nips nips2007 nips2007-196 knowledge-graph by maker-knowledge-mining

196 nips-2007-The Infinite Gamma-Poisson Feature Model

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Author: Michalis K. Titsias

Abstract: We present a probability distribution over non-negative integer valued matrices with possibly an inﬁnite number of columns. We also derive a stochastic process that reproduces this distribution over equivalence classes. This model can play the role of the prior in nonparametric Bayesian learning scenarios where multiple latent features are associated with the observed data and each feature can have multiple appearances or occurrences within each data point. Such data arise naturally when learning visual object recognition systems from unlabelled images. Together with the nonparametric prior we consider a likelihood model that explains the visual appearance and location of local image patches. Inference with this model is carried out using a Markov chain Monte Carlo algorithm. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract We present a probability distribution over non-negative integer valued matrices with possibly an inﬁnite number of columns. [sent-5, score-0.218]

2 We also derive a stochastic process that reproduces this distribution over equivalence classes. [sent-6, score-0.125]

3 This model can play the role of the prior in nonparametric Bayesian learning scenarios where multiple latent features are associated with the observed data and each feature can have multiple appearances or occurrences within each data point. [sent-7, score-0.456]

4 Together with the nonparametric prior we consider a likelihood model that explains the visual appearance and location of local image patches. [sent-9, score-0.232]

5 An image can contain views of multiple object classes such as cars and humans and each class may have multiple occurrences in the image. [sent-20, score-0.324]

6 To deal with features having multiple occurrences, we introduce a probability distribution over sparse non-negative integer valued matrices with possibly an unbounded number of columns. [sent-21, score-0.29]

7 Each matrix row corresponds to a data point and each column to a feature similarly to the binary matrix used in the Indian buffet process [7]. [sent-22, score-0.191]

8 Each element of the matrix can be zero or a positive integer and expresses the number of times a feature occurs in a speciﬁc data point. [sent-23, score-0.164]

9 This model is derived by considering a ﬁnite gamma-Poisson distribution and taking the inﬁnite limit for equivalence classes of non-negative integer valued matrices. [sent-24, score-0.249]

10 This process uses the Ewens’s distribution [5] over integer partitions which was introduced in population genetics literature and it is equivalent to the distribution over partitions of objects induced by the Dirichlet process [1]. [sent-26, score-0.305]

11 The inﬁnite gamma-Poisson model can play the role of the prior in a nonparametric Bayesian learning scenario where both the latent features and the number of their occurrences are unknown. [sent-27, score-0.275]

12 Given this prior, we consider a likelihood model which is suitable for explaining the visual appearance and location of local image patches. [sent-28, score-0.18]

13 We assume that each data point is associated with a set of latent features and each feature can have multiple occurrences. [sent-35, score-0.162]

14 Let znk denote the number of times feature k occurs in the data point Xn . [sent-36, score-0.915]

15 Given K features, Z = {znk } is a N × K non-negative integer valued matrix that collects together all the znk values so as each row corresponds to a data point and each column to a feature. [sent-37, score-1.072]

16 Given that znk is drawn from a Poisson with a feature-speciﬁc parameter λk , Z follows the distribution N K P (Z|{λk }) = n=1 k=1 λznk exp{−λk } k = znk ! [sent-38, score-1.727]

17 α Γ( K ) (2) The hyperparameter α itself is given a vague gamma prior G(α; α0 , β0 ). [sent-42, score-0.114]

18 Using the above equations we can easily integrate out the parameters {λk } as follows K Γ(mk + P (Z|α) = α K) α k=1 α Γ( K )(N + 1)mk + K N n=1 znk ! [sent-43, score-0.853]

19 Using Equation (4) the expectation of the sum of znk s is αN and p is independent of the number of features. [sent-50, score-0.853]

20 3 The inﬁnite limit and the stochastic process To express the probability distribution in (3) for inﬁnite many features K we need to consider equivalence classes of Z matrices similarly to [7]. [sent-71, score-0.222]

21 We deﬁne the left-ordered form of non-negative integer valued matrices as follows. [sent-74, score-0.197]

22 We assume that for any possible znk holds znk ≤ c − 1, where c is a sufﬁciently large integer. [sent-75, score-1.706]

23 zN k ) as the integer number associated with column k that is expressed in a numeral system with basis c. [sent-79, score-0.12]

