cvpr cvpr2013 cvpr2013-7 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Yan Pan, Hanjiang Lai, Cong Liu, Shuicheng Yan
Abstract: Data fusion, which effectively fuses multiple prediction lists from different kinds of features to obtain an accurate model, is a crucial component in various computer vision applications. Robust late fusion (RLF) is a recent proposed method that fuses multiple output score lists from different models via pursuing a shared low-rank latent matrix. Despite showing promising performance, the repeated full Singular Value Decomposition operations in RLF’s optimization algorithm limits its scalability in real world vision datasets which usually have large number of test examples. To address this issue, we provide a scalable solution for large-scale low-rank latent matrix pursuit by a divide-andconquer method. The proposed method divides the original low-rank latent matrix learning problem into two sizereduced subproblems, which may be solved via any base algorithm, and combines the results from the subproblems to obtain the final solution. Our theoretical analysis shows that withfixedprobability, theproposed divide-and-conquer method has recovery guarantees comparable to those of its base algorithm. Moreover, we develop an efficient base algorithm for the corresponding subproblems by factorizing a large matrix into the product of two size-reduced matrices. We also provide high probability recovery guarantees of the base algorithm. The proposed method is evaluated on various fusion problems in object categorization and video event detection. Under comparable accuracy, the proposed method performs more than 180 times faster than the stateof-the-art baselines on the CCV dataset with about 4,500 test examples for video event detection.
Reference: text
sentIndex sentText sentNum sentScore
1 Abstract Data fusion, which effectively fuses multiple prediction lists from different kinds of features to obtain an accurate model, is a crucial component in various computer vision applications. [sent-4, score-0.191]
2 Robust late fusion (RLF) is a recent proposed method that fuses multiple output score lists from different models via pursuing a shared low-rank latent matrix. [sent-5, score-0.926]
3 Despite showing promising performance, the repeated full Singular Value Decomposition operations in RLF’s optimization algorithm limits its scalability in real world vision datasets which usually have large number of test examples. [sent-6, score-0.265]
4 To address this issue, we provide a scalable solution for large-scale low-rank latent matrix pursuit by a divide-andconquer method. [sent-7, score-0.255]
5 The proposed method divides the original low-rank latent matrix learning problem into two sizereduced subproblems, which may be solved via any base algorithm, and combines the results from the subproblems to obtain the final solution. [sent-8, score-0.54]
6 Our theoretical analysis shows that withfixedprobability, theproposed divide-and-conquer method has recovery guarantees comparable to those of its base algorithm. [sent-9, score-0.42]
7 Moreover, we develop an efficient base algorithm for the corresponding subproblems by factorizing a large matrix into the product of two size-reduced matrices. [sent-10, score-0.428]
8 We also provide high probability recovery guarantees of the base algorithm. [sent-11, score-0.32]
9 The proposed method is evaluated on various fusion problems in object categorization and video event detection. [sent-12, score-0.313]
10 Under comparable accuracy, the proposed method performs more than 180 times faster than the stateof-the-art baselines on the CCV dataset with about 4,500 test examples for video event detection. [sent-13, score-0.206]
11 Introduction Effectively integrating multiple cues/features to obtain an accurate model, which is known as data fusion, is a popular approach to solve various tasks in computer vision and multimedia analysis, such as object classification and detection [4, 17, 19], video event detection [7, 8]. [sent-15, score-0.098]
12 Various methods that fuse multiple kinds of features have been proposed in the literature (see, e. [sent-16, score-0.099]
