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301 nips-2011-Variational Gaussian Process Dynamical Systems

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Author: Neil D. Lawrence, Michalis K. Titsias, Andreas Damianou

Abstract: High dimensional time series are endemic in applications of machine learning such as robotics (sensor data), computational biology (gene expression data), vision (video sequences) and graphics (motion capture data). Practical nonlinear probabilistic approaches to this data are required. In this paper we introduce the variational Gaussian process dynamical system. Our work builds on recent variational approximations for Gaussian process latent variable models to allow for nonlinear dimensionality reduction simultaneously with learning a dynamical prior in the latent space. The approach also allows for the appropriate dimensionality of the latent space to be automatically determined. We demonstrate the model on a human motion capture data set and a series of high resolution video sequences. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract High dimensional time series are endemic in applications of machine learning such as robotics (sensor data), computational biology (gene expression data), vision (video sequences) and graphics (motion capture data). [sent-12, score-0.258]

2 In this paper we introduce the variational Gaussian process dynamical system. [sent-14, score-0.498]

3 Our work builds on recent variational approximations for Gaussian process latent variable models to allow for nonlinear dimensionality reduction simultaneously with learning a dynamical prior in the latent space. [sent-15, score-1.109]

4 The approach also allows for the appropriate dimensionality of the latent space to be automatically determined. [sent-16, score-0.296]

5 We demonstrate the model on a human motion capture data set and a series of high resolution video sequences. [sent-17, score-0.463]

6 A standard approach is to simultaneously apply a nonlinear dimensionality reduction to the data whilst governing the latent space with a nonlinear temporal prior. [sent-19, score-0.441]

7 The key difﬁculty for such approaches is that analytic marginalization of the latent space is typically intractable. [sent-20, score-0.248]

8 Markov chain Monte Carlo approaches can also be problematic as latent trajectories are strongly correlated making efﬁcient sampling a challenge. [sent-21, score-0.248]

9 One promising approach to these time series has been to extend the Gaussian process latent variable model [1, 2] with a dynamical prior for the latent space and seek a maximum a posteriori (MAP) solution for the latent points [3, 4, 5]. [sent-22, score-1.027]

10 We refer to this class of dynamical models based on the GP-LVM as Gaussian process dynamical systems (GPDS). [sent-24, score-0.378]

11 Firstly, since the latent variables are not marginalised, the parameters of the dynamical prior cannot be optimized without the risk of overﬁtting. [sent-26, score-0.44]

12 Further, the dimensionality of the latent space cannot be determined by the model: adding further dimensions always increases the likelihood of the data. [sent-27, score-0.362]

13 As well as providing a principled approach to handling uncertainty in the latent space, this allows both the parameters of the latent dynamical process and the dimensionality of the latent space to be determined. [sent-29, score-1.008]

14 We illustrate this by modeling human motion capture data and high dimensional video sequences. [sent-31, score-0.518]

15 1 2 The Model Assume a multivariate times series dataset {yn , tn }N , where yn ∈ RD is a data vector observed at time n=1 tn ∈ R+ . [sent-33, score-0.21]

16 We are especially interested in cases where each yn is a high dimensional vector and, therefore, we assume that there exists a low dimensional manifold that governs the generation of the data. [sent-34, score-0.285]

17 We do not want to make strong assumptions about the functional form of the latent functions (x, f ). [sent-36, score-0.248]

18 Therefore, we assume that x is a multivariate Gaussian process indexed by time t and f is a different multivariate Gaussian process indexed by x, and we write xq (t) ∼ GP(0, kx (ti , tj )), q = 1, . [sent-38, score-0.664]

19 (2) (3) Here, the individual components of the latent function x are taken to be independent sample paths drawn from a Gaussian process with covariance function kx (ti , tj ). [sent-45, score-0.772]

20 Similarly, the components of f are independent draws from a Gaussian process with covariance function kf (xi , xj ). [sent-46, score-0.324]

21 More precisely, kx determines the properties of each temporal latent function xq (t). [sent-48, score-0.714]

22 For instance, the use of an Ornstein-Uhlbeck covariance function yields a Gauss-Markov process for xq (t), while the squared-exponential covariance function gives rise to very smooth and non-Markovian processes. [sent-49, score-0.578]

