nips nips2008 nips2008-226 knowledge-graph by maker-knowledge-mining

226 nips-2008-Supervised Dictionary Learning

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Author: Julien Mairal, Jean Ponce, Guillermo Sapiro, Andrew Zisserman, Francis R. Bach

Abstract: It is now well established that sparse signal models are well suited for restoration tasks and can be effectively learned from audio, image, and video data. Recent research has been aimed at learning discriminative sparse models instead of purely reconstructive ones. This paper proposes a new step in that direction, with a novel sparse representation for signals belonging to different classes in terms of a shared dictionary and discriminative class models. The linear version of the proposed model admits a simple probabilistic interpretation, while its most general variant admits an interpretation in terms of kernels. An optimization framework for learning all the components of the proposed model is presented, along with experimental results on standard handwritten digit and texture classiﬁcation tasks. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract It is now well established that sparse signal models are well suited for restoration tasks and can be effectively learned from audio, image, and video data. [sent-12, score-0.361]

2 Recent research has been aimed at learning discriminative sparse models instead of purely reconstructive ones. [sent-13, score-0.722]

3 This paper proposes a new step in that direction, with a novel sparse representation for signals belonging to different classes in terms of a shared dictionary and discriminative class models. [sent-14, score-1.058]

4 The linear version of the proposed model admits a simple probabilistic interpretation, while its most general variant admits an interpretation in terms of kernels. [sent-15, score-0.143]

5 An optimization framework for learning all the components of the proposed model is presented, along with experimental results on standard handwritten digit and texture classiﬁcation tasks. [sent-16, score-0.268]

6 1 Introduction Sparse and overcomplete image models were ﬁrst introduced in [1] for modeling the spatial receptive ﬁelds of simple cells in the human visual system. [sent-17, score-0.119]

7 In [3, 4], sparse decompositions are used with predeﬁned dictionaries for face and signal recognition. [sent-21, score-0.545]

8 In [5], dictionaries are learned for a reconstruction task, and the corresponding sparse models are used as features in an SVM. [sent-22, score-0.536]

9 In [6], a discriminative method is introduced for various classiﬁcation tasks, learning one dictionary per class; the classiﬁcation process itself is based on the corresponding reconstruction error, and does not exploit the actual decomposition coefﬁcients. [sent-23, score-0.891]

10 In [7], a generative model for documents is learned at the same time as the parameters of a deep network structure. [sent-24, score-0.226]

11 In [8], multi-task learning is performed by learning features and tasks are selected using a sparsity criterion. [sent-25, score-0.09]

12 , [10, 11, 12, 13, 14], and in neural networks [15], but not, to the best of our knowledge, in the sparse dictionary learning framework. [sent-29, score-0.662]

13 Section 2 presents a formulation for learning a dictionary tuned for a classiﬁcation task, which we call supervised dictionary learning, and Section 3 its interpretation in term of probability and kernel frameworks. [sent-30, score-1.204]

14 2 Supervised dictionary learning We present in this section the core of the proposed model. [sent-32, score-0.498]

15 In classical sparse coding tasks, one considers a signal x in Rn and a ﬁxed dictionary D = [d1 , . [sent-33, score-0.92]

16 , dk ] in Rn×k (allowing k > n, making the dictionary overcomplete). [sent-36, score-0.498]

17 In this setting, sparse coding with an ℓ1 regularization1 amounts to computing R⋆ (x, D) = min ||x − Dα||2 + λ1 ||α||1 . [sent-37, score-0.375]

18 (1) 2 α∈Rk It is well known in the statistics, optimization, and compressed sensing communities that the ℓ1 penalty yields a sparse solution, very few non-zero coefﬁcients in α, although there is no explicit analytic link between the value of λ1 and the effective sparsity that this model yields. [sent-38, score-0.25]

19 Note that sparse coding with an ℓ0 penalty is an NP-hard problem and is often approximated using greedy algorithms. [sent-41, score-0.291]

20 We consider a training set of m labeled signals (xi )m in Rn , associated with binary labels (yi ∈ {−1, +1})m . [sent-44, score-0.114]

21 i=1 i=1 Our goal is to learn jointly a single dictionary D adapted to the classiﬁcation task and a function f which should be positive for any signal in class +1 and negative otherwise. [sent-45, score-0.65]

22 We consider in this paper two different models to use the sparse code α for the classiﬁcation task: (i) linear in α: f (x, α, θ) = wT α + b, where θ = {w ∈ Rk , b ∈ R} parametrizes the model. [sent-46, score-0.189]