13 The second stream is Late Fusion, which firstly learns models from different kinds of features, and then combines the intermediate output scores of the learned models to yield a final model. [sent-24, score-0.196]
14 In this paper, we focus on the problem ofRobust Late Fusion (RLF) proposed in [21], in which the authors formulated late fusion as an optimization problem of Low-rank Latent Matrix Pursuit (LLMP). [sent-25, score-0.557]
15 As shown in Figure 1(a), RLF fuses multiple models via seeking a shared low-rank comparison matrix. [sent-26, score-0.107]
16 The formulation of LLMP is general and can also be applied to (with simple extensions) other important problems, such as rank aggregation [5] for information retrieval and collaborative filtering. [sent-27, score-0.092]
17 The main drawback of RLF-ALM is that it needs repeated operations of full Singular Value Decomposition (SVD), whose time complexity is cubic w. [sent-29, score-0.131]
18 This limits RLF-ALM’s scalability on large-scale datasets with thousands of test examples. [sent-33, score-0.107]
19 Our method has two building blocks: (1) Inspired by the recent advance in divide-and-conquer matrix factorization, we develop a divide-and-conquer ap- proach for LLMP. [sent-35, score-0.089]
20 As illustrated in Figure 1(b), our method divides the original LLMP problem into reduced-size problems, solves the subproblems sub- via any base algorithm × × for LLMP, and then combines the results from the subproblems to obtain the final solution. [sent-36, score-0.555]
21 Our theoretical analysis shows that with fixed probability, the proposed divide-andconquer algorithm has recovery those of the base algorithm. [sent-37, score-0.307]
22 scalability, guarantees comparable (2) To further improve to the we propose an efficient base algorithm for the 0Corresponding author: Cong Liu 555222224 Given 푛 score lists 푠푖 (푖 = 1푡표 푛) from different models, RLF constructs 푛 matrices 푇(푖) (푖 = 1, 2, . [sent-38, score-0.475]
23 ,푛), where each 푇(푖) encodes the pairwise relations of scores of every two test images in score list 푠푖. [sent-41, score-0.222]
24 , 푛) are used as input of LLMP to learn a shared, low-rank and skew-symmetric matrix 푇. [sent-45, score-0.089]
25 Each 푇(푖) can be reconstructed by 푇 combining with an additive sparse matrix 퐸(푖) . [sent-46, score-0.089]
26 At last, RLF can recover a final and more accurate score list from 푇. [sent-47, score-0.156]
27 Then in step 1-2, we recover 푇푆 and 푇퐴 by solving two size-reduced subproblems like the original LLMP problem, respectively. [sent-50, score-0.202]
28 In step 4, we calculate 푇퐶 by simple matrix algebra. [sent-52, score-0.117]
29 Finally, we simply combine the four parts to obtain the final matrix 푇. [sent-53, score-0.089]
30 In each iteration, the base al- gorithm factorizes a large comparison matrix 푇 ∈ ℝ푚1 ×푚2 ignotroi hthme product osf a tw laorg esi czeo-mrepdaurciseodn m maatrtircixes 푇 ( ∈i. [sent-55, score-0.259]
31 We further provide recovery guarantees of the proposed base algorithm. [sent-58, score-0.32]
32 The results on the Columbia Consumer Video (CCV) dataset with 4,658 test examples show that, under comparable accuracy, the propose method performs more than 180 times faster than the state-of-the-art RLF-ALM algorithm. [sent-60, score-0.108]
33 The strategies of feature combination can be divided into early fusion and late fusion. [sent-63, score-0.565]
34 A representative method of early fusion is MKL [1], which simultaneously learns kernel matrices and their associated combination weights. [sent-64, score-0.371]
35 [14] used finite Gaussian mixture to model the distribution of the intermediate output scores, and then combined the scores via likelihood ratio test. [sent-69, score-0.106]
36 [18] used a non-Bayesian probabilistic framework to fuse multiple output scores by minimizing the misclassification rates under the ℓ1 constraint. [sent-71, score-0.113]
37 However, the optimization algorithm in [21] needs repeated full SVD operations whose time complexity is cubic w. [sent-73, score-0.162]
38 the number of text examples, which limits its scalability on large-scale real world applications. [sent-76, score-0.117]
39 Our work is motivated by the recent progress in efficient algorithms for large-scale low-rank matrix learning. [sent-77, score-0.089]