23 In our experiments, we will focus on the squared exponential covariance function (RBF), the Mat´ rn 3/2 which is only once differentiable, and a periodic covariance function e [9, 10] which can be used when data exhibit strong periodicity. [sent-50, score-0.444]

24 These covariance functions take the form: √ √ (ti −tj )2 − 3|ti −tj | 3|ti − tj | 2 2 − 2 lt kx(rbf) (ti , tj ) = σrbf e (2lt ) , kx(mat) (ti , tj ) = σmat 1 + e , lt 1 2 kx(per) (ti , tj ) = σper e− 2 sin2 ( 2π (ti −tj )) T lt . [sent-51, score-0.762]

25 (4) The covariance function kf determines the properties of the latent mapping f that maps each low dimensional variable xn to the observed vector yn . [sent-52, score-0.725]

26 We wish this mapping to be a non-linear but smooth, and thus a suitable choice is the squared exponential covariance function 1 2 kf (xi , xj ) = σard e− 2 Q q=1 wq (xi,q −xj ,q )2 , (5) which assumes a different scale wq for each latent dimension. [sent-53, score-0.616]

27 This, as in the variational Bayesian formulation of the GP-LVM [8], enables an automatic relevance determination procedure (ARD), i. [sent-54, score-0.282]

28 Similarly, the matrix F ∈ RN ×D will denote the mapping latent variables, i. [sent-58, score-0.284]

29 fnd = fd (xn ), associated with observations Y from (1). [sent-60, score-0.259]

30 Analogously, X ∈ RN ×Q will store all low dimensional latent variables xnq = xq (tn ). [sent-61, score-0.572]

31 Further, we will refer to columns of these matrices by the vectors yd , fd , xq ∈ RN . [sent-62, score-0.559]

32 Later we also use a similar convention for the covariance functions by often writing them as kf and kx . [sent-65, score-0.467]

33 In the next section we describe how efﬁcient variational approximations can be applied to marginalize X by extending the framework of [8]. [sent-71, score-0.282]

34 We now invoke the variational Bayesian methodology to approximate the integral. [sent-75, score-0.282]

35 Following a standard procedure [11], we introduce a variational distribution q(Θ) and compute the Jensen’s lower bound Fv on the logarithm of (9), Fv (q, θ) = q(Θ) log p(Y |F )p(F |X)p(X |t) dXdF, q(Θ) (10) where θ denotes the model’s parameters. [sent-76, score-0.311]

36 More precisely, we augment the joint probability model in (6) by including M extra samples of the GP latent mapping f , known as inducing points, so that um ∈ RD is such a sample. [sent-79, score-0.354]

38 By dropping X from our expressions, we write the augmented GP prior analytically (see [9]) as −1 −1 p(fd |ud , X) = N fd |KN M KM M ud , KN N − KN M KM M KM N . [sent-82, score-0.531]

39 Titsias and Lawrence [8] assume full independence for q(X) and the variational covariances are diagonal matrices. [sent-84, score-0.282]

40 Here, in contrast, the posterior over the latent variables will have strong correlations, so Sq is taken to be a N × N full covariance matrix. [sent-85, score-0.394]

41 Optimization of the variational lower bound provides an approximation to the true posterior p(X|Y ) by q(X). [sent-86, score-0.311]

42 In the augmented probability model, the “difﬁcult” term p(F |X) appearing in (10) is now replaced with (12) and, eventually, it cancels out with the ﬁrst factor of the variational distribution (13) so that F can be marginalised out analytically. [sent-87, score-0.32]

43 All the information regarding data point correlations is captured in the KL term and the connection with the observations comes through the variational distribution. [sent-91, score-0.282]

44 However, when not factorizing q(X) across data points yields O(N 2 ) variational parameters to optimize. [sent-94, score-0.282]

45 2 Reparametrization and Optimization The optimization involves the model parameters θ = (β, θf , θx ), the variational parameters {µq , Sq }Q from q=1 ˜ q(X) and the inducing points3 X. [sent-97, score-0.352]

46 Optimization of the variational parameters appears challenging, due to their large number and the correlations between them. [sent-98, score-0.282]

47 However, by reparametrizing our O N 2 variational parameters according to the framework described in [12] we can obtain a set of O(N ) less correlated variational parameters. [sent-99, score-0.564]

48 Speciﬁcally, we ﬁrst take the derivatives of the variational bound (14) w. [sent-100, score-0.311]

49 in human motion capture data several walks from a subject). [sent-110, score-0.279]

