]
27 This method increases the redundancy of the information encoded in the learned dictionary as the informationcommonto two ormore groups (acommon scenario in many classification problems) will need to be stored separately within each block. [sent-95, score-0.68]
28 Since one dictionary block is assigned to each class, the redundancy induced in the dictionary needs to be reduced for greater efficiency. [sent-96, score-1.36]
29 This is done by removing dictionary elements whose mutual dot product has an absolute value greater than an arbitrary-chosen threshold (e. [sent-97, score-0.584]
30 In particular, our proposal to encode data from a single class using multiple blocks obviates the need to even incorporate an explicit inter-block coherence minimization term unlike [13]. [sent-102, score-0.524]
31 3 in) 1 [ 5w],he thne na (2k − 1)μB < 1− (na − 1)μS, (6) where na and k are the size and the rank of a block, respectively, and μB and μS are inter- and intra-block coherence defined in Section 2. [sent-107, score-0.506]
32 A way to achieve minimum μS is to make atoms orthonormal within each block [11, 3]. [sent-110, score-0.478]
33 However, such dictionaries (over-complete dictionary with union of orthonormal basis) do not perform as well as those with more flexible structure [14]. [sent-111, score-0.688]
34 For example, in SRC-based face recognition, each block contains atoms representing faces of the same person. [sent-112, score-0.424]
35 It does not make sense to impose strict orthogonality on each block. [sent-113, score-0.173]
36 Therefore, rather than imposing strong orthogonality constraint on each block, we propose a dictionary learning algorithm that minimizes only intrablock coherence. [sent-114, score-0.856]
37 The proposed dictionary learning framework learns the dictionary D by minimizing the objective function given in Eq. [sent-115, score-1.153]
38 (16) measures the mutual coherence within each block of D. [sent-117, score-0.537]
39 The corresponding sparse coding can be either BGSC or R-BGSC. [sent-118, score-0.171]
40 Besides the novel sparse coding algorithms, BGSC and R-BGSC, there are three specific features that distinguish our dictionary framework from existing methods: 1. [sent-119, score-0.716]
41 4 minimizes the intrablock coherence as one of its main objectives. [sent-122, score-0.395]
42 Our framework does not require to assign a class or a group of data to block(s) in the dictionary as in [13]. [sent-124, score-0.717]
43 This allows some blocks of the dictionary to be shared by different classes. [sent-125, score-0.777]
44 The dictionary is trained simultaneously with respect to each group of training samples X(g) using our proposed block/group regularized SC algorithm. [sent-127, score-0.757]
45 We will start with sparse coding algorithms first and work our way towards the full dictionary learning algorithm. [sent-133, score-0.749]
46 We denote scalars with lower-case letters, vectors with bold lower-case letters, matrices with upper-case letters, and the i-th block and group of a matrix (or vector) with Z[i] , and Z(i) , respectively. [sent-134, score-0.356]
47 imization steps only for one specific group of data X and its corresponding sparse coefficients C. [sent-165, score-0.34]
48 (13), the block sparsity of C depends on the value of λ. [sent-265, score-0.242]
49 When there is no block structure on D, BGSC is equivalent to the group sparse coding (GSC) in [1]. [sent-273, score-0.59]
50 Intra-Block Coherence Suppression Dictionary Learning The intra-block coherence is defined as μS(D) = miax? [sent-302, score-0.256]
51 =j (15) set of the atoms in block μB tis o fde tfhineed a as μB n(D b)l c=k ? [sent-318, score-0.424]
52 As mentioned in the Introduction, it is necessary to have a dictionary updating algorithm that minimizes the intrablock coherence. [sent-324, score-0.684]
53 =r where cr is the r-th row of C and dr is in block b. [sent-353, score-0.363]
54 It is clear from the first three terms of the above equation why group-regularized SC algorithms tend to generate high intra-block coherence blocks. [sent-356, score-0.256]
55 As we can see, the value of dr depends not only on how much it is being used to encode X (1st and 3rd term) but also on how much other dk’s are being used to encode X. [sent-357, score-0.187]
56 Since, BGSC and R-BGSC minimize the number of blocks to be used for encoding X, the atoms dr are likely in the same block as dk. [sent-358, score-0.753]
57 For example, if the coefficient C of X has only one non-zero block, then the atoms dr and dk, which correspond to the non-zero rows of coefficients c’s in the above equation, are all in the same block. [sent-359, score-0.441]
58 This justifies putting the intra-block coherence suppressing regularizer term in Eq. [sent-361, score-0.363]
59 badly written ’9’s that look like a ’7’, into one group and hence allow them to act as one different class and to be used to train the dictionary blocks corresponding to the wrong classes. [sent-367, score-1.022]
60 Note that it is not uncommon to add a post-processing step to make atoms in D unit vectors or requiring ? [sent-371, score-0.182]
61 We collected 15 groups of data for each digit where each group contained 50 randomly chosen images from the same class. [sent-382, score-0.26]
62 Generate a random dictionary D with nb blocks and each block contains na columns (atoms) (a total of nb na columns). [sent-384, score-1.493]
63 Iteratively compute coefficients using BGSC and update the dictionary using ICS-DL algorithm. [sent-386, score-0.683]
64 Use the coefficients of the training data to train 10 onevs-all linear SVMs[2]. [sent-388, score-0.204]