40 The main idea is to divide a large-scale matrix learning problem into several size-reduced subproblems, solve the subproblems in parallel, and then combine the results from the subproblems. [sent-80, score-0.258]
41 With fixed probability, their divide-and-conquer algorithm has estimation error bound guarantees compared to its base algorithm that directly solves the original large-scale problem. [sent-81, score-0.236]
42 In contrast to RPCA with a single low-rank and sparse decomposition constraint, our work focus on a divide-and-conquer 555222335 approach for the LLMP problem that seeks to learn a shared × skew-symmetric comparison matrix with multiple low-rank and sparse decomposition constraints. [sent-82, score-0.213]
43 Problem Formulation The robust late fusion problem [21] is to fuse 푛 confidence score lists {푠(푖) }푖푛=1 . [sent-84, score-0.698]
44 For each 푠(푖), we convert it into a 푚 푚 comparison (푠1(푖), 푠2(푖), matrix 푇(푖), 푠푚(푖))푇 in which each 푇푗(,푖푘) 푇푗(,푖푘) = is defined as: 푠푖푔푛(푠(푗푖) − 푠푘(푖)). [sent-92, score-0.089]
45 (1) Each comparison matrix 푇(푖) is an isotonic representation that encodes the relative relationship among the test examples. [sent-93, score-0.148]
46 Some entries in 푇(푖) correctly reflect the relative relationship among the test examples, while other entries may be incorrect due to the wrong predictions in the score list 푠(푖) . [sent-94, score-0.241]
47 푇(푖) can be viewed as a combination of two parts: a shared low-rank part that reflects the correct relationship among the test examples, and a sparse part 퐸(푖) that encodes the irregular corruptions made by the incorrect entries. [sent-95, score-0.151]
48 Ideally, assume there exists a score list 푠 that correctly explains the relations of the test examples, then 푇 can be viewed as a matrix which encodes the relations in 푠, i. [sent-97, score-0.271]
49 However, if there is large variation in the scores, the matrix 푇 might have an unknown rank higher than 2. [sent-101, score-0.145]
50 This motivates us to formulate RLF as an optimization problem of pursuing a shared, low-rank and skew-symmetric latent matrix: ∑푛 푇,{퐸m(i푖n)}푖푛=1푟푎푛푘(푇) + 휆푖∑=1∣∣퐸(푖)∣∣0 푠. [sent-102, score-0.155]
51 Gereive ∣∣n푇 ∣th∣e learned matrix 푇, we can easily obtain the score list 푠 by 푠 = 푚1푇푒푇. [sent-115, score-0.212]
52 More importantly, LLMP is general and can also be applied to other family of problems, such as rank aggregation [5] for information retrieval and collaborative filtering. [sent-117, score-0.092]
53 A Divide-and-Conquer Solution The existing algorithms for LLMP, such as RLFALM [21], need repeated SVD operations on the whole comparison matrix, which have cubic time complexity w. [sent-120, score-0.131]
54 Inspired by the recent advance in divide-and-conquer algorithms [13] for RPCA problems [3], we propose a scalable divide-and-conquer method for the LLMP optimization problem. [sent-125, score-0.091]
55 The main idea is to divide the optimization problem (3) into smallsize subproblems, each of which can be easily solved by a base algorithm or by simple algebra, and then combine the results to get a low-rank and skew-symmetric comparison matrix. [sent-126, score-0.201]
56 푘 }W, ea sample the submatrices 푇푆, and and then each matrix is partitioned into four parts: 푇푆(푖) 퐸푆(푖), [푇푇푆퐵 푇 퐶퐴 ],(4) T(i)=[푇푇푆퐵((푖푖)) 푇푇퐴퐶((푖푖))], E(i)=[퐸퐸(푆퐵(푖푖)) 퐸퐸((퐴퐶푖푖))]. [sent-143, score-0.124]
57 (2) Recovering the seed matrix We recover 푇the seed matrix 푇푆 by solving the following subproblem: 푇푆,{m퐸푆i(푖n)}푖푛=1∣∣푇푆∣∣∗+ 휆푖∑=푛1∣∣퐸푆(푖)∣∣1 푠. [sent-145, score-0.285]
58 Obviously this is a size-reduced problem of (3) which can be solved by any base algorithm for LLMP (i. [sent-151, score-0.17]
59 Randomly sample a set of 푘 indices to partition each (푇푆(푖), 푇퐴(푖), 푇퐵(푖), 푇퐶(푖)); 퐸푆(푖) by (5); matrix of 푇(푖) into 4 parts 2. [sent-161, score-0.089]
60 , kernel matrices learning) [20], and it has been generalized for arbitrary matrices [6]. [sent-187, score-0.175]
61 It is worth noting that, although our framework is based on Generalized Nystrom Method, the involved subproblems (i. [sent-188, score-0.169]
62 update the+ multiplier 푌푡+(푖1) 푌푡(푖) + 휇푡(푇(푖) 8. [sent-208, score-0.09]