50 We handle this by allowing a different temporal latent function for each of the independent sequences, so that X (s) is the set of latent variables corresponding to the sequence s. [sent-116, score-0.533]

51 In this setting, each block of observations Y (s) is generated from its corresponding X (s) according to Y (s) = F (s) + , where the latent function which governs this mapping is shared across all sequences and is Gaussian noise. [sent-124, score-0.407]

52 It should be capable of generating completely new sequences or reconstructing missing observations from partially observed data. [sent-126, score-0.222]

53 We will use the term “variational parameters” to refer only to the parameters of q(X) although the inducing points are also variational parameters. [sent-138, score-0.352]

54 1 Predictions Given Only the Test Time Points To approximate the predictive density, we will need to introduce the underlying latent function values F∗ ∈ RN∗ ×D (the noisy-free version of Y∗ ) and the latent variables X∗ ∈ RN∗ ×Q . [sent-140, score-0.542]

55 (16), it is approximated by a Gaussian variational distribution q(X∗ ), Q q(X∗ ) = Q q=1 Q p(x∗,q |xq )q(xq )dxq = q(x∗,q ) = q=1 p(x∗,q |xq ) q(xq ) , (18) q=1 where p(x∗,q |xq ) is a Gaussian found from the conditional GP prior (see [9]) and q(X) is also Gaussian. [sent-145, score-0.312]

56 We can, thus, work out analytically the mean and variance for (18), which turn out to be: µx∗,q = K∗N µq ¯ (19) var(x∗,q ) = K∗∗ − K∗N (Kt + Λ−1 )−1 KN ∗ q (20) where K∗N = kx (t∗ , t), K∗N = K∗N and K∗∗ = kx (t∗ , t∗ ). [sent-146, score-0.45]

57 To obtain an approximation, we ﬁrstly need to apply variational inference and approximate p(X∗ |Y∗p , Y ) with a Gaussian distribution. [sent-159, score-0.282]

58 This requires the optimisation of a new variational lower bound that accounts for the contribution of the partially observed data Y∗p . [sent-160, score-0.435]

59 Moreover, the variational optimisation requires the deﬁnition of the variational distribution q(X∗ , X) which needs to be optimised and is fully correlated across X and X∗ . [sent-162, score-0.611]

60 A much faster but less accurate method would be to decouple the test from the training latent variables by imposing the factorisation q(X∗ , X) = q(X)q(X∗ ). [sent-164, score-0.319]

61 5 4 Handling Very High Dimensional Datasets Our variational framework avoids the typical cubic complexity of Gaussian processes allowing relatively large training sets (thousands of time points, N ). [sent-166, score-0.353]

62 5 Experiments We consider two different types of high dimensional time series, a human motion capture data set consisting of different walks and high resolution video sequences. [sent-174, score-0.576]

63 Matlab source code for repeating the following experiments and links to the video ﬁles are available on-line from http://staffwww. [sent-176, score-0.205]

64 1 Human Motion Capture Data We followed [14, 15] in considering motion capture data of walks and runs taken from subject 35 in the CMU motion capture database. [sent-183, score-0.416]

65 This results in 2,613 separate 59-dimensional frames split into 31 training sequences with an average length of 84 frames each. [sent-186, score-0.303]

66 the algorithm learns a common latent space for these motions. [sent-190, score-0.248]

67 We can also indirectly compare with the binary latent variable model (BLV) of [14] which used a slightly different data preprocessing. [sent-194, score-0.248]

68 We performed two runs, once using the Mat´ rn covariance function for the dynamical prior and once using the RBF. [sent-197, score-0.412]

69 From table 1 we see that the e variational Gaussian process dynamical system considerably outperforms the other approaches. [sent-198, score-0.498]

70 The appropriate latent space dimensionality for the data was automatically inferred by our models. [sent-199, score-0.296]

71 The model which employed an RBF covariance to govern the dynamics retained four dimensions, whereas the model that used the Mat´ rn e kept only three. [sent-200, score-0.22]

72 The other latent dimensions were completely switched off by the ARD parameters. [sent-201, score-0.314]

73 The best performance for the legs and the body reconstruction was achieved by the VGPDS model that used the Mat´ rn e and the RBF covariance function respectively. [sent-202, score-0.266]