23 (ii) bilinear in x and α: f (x, α, θ) = xT Wα + b, where θ = {W ∈ Rn×k , b ∈ R}. [sent-47, score-0.171]

24 In this case, the model is bilinear and f acts on both x and its sparse code α. [sent-48, score-0.358]

25 Note that one can interpret W as a linear ﬁlter encoding the input signal x into a model for the coefﬁcients α, which has a role similar to the encoder in [18] but for a discriminative task. [sent-50, score-0.395]

26 Such a constraint is classical in sparse coding [2]. [sent-52, score-0.343]

27 2 (5) In this setting, the classiﬁcation procedure of a new signal x with an unknown label y, given a learned dictionary D and parameters θ, involves supervised sparse coding: min S ⋆ (x, D, θ, y), (6) y∈{−1;+1} The learning procedure of Eq. [sent-57, score-1.058]

28 (4) minimizes the sum of the costs for the pairs (xi , yi )m and cori=1 responds to a generative model. [sent-58, score-0.273]

29 We will refer later to this model as SDL-G (supervised dictionary 1 2 P The ℓ1 norm of a vector x of size n is deﬁned as ||x||1 = n |x[i]|. [sent-59, score-0.56]

30 , m D w αi xi yi Figure 1: Graphical model for the proposed generative/discriminative learning framework. [sent-65, score-0.286]

31 Note the explicit incorporation of the reconstructive and discriminative component into sparse coding, in addition to the classical reconstructive term (see [9] for a different classiﬁcation component). [sent-67, score-1.064]

32 This leads to: m C(S ⋆ (xi , D, θ, −yi ) − S ⋆ (xi , D, θ, yi )) + λ2 ||θ||2 . [sent-70, score-0.144]

33 2 min D,θ (7) i=1 As detailed below, this problem is more difﬁcult to solve than (4), and therefore we adopt instead a mixed formulation between the minimization of the generative Eq. [sent-71, score-0.228]

34 (4) and its discriminative version (7), (see also [13])—that is, m µC(S ⋆ (xi , D, θ, −yi ) − S ⋆ (xi , D, θ, yi )) + (1 − µ)S ⋆ (xi , D, θ, yi ) + λ2 ||θ||2 , 2 (8) i=1 where µ controls the trade-off between the reconstruction from Eq. [sent-72, score-0.63]

35 This is the proposed generative/discriminative model for sparse signal representation and classiﬁcation from learned dictionary D and model θ. [sent-75, score-0.854]

36 We will refer to this mixed model as SDL-D, (supervised dictionary learning, discriminative). [sent-76, score-0.562]

37 All of these formulations admit a straightforward multiclass extension, using softmax discriminative p cost functions Ci (x1 , . [sent-78, score-0.345]

38 , xp ) = log( j=1 exj −xi ), which are multiclass versions of the logistic function, and learning one model θ i per class. [sent-81, score-0.139]

39 Before presenting the optimization procedure, we provide below two interpretations of the linear and bilinear versions of our formulation in terms of a probabilistic graphical model and a kernel. [sent-84, score-0.315]

40 • The signals xi are generated according to a Gaussian probability distribution conditioned on D 2 and αi , p(xi |αi , D) ∝ e−λ0 ||xi −Dαi ||2 . [sent-90, score-0.182]

41 All the xi ’s are considered independent from each other. [sent-91, score-0.119]

42 • The labels yi are generated according to a probability distribution conditioned on w and αi , and T T T given by p(yi = ǫ|αi , W) = e−ǫw αi / e−W αi + eW αi . [sent-92, score-0.144]

43 Given D and w, all the triplets (αi , xi , yi ) are independent. [sent-93, score-0.263]

44 , [12, 13]), amounts to ﬁnding the maximum likelihood estimates for D and w according to the joint distribution p({xi , yi }m , D, W), where the xi ’s and the yi ’s are the training signals and their labels respeci=1 tively. [sent-96, score-0.561]

45 It can easily be shown (details omitted due to space limitations) that there is an equivalence between this generative training and our formulation in Eq. [sent-97, score-0.172]

46 3 Although joint generative modeling of x and y through a shared representation has shown great promise [10], we show in this paper that a more discriminative approach is desirable. [sent-99, score-0.382]

47 The same kind of MAP approximation relates this discriminative training formulation to the discriminative model of Eq. [sent-101, score-0.651]