24 mk > 0 for k ≤ K+ , while the rest K −K+ features are unrepresented i. [sent-92, score-0.224]

25 The inﬁnite limit of (6) is derived by following a similar strategy with the one used for expressing the distribution over partitions of objects as a limit of the Dirichlet-multinomial pair [6, 9]. [sent-95, score-0.145]

26 This expression deﬁnes an exchangeable joint distribution over non-negative integer valued matrices with inﬁnite many columns in a left-ordered form. [sent-100, score-0.296]

27 1 The stochastic process The distribution in Equation (7) can be derived from a simple stochastic process that constructs the matrix Z sequentially so as the data arrive one at each time in a ﬁxed order. [sent-103, score-0.115]

28 We sample feature occurrences from the set of unrepresented features as follows. [sent-106, score-0.312]

29 Firstly, we draw an integer number 1 g1 from the negative binomial N B(g1 ; α, 2 ) which has a mean value equal to α. [sent-107, score-0.14]

30 g1 is the total number of feature occurrences for the ﬁrst data point. [sent-108, score-0.206]

31 + z1K1 = g1 and 1 ≤ K1 ≤ g1 , by drawing from Ewens’s distribution [5] over integer partitions which is given by P (z11 , . [sent-117, score-0.172]

32 , (8) (1) where vi is the multiplicity of integer i in the partition (z11 , . [sent-125, score-0.174]

33 The Ewens’s distribution is equivalent to the distribution over partitions of objects induced by the Dirichlet process and the Chinese restaurant process since we can derive the one from the other using simple combinatorics arguments. [sent-129, score-0.154]

34 The difference between them is that the former is a distribution over integer partitions while the latter is a distribution over partitions of objects. [sent-130, score-0.242]

35 For each feature k, with k ≤ Kn−1 , we choose znk based on the popularity of this feature in the previous n − 1 data 1 The partition of a positive integer is a way of writing this integer as a sum of positive integers where order does not matter, e. [sent-132, score-1.237]

36 This popularity is expressed by the total number of occurrences for the feature k which is n−1 n given by mk = i=1 zik . [sent-136, score-0.34]

37 Particularly, we draw znk from N B(znk ; mk , n+1 ) which has a mean mk value equal to n . [sent-137, score-1.111]

38 Similarly to the ﬁrst data point, we ﬁrst draw an n integer gn from N B(gn ; α, n+1 ), and subsequently we select a partition of that integer by drawing from the Ewens’s formula. [sent-139, score-0.305]

39 However, if we consider only the left-ordered class of matrices generated by the stochastic process then we obtain the exchangeable distribution in Equation (7). [sent-149, score-0.139]

40 2 Conditional distributions When we combine the prior P (Z|α) with a likelihood model p(X|Z) and we wish to do inference over Z using Gibbs-type sampling, we need to express the conditionals of the form P (znk |Z−(nk) , α) where Z−(nk) = Z \ znk . [sent-152, score-0.947]

41 When m−n,k > 0, the conditional of znk is given by e N B(znk ; m−n,k , NN ). [sent-158, score-0.876]

42 This conditional draws an integer number from N B(gn ; a, NN ) and then determines the occurrences for the new features by choosing a +1 partition of the integer gn using the Ewens’s distribution. [sent-160, score-0.501]

43 4 A likelihood model for images An image can contain multiple objects of different classes. [sent-163, score-0.141]

44 Unsupervised learning should deal with the unknown number of object classes in the images and also the unknown number of occurrences of each class in each image separately. [sent-167, score-0.281]

45 If object classes are the latent features, what we wish to infer is the underlying feature occurrence matrix Z. [sent-168, score-0.22]

46 Each image n is represented by dn local patches that are detected in the image so as Xn = (Yn , Wn ) = {(yni , wni ), i = 1, . [sent-171, score-0.345]

47 yni is the two-dimensional location of patch i and wni is an indicator vector (i. [sent-175, score-0.305]

48 is binary and satisﬁes L ℓ ℓ=1 wni = 1) that points into a set of L possible visual appearances. [sent-177, score-0.117]

