nips nips2012 nips2012-89 knowledge-graph by maker-knowledge-mining

89 nips-2012-Coupling Nonparametric Mixtures via Latent Dirichlet Processes

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Author: Dahua Lin, John W. Fisher

Abstract: Mixture distributions are often used to model complex data. In this paper, we develop a new method that jointly estimates mixture models over multiple data sets by exploiting the statistical dependencies between them. Speciﬁcally, we introduce a set of latent Dirichlet processes as sources of component models (atoms), and for each data set, we construct a nonparametric mixture model by combining sub-sampled versions of the latent DPs. Each mixture model may acquire atoms from different latent DPs, while each atom may be shared by multiple mixtures. This multi-to-multi association distinguishes the proposed method from previous ones that require the model structure to be a tree or a chain, allowing more ﬂexible designs. We also derive a sampling algorithm that jointly infers the model parameters and present experiments on both document analysis and image modeling. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Speciﬁcally, we introduce a set of latent Dirichlet processes as sources of component models (atoms), and for each data set, we construct a nonparametric mixture model by combining sub-sampled versions of the latent DPs. [sent-6, score-0.55]

2 Each mixture model may acquire atoms from different latent DPs, while each atom may be shared by multiple mixtures. [sent-7, score-0.765]

3 This assumption does not hold in the cases with multiple groups of data, where samples in different groups are generally not exchangeable. [sent-15, score-0.202]

4 Our primary goal here is to describe multiple groups of data through coupled mixture models. [sent-28, score-0.224]

5 Sharing statistical properties across different groups allows for more reliable model estimation, especially 1 when the observed samples in each group are limited or noisy. [sent-29, score-0.188]

6 (2) The marginal distribution of atoms for each group remains a DP. [sent-32, score-0.232]

7 For example, the prior weight of a common atom can vary across groups. [sent-34, score-0.401]

8 Speciﬁcally, we express mixture models for each group as a stochastic combination over a set of latent DPs. [sent-40, score-0.319]

9 The multi-to-multi association between data groups and latent DPs provides much greater ﬂexibility to model conﬁgurations, as opposed to prior work (we provide a detailed comparison in section 3. [sent-41, score-0.304]

10 α + (n − 1) α + (n − 1) (3) Here, θ /i denotes all component parameters except θi , K/i denotes the number of distinct atoms among them, and m/i (k) denotes the number of occurrences of the atom φk . [sent-61, score-0.512]

11 Then, with a probability proportional to m/i (k)f (xi ; φk ), we set θi = φk , and with a probability proportional to αf (xi ; B), we draw an new atom from B(·|xi ), which is the posterior parameter distribution given xi . [sent-70, score-0.365]

12 2 (4) H1 q11 H2 q21 q31 D1 q22 q32 D2 ✓1i D4 ✓3i x2i Hs B ML Groups ct zti k rtk Q ✓4i x3i n2 n1 ↵s q42 D3 ✓2i x1i Latent DPs xti Atoms 1 1 nt M x4i n3 Figure 2: The reformulated model for Gibbs n4 sampling contains latent DPs, groups of data, and atoms. [sent-87, score-1.023]

13 Each sample xti is attached a label zti that assigns it an atom φzti . [sent-88, score-0.746]

14 To generate zti , we draw a latent DP (from Mult(ct )) and choose a label therefrom. [sent-89, score-0.407]

15 In sampling, Hs is integrated out, resulting in mutual dependency between zti , as in the Chinese restaurant process. [sent-90, score-0.196]

16 Figure 1: This shows the graphical model of the coupled DP formulation on a case with four groups and two latent DPs. [sent-91, score-0.355]

17 Each mixture model Dt inherits atoms from Hs with a probability qts , resulting in Eq. [sent-92, score-0.498]

18 Given a sub-sampling probability q, one draws a binary value rk with Pr(rk = 1) = q for each atom φk to decide whether to retain it, resulting in a DP as Sq (D) k:rk =1 πk δφk ∼ DP(αqB). [sent-97, score-0.347]