48 (8) is a classical trade-off between generative and discriminative (e. [sent-104, score-0.4]

49 , [12, 13]), where generative components are often added to discriminative frameworks to add robustness, e. [sent-106, score-0.378]

50 2 A kernel interpretation of the bilinear model Our bilinear model with f (x, α, θ) = xT Wα + b does not admit a straightforward probabilistic interpretation. [sent-110, score-0.484]

51 On the other hand, it can easily be interpreted in terms of kernels: Given two signals x1 and x2 , with coefﬁcients α1 and α2 , using the kernel K(x1 , x2 ) = αT α2 xT x2 in a logistic 1 1 regression classiﬁer amounts to ﬁnding a decision function of the same form as f . [sent-111, score-0.171]

52 [5] learn a dictionary adapted to reconstruction on a training set, then train an SVM a posteriori on the decomposition coefﬁcients α. [sent-114, score-0.745]

53 4 Optimization procedure Classical dictionary learning techniques (e. [sent-117, score-0.535]

54 , [1, 5, 19]), address the problem of learning a reconstructive dictionary D in Rn×k well adapted to a training set, which is presented in Eq. [sent-119, score-0.89]

55 It can be seen as an optimization problem with respect to the dictionary D and the coefﬁcients α. [sent-121, score-0.53]

56 Equation (4) can be rewritten as: m S(xj , αj , D, θ, yi ) + λ2 ||θ||2 , s. [sent-128, score-0.144]

57 2 min D,θ,α (9) i=1 Block coordinate descent consists therefore of iterating between supervised sparse coding, where D and θ are ﬁxed and one optimizes with respect to the α’s and supervised dictionary update, where the coefﬁcients αi ’s are ﬁxed, but D and θ are updated. [sent-134, score-0.908]

58 To reach a local minimum for this difﬁcult non-convex optimization problem, we have chosen a continuation method, starting from the generative case and ending with the discriminative one as in [6]. [sent-140, score-0.451]

59 1 Supervised sparse coding The supervised sparse coding problem from Eq. [sent-143, score-0.683]

60 The ﬁxed-point continuation method (FPC) from 3 We are also investigating how to properly estimate D by marginalizing over α instead of maximizing with respect to α. [sent-145, score-0.093]

61 Input: n (signal dimensions); (xi , yi )m (training signals); k (size of the dictionary); λ0 , λ1 , λ2 i=1 (parameters); 0 ≤ µ1 ≤ µ2 ≤ . [sent-146, score-0.144]

62 , µm , Loop: Repeat until convergence (or a ﬁxed number of iterations), • Supervised sparse coding: Solve, for all i = 1, . [sent-156, score-0.164]

63 α⋆ = arg minα S(α, xi , D, θ, +1) i,+ (10) • Dictionary and parameters update: Solve m µC (S(α⋆ , xi , D, θ, −yi ) − S(α⋆ , xj , D, θ, yi )) + i,− i,+ min D,θ i=1 (1 − µ)S(α⋆ i , xi , D, θ, yi ) + λ2 ||θ||2 i,y 2 s. [sent-160, score-0.72]

64 (11) Figure 2: SDL: Supervised dictionary learning algorithm. [sent-163, score-0.498]

65 (11) is not convex in general (except when µ is close to 0), but a local minimum can be obtained using projected gradient descent (as in the general literature on dictionary learning, this local minimum has experimentally been found to be good enough in terms of classiﬁcation performance). [sent-169, score-0.526]

66 i,+ Partial derivatives when using our model with multiple classes or with the bilinear models f (x, α, θ) = xT Wα + b are not presented in this paper due to space limitations. [sent-174, score-0.225]

67 We also compare the performance of the linear (L) and bilinear (BL) models. [sent-176, score-0.196]

68 λ1 We deﬁne ﬁrst the sparsity parameter κ = λ0 , which dictates how sparse the decompositions are. [sent-192, score-0.274]

69 For reconstructive tasks, a typical value often used in the literature (e. [sent-195, score-0.29]

70 Nevertheless, for discriminative tasks, increasing the number of parameters is likely to lead to overﬁtting, and smaller values like k = 64 or k = 32 are preferred. [sent-198, score-0.299]

71 As in logistic regression or support vector machines, this parameter is crucial when the number of training samples is small. [sent-200, score-0.096]

72 Then, a scale factor making the costs S ⋆ discriminative for µ > 0 can be chosen during the optimization process: Given a set of computed costs m S ⋆ , one can compute a scale factor γ ⋆ such that γ ⋆ = arg minγ i=1 C({γ(S ⋆ (xi , D, θ, −yi ) − ⋆ S (xi , D, θ, yi )). [sent-204, score-0.542]