49 We will describe the probabilistic model starting from the joint distribution of all variables which is given by joint = p(α)P (Z|α)p({θk }|Z)× dn N p(πn |Zn )p(mn , Σn |Zn ) n=1 P (sni |πn )P (wni |sni , {θk })p(yni |sni , mn , Σn ) . [sent-179, score-0.192]

50 (11) i=1 2 Features of this kind are the unrepresented features (k > K+ ) as well as all the unique features that occur only in the data point n (i. [sent-180, score-0.157]

51 Z α {θk } (mn, Σn) πn sni yni dn wni N Figure 1: Graphical model for the joint distribution in Equation (11). [sent-183, score-0.425]

52 Firstly, we generate α from its prior and then we draw the feature occurrence matrix Z using the inﬁnite gamma-Poisson prior P (Z|α). [sent-186, score-0.186]

53 , θkL } describes the appearance of the local patches W for the feature (object class) k. [sent-191, score-0.166]

54 Note that the feature appearance parameters {θk } depend on Z only through the number of represented features K+ which is obtained by counting the non-zero columns of Z. [sent-193, score-0.169]

55 To see how this mixture model arises, notice that a local patch in image n belongs to a certain occurrence of a feature. [sent-195, score-0.239]

56 We use the double index kj to denote the j occurrence of feature k where j = 1, . [sent-196, score-0.134]

57 This mixture model has k K+ Mn = k=1 znk components, i. [sent-200, score-0.891]

58 as many as the total number of feature occurrences in image n. [sent-202, score-0.271]

59 The assignment variable sni = {skj }, which takes Mn values, indicates the feature occurrence of ni patch i. [sent-203, score-0.347]

60 The parameters (mn , Σn ) determine the image-speciﬁc distribution for the locations {yni } of the local patches in image n. [sent-206, score-0.188]

61 We assume that each occurrence of a feature forms a Gaussian cluster of patch locations. [sent-207, score-0.179]

62 Thus yni follows a image-speciﬁc Gaussian mixture with Mn components. [sent-208, score-0.162]

63 mn and Σn collect all the means and covariances of the clusters in the image n. [sent-211, score-0.171]

64 Given that any object can be anywhere in the image and have arbitrary scale and orientation, (mnkj , Σnkj ) should be drawn from a quite vague prior. [sent-212, score-0.138]

65 We use a conjugate normal-Wishart prior for the pair (mnkj , Σnkj ) so as znk N (mnkj |µ, τ Σnkj )W (Σ−1 |v, V ), nkj p(mn , Σn |Zn ) = (12) k:znk >0 j=1 where (µ, τ, v, V ) are the hyperparameters shared by all features and images. [sent-213, score-1.065]

66 The assignment sni which determines the allocation of a local patch in a certain feature occurrence follows a multinokj znk mial: P (sni |πn ) = k:znk >0 j=1 (πnkj )sni . [sent-214, score-1.189]

67 Similarly the observed data pair (wni , yni ) of a local image patch is generated according to K+ L wℓ θkℓni P (wni |sni , {θk }) = k=1 ℓ=1 Pznk j=1 skj ni and znk skj ni [N (yni |mnkj , Σnkj )] p(yni |sni , mn , Σn ) = . [sent-215, score-1.37]

68 1 MCMC inference Inference with our model involves expressing the posterior P (S, Z, α|X) over the feature occurrences Z, the assignments S and the parameter α. [sent-220, score-0.295]

69 Note that the joint P (S, Z, α, X) factorizes N according to p(α)P (Z|α)P (W |S, Z) n=1 P (Sn |Zn )p(Yn |Sn , Zn ) where Sn denotes the assignments associated with image n. [sent-221, score-0.134]

70 A single step can K+ new old change an element of Z by one so as |znk − znk | ≤ 1. [sent-225, score-0.853]

71 Initially Z is such that Mn = k=1 znk ≥ 1, for any n which means that at least one mixture component explains the data of each image. [sent-226, score-0.891]