19 Given D = k=1 πk δφk ∼ DP (αB), perturbing the locations of each atom fol∞ lowing a probabilistic transition kernel T also yields a new DP, given by T (D) k=1 πk δT (φk ) . [sent-100, score-0.325]

20 3 Coupled Nonparametric Mixture Models Our primary goal is to develop a joint formulation over group-wise DP mixture models where components are shared across different groups and the weights and parameters of shared components vary across groups. [sent-102, score-0.314]

21 The generative formulation is then described as follows: First, generate ML latent DPs independently, as Hs ∼ DP (αs B), for s = 1, . [sent-106, score-0.23]

22 (6) Second, generate M dependent DPs, each for a group of data, by combining the sub-sampled versions of the latent DPs through stochastic convex combination. [sent-110, score-0.302]

23 (7) Intuitively, for each group of data (say the t-th), we choose a subset of atoms from each latent source and bring them together to generate Dt . [sent-121, score-0.438]

24 Here, qts is the prior probability that an atom in Hs will be inherited by Dt . [sent-122, score-0.673]

25 Particularly, the atom parameter would be an adapted version from Tt (φk , ·) instead of φk itself, when the atom φk is inherited by Dt . [sent-126, score-0.716]

26 (8) Here, xt,i is the i-th data sample in the t-th group, and θt,i is the associated atom parameter. [sent-134, score-0.346]

27 (7) has Dt ∼ DP(βt B), with βt = ML s=1 αs qts . [sent-138, score-0.259]

28 Two models are strongly coupled, if there exists a subset of latent DPs, from which both inherit atoms with high probabilities, while their coupling is much weaker if the associated q-values are set differently. [sent-146, score-0.47]

29 Generally, higher values of αs lead to more atoms being associated with the data, resulting in ﬁner clusters. [sent-150, score-0.187]

30 Another important factor is ML , the number of latent DPs. [sent-151, score-0.18]

31 A large number of latent DPs provides ﬁne-grained control of the model conﬁguration at the cost of increased complexity. [sent-152, score-0.18]

32 Our model allows the mixture model for each group to inherit from multiple sources, making it applicable to more general contexts. [sent-159, score-0.192]

33 Though motivated differently, this construction can be reduced to a formulation in the form Dt = j∈Rt ctj Hj , where Rt is the subset of latent DPs used for Dt . [sent-167, score-0.266]

34 Consequently, the combination coefﬁcients have to satisfy (ctj )j∈Rt ∼ Dir((αj )j∈Rt ), implying that the relative weights of two latent sources are restricted to be the same in all groups that inherit from both. [sent-172, score-0.37]

35 In contrast, the approach here allows the weights of latent DPs to vary across groups. [sent-173, score-0.233]

36 Also, SNΓP doesn’t allow atom parameters to vary across groups. [sent-174, score-0.378]

37 (2) Each group maintains a distribution over the latent DPs to choose from, which reﬂects the different contributions of these sources. [sent-178, score-0.246]

38 In particular, each group maintains indicators of whether particular atoms are inherited, and as a consequence, the ones that are deemed irrelevant are put out of scope. [sent-180, score-0.232]

39 (4) As there are multiple latent DPs, for each atom, there is uncertainty about where it comes from. [sent-181, score-0.18]

40 We have a speciﬁc step that takes this into account, which allows reassigning an atom to different sources. [sent-182, score-0.325]

41 Recall that there are M groups of data, and ML latent DPs to link between them. [sent-184, score-0.281]

42 Note here that the index k is a globally unique identiﬁer of the atom, which would not be changed during atom relocation. [sent-190, score-0.325]

43 Since an atom may correspond to multiple data samples. [sent-191, score-0.325]

44 Instead of instantiating the parameter θti for each data sample xti , we attach to xti an indicator zti that associates the sample to a particular atom. [sent-192, score-0.681]

45 To facilitate the sampling process, for each atom φk , we maintain an indicator sk specifying the latent DP that contains it, and a set of counters {mtk }, where mtk equals the number of associated data samples in t-th group. [sent-194, score-0.669]