63 , the algorithm in [21]), a challenging issue arising in solving (9) is that, when applied to subproblems with large 푚1 and 푚2, the repeated full SVD operations are still computationally expensive. [sent-214, score-0.3]
64 To address this issue, we propose to factorize the large comparison matrix ∈ ℝ푚1 ×푚2 into tfahec product eo lfa trwgoe csiozmep-raerdiusocend m maatrtirxice 푇s ∈(i. [sent-215, score-0.089]
65 This kind of fixed ra×nk 푟 prior i푟s ×us 푚ed i,n 푟 r ≪ece 푚nt푖 푛w(o푚rks [12, 11] to speed up low-rank matrix learning problems such as RPCA [3] and LRR [10]. [sent-218, score-0.089]
66 There are two advantages from the factorization: (1) the resulting optimization problem only has 푂((푚1 +푚2)푟) vari- 푇 × ables, and (2) only size-reduced SVD operations are needed in each iteration, which is much cheaper than the full SVD operations. [sent-219, score-0.118]
67 Theoretical Analysis A natural question arising here is whether there is any theoretical guarantee on the recovery errors of the proposed divide-and-conquer method compared to the errors of the base algorithm. [sent-241, score-0.411]
68 Our analysis is based on the standard matrix coherence assumption. [sent-243, score-0.089]
69 certain conditions, the solution to (11) guarantees low recovery errors with high probability. [sent-255, score-0.181]
70 The second theorem shows that, with fixed probability, the recovery errors of the proposed divide-and-conquer method is bounded by the errors of the solution to (11). [sent-263, score-0.186]
71 Let 푇0 be the ground truth of the rank푟0 latent matrix, and 푇 be the output of Algorithm 1. [sent-265, score-0.1]
72 푇 is partitioned as in (4), and correspondingly 푇0 is partitioned into (푇0,푆 , 푇0,퐴 , 푇0,퐵 , 푇0,퐶). [sent-266, score-0.101]
73 Experimental Evaluation In this section, we evaluate the accuracy and running time of the proposed divide-and-conquer method on various real world and simulated datasets. [sent-271, score-0.147]
74 We compare the performance of three algorithms for the LLMP problem, including the proposed base algorithm LLMP-Base, the proposed divide-and-conquer algorithm DC-LLMP, and the state-of- RLF-ALM1 the-art algorithm [21]. [sent-272, score-0.17]
75 Note that the original RLF-ALM algorithm in [21] enforces the learned low-rank matrix to be rank-2 as a convergence condition. [sent-276, score-0.089]
76 As explained in Section 3, in practice, the matrix 푇 might have an unknown rank higher than 2 due to large variation in scores. [sent-277, score-0.145]
77 Experiments on real world datasets We evaluate the proposed method on object categorization and video event detection tasks. [sent-283, score-0.136]
78 We use one-vs-all SVM to generate the output score lists from different kinds of features. [sent-285, score-0.23]
79 In all the experiments, the running time results of DC-LLMP is the average of 5 trials. [sent-294, score-0.074]
80 MAP and running time comparison results on Oxford Flower 17 dataset. [sent-298, score-0.074]
81 These distance matrices are computed from seven kinds of features: HOG, color, shape, texture, clustered HSV values, SIFT on the foreground internal region (SIFTint) and on the foreground boundary (SIFTbdy) (see [15] for details). [sent-330, score-0.119]
82 The running time and MAP are first 2Although we can simply set 휆 = 1/√푚2 by Theorem 1, we found that choosing 휆 by cross validation leads to better performance. [sent-332, score-0.105]
83 And all the three robust late fusion methods have superior performance on MAP over the other three early and late fusion baselines. [sent-335, score-1.091]
84 Results on video event detection We evaluate the proposed method on video event detection using the Columbia Consumer Video dataset [7] which contains 9,3 17 video over 20 classes of semantic events. [sent-347, score-0.231]
85 (2) Under comparable MAP, DCLLMP performs more than 30 times faster than its base algorithm LLMP-Base, and it performs more than 180 times faster than the state-of-the-art RLF-ALM. [sent-359, score-0.283]
86 We can conclude that DC-LLMP has superior running time performance in real world vision tasks which usually have thousands of test examples, and it is a practical and scalable method for largescale low-rank latent matrix pursuit. [sent-360, score-0.354]