72 The proposal distribution for changing znk s ensures that this constraint is satisﬁed. [sent-227, score-0.874]

73 Suppose we wish to sample a new value for znk using the joint model p(S, Z, α, X). [sent-228, score-0.893]

74 Simply witting P (znk |S, Z−(nk) , α, X) is not useful since when znk changes the number of states the assignments Sn can take also changes. [sent-229, score-0.903]

75 This is clear since znk is a structural variable that affects the number of K+ components Mn = k=1 znk of the mixture model associated with image n and assignments Sn . [sent-230, score-1.859]

76 On the other hand the dimensionality of the assignments S−n = S \ Sn of all other images is not affected when znk changes. [sent-231, score-0.922]

77 Sampling from P (Sn |S−n , Z, W, Yn ) is carried out by applying local Gibbs sampling moves and global Metropolis moves that allow two occurrences of different features to exchange their data clusters. [sent-235, score-0.269]

78 The discrete patch appearances correspond to pixels and can take 20 possible grayscale values. [sent-240, score-0.163]

79 To generate an image we ﬁrst decide to include each feature with probability 0. [sent-242, score-0.127]

80 Then for each included feature we randomly select the number of occurrences from the range [1, 3]. [sent-244, score-0.206]

81 For each feature occurrence we select the pixels using the appearance multinomial and place the respective feature shape in a random location so that feature occurrences do not occlude each other. [sent-245, score-0.486]

82 The ﬁrst row of Figure 2 shows a training image (left), the locations of pixels (middle) and the discrete appearances (right). [sent-246, score-0.237]

83 The third row of Figure 2 shows how K+ (left) and the sum of all znk s (right) evolve through the ﬁrst 500 MCMC iterations. [sent-251, score-0.912]

84 The algorithm in the ﬁrst 20 iterations has training image n locations Yn appearances Wn 1331 3230 0212 Figure 2: The ﬁrst row shows a training image (left), the locations of pixels (middle) and the discrete appearances (right). [sent-252, score-0.409]

85 The second row shows the localizations of all feature occurrences in three images. [sent-253, score-0.29]

86 The third row shows how K+ (left) and the sum of all znk s (right) evolve through the ﬁrst 500 MCMC iterations. [sent-255, score-0.912]

87 Figure 3: The left most plot on the ﬁrst row shows the locations of detected patches and the bounding boxes in one of the annotated images. [sent-256, score-0.193]

88 The remaining ﬁve plots show examples of detections and localizations of the three most dominant features (including the car-category) in ﬁve non-annotated images. [sent-257, score-0.117]

89 For the state (Z, S) that is most frequently visited, the second row of Figure 2 shows the localizations of all different feature occurrences in three images. [sent-260, score-0.29]

90 Note that ellipses with the same color correspond to the different occurrences of the same feature. [sent-262, score-0.168]

91 We used the patch detection method presented in [8] and we constructed a dictionary of 200 visual appearances by clustering the SIFT [8] descriptors of the patches using K-means. [sent-264, score-0.212]

92 Locations of detected patches are shown in the ﬁrst row (left) of Figure 3. [sent-265, score-0.113]

93 These znk s values plus the assignments of all patches inside the boxes do not change during sampling. [sent-269, score-0.986]

94 Also the patches that lie outside the boxes in all annotated images are not allowed to be part of car occurrences. [sent-270, score-0.122]

95 To keep the plots uncluttered, Figure 3 shows the detections and localizations of only the three most dominant features (including the car-category) in ﬁve non-annotated images. [sent-273, score-0.117]

96 The red ellipses correspond to different occurrences of the car-feature, the green ones to a tree-feature and the blue ones to a street-feature. [sent-274, score-0.168]

97 6 Discussion We presented the inﬁnite gamma-Poisson model which is a nonparametric prior for non-negative integer valued matrices with inﬁnite number of columns. [sent-275, score-0.249]

98 We discussed the use of this prior for unsupervised learning where multiple features are associated with our data and each feature can have multiple occurrences within each data point. [sent-276, score-0.345]

99 For example, an interesting application can be Bayesian matrix factorization where a matrix of observations is decomposed into a product of two or more matrices with one of them being a non-negative integer valued matrix. [sent-278, score-0.197]

100 Inﬁnite latent feature models and the Indian buffet process. [sent-314, score-0.138]

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