73 Since we are following a continuation path from µ = 0 to µ = 1, the optimal value of µ is found along the path by measuring the classiﬁcation performance of the model on a validation set during the optimization. [sent-207, score-0.123]

74 1 Digits recognition In this section, we present experiments on the popular MNIST [20] and USPS handwritten digit datasets. [sent-209, score-0.106]

75 USPS is composed of 7291 training images and 2007 test images of size 16 × 16. [sent-211, score-0.133]

76 For a given image x, the test procedure consists of selecting the class which receives the most votes from the pairwise classiﬁers. [sent-222, score-0.117]

77 4% [21] for USPS, using methods tailored to these tasks, whereas ours is generic and has not been tuned for the handwritten digit classiﬁcation domain. [sent-230, score-0.106]

78 The purpose of our second experiment is not to measure the raw performance of our algorithm, but to answer the question “are the obtained dictionaries D discriminative per se? [sent-231, score-0.523]

79 To do so, we have trained on the USPS dataset 10 binary classiﬁers, one per digit in a one vs all fashion on the training set. [sent-233, score-0.157]

80 For a given value of µ, we obtain 10 dictionaries D and 10 sets of parameters θ, learned by the SDL-D L model. [sent-234, score-0.329]

81 To evaluate the discriminative power of the dictionaries D, we discard the learned parameters θ and use the dictionaries as if they had been learned in a reconstructive REC model: For each dictionary, 2. [sent-235, score-1.185]

82 0 Figure 3: On the left, a reconstructive and a discriminative dictionary. [sent-245, score-0.558]

83 On the right, average error rate in percents obtained by our dictionaries learned in a discriminative framework (SDL-D L) for various values of µ, when used at test time in a reconstructive framework (REC-L). [sent-246, score-0.903]

84 26 Gain 0% 2% 4% 6% 15% 25% Table 2: Error rates for the texture classiﬁcation task using various methods and sizes m of the training set. [sent-282, score-0.158]

85 we decompose each image from the training set by solving the simple sparse reconstruction problem from Eq. [sent-284, score-0.341]

86 Repeating the sparse decomposition procedure on the test set permits us to evaluate the performance of these learned linear SVMs. [sent-287, score-0.32]

87 We see that using the dictionaries obtained with discrimative learning (µ > 0, SDL-D L) dramatically improves the performance of the basic linear classiﬁer learned a posteriori on the α’s, showing that our learned dictionaries are discriminative per se. [sent-289, score-0.957]

88 Figure 3 also shows a dictionary adapted to the reconstruction of the MNIST dataset and a discriminative one, adapted to “9 vs all”. [sent-290, score-0.965]

89 2 Texture classiﬁcation In the digit recognition task, our bilinear framework did not perform better than the linear one L. [sent-292, score-0.255]

90 We have chosen to consider two texture images from the Brodatz dataset, presented in Figure 4, and to build two classes, composed of 12 × 12 patches taken from these two textures. [sent-296, score-0.201]

91 We have compared the classiﬁcation performance of all our methods, including BL, for a dictionary of size k = 64 and κ = 0. [sent-297, score-0.498]

92 The training set was composed of patches from the left half of each texture and the test sets of patches from the right half, so that there is no overlap between them in the training and test set. [sent-299, score-0.315]

93 Future work will be devoted to adapting the proposed framework to shift-invariant models that are standard in image processing tasks, but not readily generalized to the sparse dictionary learning setting. [sent-307, score-0.714]

94 Right: reconstructive and discriminative dictionaries Acknowledgments This paper was supported in part by ANR under grant MGA. [sent-310, score-0.789]

95 Sparse coding with an overcomplete basis set: A strategy employed by v1? [sent-318, score-0.194]

96 Image denoising via sparse and redundant representations over learned dictionaries. [sent-323, score-0.29]

97 Sparse representations for image classiﬁcation: Learning discriminative and reconstructive non-parametric dictionaries. [sent-371, score-0.641]

98 A ﬁxed-point continuation method for l1-regularized minimization with applications to compressed sensing. [sent-426, score-0.095]

99 Efﬁcient learning of sparse representations with an energy-based model. [sent-433, score-0.195]

100 The K-SVD: An algorithm for designing of overcomplete dictionaries for sparse representations. [sent-440, score-0.462]

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