87 To answer this question, We conducted experiments on the CCV dataset to observe the effects of different 푟 and 푘. [sent-391, score-0.082]
88 The results in Figure 2(b) indicate that when 푟 ≥ 10, DC-LLMP and its base algoirintdhimca tLeL tMhaPt- wBhaesne h푟a v≥e comparable MMPA aPn dva iltuse bsa stoe athlgosoeof RLF-ALM. [sent-393, score-0.217]
89 This justifies that DCLLMP has comparable performance with the algorithms directly working on full matrices under relative small values of 푘, which is a key factor that leads to the speed-ups of DC-LLMP. [sent-401, score-0.105]
90 Simulation results To further investigate the characteristics of the proposed algorithms, we conduct simulated experiments to observe the effects on the running time and MAP with (1) different input data size (i. [sent-404, score-0.22]
91 In our simulation, we firstly generate 푛 score lists 푓1 , 푓2, . [sent-407, score-0.134]
92 Then we add random noise (sampled uniformly from [−100, 100]) to 푠% entries ofeach score lliestd. [sent-411, score-0.102]
93 After that, we use each score list to construct the corresponding input matrix 푇(푖) . [sent-413, score-0.212]
94 Effects of different input data size The first simulation is to explore the effects on the running time w. [sent-415, score-0.211]
95 In practice, we usually assume matrix 푇 have small rank 푟0, and set 푘 to be 푟0 + 푝 with a small constant 푝. [sent-419, score-0.145]
96 Effects of different proportion of corruptions Figure 4(c) show the comparison results on MAP with different proportion of corrupted entries in each score list, assuming other parameters being fixed. [sent-430, score-0.175]
97 Conclusions In this paper, we developed DC-LLMP, a scalable divideand-conquer method for large low-rank latent matrix pursuit. [sent-434, score-0.214]
98 DC-LLMP divides the original low-rank latent matrix learning problem into size-reduced subproblems which can be solved cheaply by a base algorithm, and obtains the fi555232880 )imeTc(s1250 20DLRCMF−LPA nBLMu4maP0sebrofsa6m0ples(m)801 )(cesmiT32105 0 RLD5CFM−PLA nBMuaPmsebr1o0fmdels(n1)520AMP0 . [sent-435, score-0.54]
99 The theoretical analysis showed that the recovery errors of DC-LLMP are bounded by the recovery errors of the base algorithm with fixed probability. [sent-451, score-0.453]
100 Furthermore, we developed an efficient base algorithm to solve the corresponding subproblems. [sent-452, score-0.17]
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Author: Yan Pan, Hanjiang Lai, Cong Liu, Shuicheng Yan
Abstract: Data fusion, which effectively fuses multiple prediction lists from different kinds of features to obtain an accurate model, is a crucial component in various computer vision applications. Robust late fusion (RLF) is a recent proposed method that fuses multiple output score lists from different models via pursuing a shared low-rank latent matrix. Despite showing promising performance, the repeated full Singular Value Decomposition operations in RLF’s optimization algorithm limits its scalability in real world vision datasets which usually have large number of test examples. To address this issue, we provide a scalable solution for large-scale low-rank latent matrix pursuit by a divide-andconquer method. The proposed method divides the original low-rank latent matrix learning problem into two sizereduced subproblems, which may be solved via any base algorithm, and combines the results from the subproblems to obtain the final solution. Our theoretical analysis shows that withfixedprobability, theproposed divide-and-conquer method has recovery guarantees comparable to those of its base algorithm. Moreover, we develop an efficient base algorithm for the corresponding subproblems by factorizing a large matrix into the product of two size-reduced matrices. We also provide high probability recovery guarantees of the base algorithm. The proposed method is evaluated on various fusion problems in object categorization and video event detection. Under comparable accuracy, the proposed method performs more than 180 times faster than the stateof-the-art baselines on the CCV dataset with about 4,500 test examples for video event detection.
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