74 2 Modeling Raw High Dimensional Video Sequences For our second set of experiments we considered video sequences. [sent-204, score-0.205]

75 This also allows us to directly sample video from the learned model. [sent-207, score-0.205]

76 Firstly, we used the model to reconstruct partially observed frames from test video sequences4 . [sent-208, score-0.438]

77 For the ﬁrst video discussed here we gave as partial information approximately 50% of the pixels while for the other two we gave approximately 40% of the pixels on each frame. [sent-209, score-0.343]

78 6 Table 1: Errors obtained for the motion capture dataset considering nearest neighbour in the angle space (NN) and in the scaled space(NN sc. [sent-219, score-0.245]

79 We also considered an HD video of dimensionality 9 × 105 that shows an artiﬁcially created scene of ocean waves as well as a 230, 400−dimensional video showing a dog running for 60 frames. [sent-265, score-0.661]

80 For the ﬁrst two videos we used the Mat´ rn and RBF covariance e functions respectively to model the dynamics and interpolated to reconstruct blocks of frames chosen from the whole sequence. [sent-267, score-0.376]

81 For the ‘dog’ dataset we constructed a compound kernel kx = kx(rbf) + kx(periodic) , where the RBF term is employed to capture any divergence from the approximately periodic pattern. [sent-268, score-0.368]

82 The number of latent dimensions selected by our model is in parenthesis. [sent-273, score-0.314]

83 As a second task, we used our generative model to create new samples and generate a new video sequence. [sent-281, score-0.205]

84 This is most effective for the ‘dog’ video as the training examples were approximately periodic in nature. [sent-282, score-0.346]

85 The results show a smooth transition from training to test and amongst the test video frames. [sent-285, score-0.238]

86 The resulting video of the dog continuing to run is sharp and high quality. [sent-286, score-0.325]

87 The full video is available in the supplementary material. [sent-289, score-0.205]

88 6 Discussion and Future Work We have introduced a fully Bayesian approach for modeling dynamical systems through probabilistic nonlinear dimensionality reduction. [sent-290, score-0.247]

89 Marginalizing the latent space and reconstructing data using Gaussian processes 7 (a) (b) (e) (c) (f) (d) (g) (h) Figure 1: (a) and (c) demonstrate the reconstruction achieved by VGPDS and NN respectively for the most challenging frame (b) of the ‘missa’ video, i. [sent-291, score-0.424]

90 Finally, we demonstrate the ability of the model to automatically select the latent dimensionality by showing the initial lengthscales (ﬁg: (g)) of the ARD covariance function and the values obtained after training (ﬁg: (h)) on the ‘dog’ data set. [sent-296, score-0.475]

91 (a) (b) (c) Figure 2: The last frame of the training video (a) is smoothly followed by the ﬁrst frame (b) of the generated video. [sent-297, score-0.342]

92 Our method’s effectiveness has been demonstrated in two tasks; ﬁrstly, in modeling human motion capture data and, secondly, in reconstructing and generating raw, very high dimensional video sequences. [sent-300, score-0.558]

93 Lawrence, “Probabilistic non-linear principal component analysis with Gaussian process latent variable models,” Journal of Machine Learning Research, vol. [sent-308, score-0.302]

94 Lawrence, “Gaussian process latent variable models for visualisation of high dimensional data,” in Advances in Neural Information Processing Systems, pp. [sent-313, score-0.394]

95 Hertzmann, “Gaussian process dynamical models for human motion,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. [sent-327, score-0.258]

96 Lawrence, “Hierarchical Gaussian process latent variable models,” in Proceedings of the International Conference in Machine Learning, pp. [sent-333, score-0.302]

97 Lawrence, “Bayesian Gaussian process latent variable model,” in Proceedings of the Thirteenth International Conference on Artiﬁcial Intelligence and Statistics, pp. [sent-350, score-0.302]

98 Archambeau, “The variational Gaussian approximation revisited,” Neural Computation, vol. [sent-376, score-0.282]

99 Roweis, “Modeling human motion using binary latent variables,” in Advances in Neural Information Processing Systems, vol. [sent-391, score-0.406]

100 Lawrence, “Learning for larger datasets with the Gaussian process latent variable model,” in Proceedings of the Eleventh International Conference on Artiﬁcial Intelligence and Statistics, pp. [sent-395, score-0.302]

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