65 Compute the sparse coefficients of the test samples using either BGSC or R-BGSC. [sent-390, score-0.258]
66 are 333787880 Table 1demonstrates the impact of the dictionary’s block structure on the error rates. [sent-393, score-0.315]
67 For the experiment in the last column of Table 1, we assign two blocks to each digit. [sent-397, score-0.239]
68 The results show that the error rates are similar when the number of blocks (nb) is greater than 10 even though the number of classes of this dataset is 10. [sent-398, score-0.305]
69 2(a), the sparse coefficients of most of the training data have 3 to 6 active blocks when nb = 20. [sent-401, score-0.585]
70 The last column of Table 1 shows that the hard assignment of blocks to classes results in higher error rate even though the size of the dictionary is twice as large as those of the first three experiments in Table 1. [sent-402, score-0.83]
71 As mentioned in the Introduction, we did not assign blocks to classes and prefer using more blocks for encoding data with larger variability. [sent-403, score-0.447]
72 We can see that ’7’ and ’9’ share two blocks of dictionary due to their similarity. [sent-407, score-0.753]
73 However, they each have an exclusive block with large coefficients (darker in color) to allow them to encode the difference. [sent-408, score-0.413]
74 52 †: Assign each digit to two blocks of the dictionary. [sent-417, score-0.297]
75 When β = 0, our ICS-DL algorithm does not suppress intra-block coherence and is hence equivalent to the dictionary learning algorithm in [1]. [sent-421, score-0.873]
76 We stopped the training roughly after 200 iterations when the dictionary update did not change much. [sent-427, score-0.597]
77 The results in Table 2 suggest that suppressing the intra-block coherence can indeed improve the performance. [sent-428, score-0.324]
78 In the extreme case when imposing a strict orthogonality on the blocks using the UOB-DL, the error rate increases to 4. [sent-430, score-0.493]
79 These results provide an empirical support for not using strict orthogonality constraint. [sent-432, score-0.173]
80 However, our ICS-DL algorithm does not impose any inter-block orthogonality constraint on the dictionary as SISF-DL does. [sent-434, score-0.649]
81 12 To further demonstrate the intra-block coherence suppressing property of our ICS-DL algorithm, we plot the intra-block coherence values of the dictionaries trained with β = 0 and β = 200, respectively, in Fig. [sent-444, score-0.645]
82 Solid and dotted lines indicate the coherence and error, respectively. [sent-447, score-0.297]
83 The red solid line demonstrates that our ICS-DL method can keep the intra-block coherence at a low value. [sent-448, score-0.292]
84 On the contrary, without the intra-block coherence suppression term, the blue solid line shows that the coherence value becomes comparably large with increasing number of iterations. [sent-449, score-0.583]
85 Once we have a trained dictionary, we used the coefficients of training samples to train ten linear SVMs. [sent-451, score-0.236]
86 Another way to obtain coefficients of training samples is to recompute them individually. [sent-453, score-0.197]
87 The dictionary was trained with β = 400, λtrain = 0. [sent-456, score-0.545]
88 Finally, we compared our results with other state-of-theart results using dictionary learning algorithms ([13, 12]) shown in Table 3. [sent-466, score-0.578]
89 We also compare with the UOBDL ([11]) which imposes strict orthogonality constraint on blocks. [sent-467, score-0.173]
90 The results show that our algorithms outperform other dictionary learning methods, even the one specially tailored for hand-written digits recognition [8]. [sent-468, score-0.578]
91 Although Table 2 suggests that suppressing intra-block coherence of D improves the classification performance, imposing a strict orthogonality on the blocks, however, does not result in any improvement. [sent-469, score-0.558]
92 (b) Intra-block coherence (solid) and error rates (dotted) of two dictionaries (red for β = 200 and blue for β = 0). [sent-482, score-0.418]
93 The scenarios differ in terms of how the training samples are organized for computing the coefficients and which proposed SC algorithms were used. [sent-485, score-0.197]
94 First column in the legend (separated by ’ | ’) indicates how the coefficients of the training samples are computed, in groups (G) or individually (I) . [sent-486, score-0.254]
95 Tumhen s ienco thned l ecgoelunmdn (s ienpdaicraatteeds wbyh i’c|h’) SinCd algorithm i tsh eus cedoe tffoi compute hthee coefficients of the training samples. [sent-487, score-0.19]
96 However, due to the amount and complexity of this dataset, we were not able to fully exploit different dictionary structures and parameters to obtain a reasonable result. [sent-490, score-0.545]
97 Conclusion We have proposed a novel dictionary learning framework that includes two novel block/group regularized sparse coding algorithms and one novel dictionary learning algorithm. [sent-496, score-1.366]
98 Experimental comparisons with several state-of-the-art dictionary learning methods are favorable, and in particular, for hand-written digit recognition experiment, the proposed framework outperformed these state-of-the-art dictionary learning algorithms. [sent-497, score-1.245]
99 5Our dictionary size is 5 times smaller than what was used in SISF-DL. [sent-540, score-0.545]
100 Classification and clustering via dictionary learning with structured incoherence and shared features. [sent-590, score-0.602]
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