46 We also maintain a set Is for Hs (the s-th latent DP), which contains the indices of all atoms therein. [sent-195, score-0.346]

47 It consists of four steps: (1) Generate latent DPs: for each s = 1, . [sent-198, score-0.18]

48 (3) Decide inheritance: for each atom φk , we draw a binary variable rtk with Pr(rtk = 1) = qtsk to indicate whether φk is inherited by the t-th group. [sent-209, score-0.644]

49 Here sk is the index of the latent DP which φk is from. [sent-210, score-0.208]

50 (4) Generate data: to generate xti , we ﬁrst choose a latent DP by drawing u ∼ Mult(ct1 , . [sent-211, score-0.466]

51 , ctML ), then draw an atom from Hu , using it to produce xti . [sent-214, score-0.625]

52 Based on this formulation, we derive the following Gibbs sampling steps to update the atom parameters and other hidden variables. [sent-215, score-0.349]

53 Recall that each data sample xti is associated with a label variable zti that indicates the atom accounting for xti . [sent-217, score-1.027]

54 To draw zti , we ﬁrst have to choose a particular latent DP as the source (we denote the index of this DP by uti ). [sent-218, score-0.488]

55 Let z/ti denote all labels except zti , and rt denote the inheritance indicators. [sent-219, score-0.279]

56 Then, we get the likelihood of xti (with Hs integrated out) as p(xti |uti = s, rt , z/ti ) = 1 wst/i + qts αs m∗k/ti f (xti ; φk ) + qts αs f (xti ; B) . [sent-220, score-0.815]

57 (10) k∈Is :rtk =1 Here, m∗k/ti is the total number of samples associated with φk in all groups (except for xti ), wst/i = k∈Is :rtk =1 m∗k/ti , f (xti ; φk ) is the pdf at xti w. [sent-221, score-0.642]

58 , ctML ) are the group-speciﬁc prior over latent sources. [sent-230, score-0.203]

59 Once a latent DP is chosen (using the formula above), we can then draw a particular atom. [sent-231, score-0.22]

60 This is similar to the Chinese restaurant process: with a probability proportional to m∗k/ti f (xti ; φk ), we set zti = k, and with a probability proportional to qts αs f (xti ; B), we draw a new atom from B(·|xi ). [sent-232, score-0.82]

61 Only the atoms that is contained in Hs and has rtk = 1 (inherited by Dt ) can be drawn at this step. [sent-233, score-0.379]

62 We have to modify relevant quantities accordingly, such as mtk , ws , and Is , when a label zti is changed. [sent-234, score-0.252]

63 Moreover, when a new atom φk is created, it will be initially assigned to the latent DP that generates it (i. [sent-235, score-0.505]

64 If an atom φk is associated with some data in the t-th group, then we know for sure that it is inherited by Dt , and thus we can set rtk = 1. [sent-239, score-0.625]

65 However, if φk is not observed, it doesn’t imply rtk = 0. [sent-240, score-0.213]

66 For such an atom (suppose it is from Hs ), we have γ(τs/t , nt ) Pr(rtk = 1|others) qts · p(zt |rtk = 1, others) qts = = . [sent-241, score-0.878]

67 Pr(rtk = 0|others) (1 − qts ) · p(zt |rtk = 0, others) 1 − qts γ(τs/t + m∗k/t , nt ) (12) Here, τs/t = qts αs + k ∈Is −{k} m∗k/t and m∗k /t is the number of samples associated with k in all other groups (excluding the ones in the t-th group). [sent-242, score-0.934]

68 , ctML ) reﬂect the relative contribution of each latent DP to the t-th group. [sent-253, score-0.18]

69 In this model, each atom is almost surely from a unique latent DP (i. [sent-265, score-0.505]

70 This leads to an important question: How to we assign atoms to latent DPs? [sent-268, score-0.346]

71 Initially, an atom is assigned to the latent DP from which it is generated. [sent-269, score-0.505]

72 Here, we treat the assignment of each atom as a variable. [sent-271, score-0.325]

73 Consider an atom φk , with sk indicating its corresponding source DP. [sent-272, score-0.353]

74 Then, we have p(sk = j|others) = qts t:rtk =1 (1 − qts ). [sent-273, score-0.518]

75 (14) t:rtk =0 When an atom φk that was in Hs is reassigned to Hs , we have to move the index k from Is to Is . [sent-274, score-0.325]

76 To exploit the statistical dependency between groups, we further introduce a set of latent DPs to link between these mixtures, as described 6 above. [sent-284, score-0.18]

77 We perform experiments on several conﬁgurations, with different ways to connect between latent sources and data groups, as illustrated in Figure 3. [sent-288, score-0.216]

78 (1) Single Latent DP (S-LDP): there is only one latent DP connecting to all groups, with q-values set to 0. [sent-289, score-0.18]

79 Though with a structure similar to HDP, the formulation is actually different: HDP generates group-speciﬁc mixtures by using the latent DP as the base measure, while our model involves explicit sub-sampling. [sent-291, score-0.204]

80 (2) Multi Latent DP (M-LDP): there are two types of latent DPs – local and global ones. [sent-292, score-0.205]

81 The local latent DPs are introduced to help sharing statistical strength among the groups close to each other, so as to capture the intuition that papers published in consecutive years are more likely to share topics than those published in distant years. [sent-293, score-0.482]

82 The inheritance probability from a local latent DP Hs to Dt is set as qts = exp(−|t − s|/σ). [sent-294, score-0.545]

83 Also, recognizing that some topics may be shared across the entire corpus, we also introduce a global latent DP, from which every group inherit atoms with the same probability, which allows distant groups to be connected. [sent-295, score-0.655]

84 For comparison, we also consider another setting of q-values under the M-LDP structure: to set qts = I(|t − s| ≤ σ), that is to connect Dt and Hs only when |t − s| ≤ σ, with qts = 1. [sent-297, score-0.518]

85 We place a weak prior over αs for each latent DP, as αs ∼ Gamma(0. [sent-304, score-0.203]

86 This suggests that the sharing of statistical strength through local latent DPs improves the reliability of the estimation, especially when the training data are limited. [sent-316, score-0.248]

87 For example, the papers published in consecutive years tend to share lots of topics, however, the topics may not be as similar when you compare papers published recently to those a decade ago. [sent-319, score-0.185]

88 A set of local latent DPs may capture such relations more effectively than a single global one. [sent-320, score-0.205]

89 This way encourages each latent DP to be locally focused, while allowing the atoms therein to be shared across the entire corpus. [sent-323, score-0.388]

90 The SNΓP, instead, provides no mechanism to vary the contributions of the latent DPs, and has to make a hard limit of their spans to achieve locality. [sent-325, score-0.212]

91 Whereas this issue could be addressed through multiple level of latent nodes with different spans, it will increase the complexity, and thus the risk of overﬁtting. [sent-326, score-0.18]

92 For M-LDP, recall that we set qts = exp(−|t − s|/σ). [sent-327, score-0.259]

93 Optimal performance is attained when the choice of σ balances the need to share atoms and the desire to keep the latent DPs locally focused. [sent-330, score-0.346]

94 5, and a set of local latent DPs, each for a category. [sent-352, score-0.205]

95 The prior probability of inheriting from the corresponding latent DP is 1. [sent-353, score-0.203]

96 Whereas no prior knowledge about the similarity between categories is assumed, the latent DPs incorporated in this way still provide a mechanism for local coupling. [sent-356, score-0.25]

97 This is due to the more ﬂexible way to conﬁgure local coupling that allows the weights of latent DPs to vary. [sent-360, score-0.255]

98 6 Conclusion We have presented a principled approach to modeling grouped data, where mixture models for different groups are coupled via a set of latent DPs. [sent-361, score-0.431]

99 The proposed framework allows each mixture model to inherit from multiple latent DPs, and each latent DP to contribute differently to different groups, thus providing great ﬂexibility for model design. [sent-362, score-0.486]

100 Construction of dependent dirichlet processes based on poisson processes. [sent-427, score-0.202]

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