Gaussian Process Dynamical Models Jack M Wang, David J Fleet, Aaron Hertzmann, NIPS 2005

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1 Gaussian Process Dynamical Models Jack M Wang, David J Fleet, Aaron Hertzmann, NIPS 2005 Presented by Piotr Mirowski CBLL meeting, May 6, 2009 Courant Institute of Mathematical Sciences, New York University 1

2 Gaussian Processes Introduction to Gaussian Processes Interlude Gaussian Process Latent Variable Models Gaussian Process Dynamical Models Application to motion capture data 2

3 Gaussian Distribution [Rasmussen, 2006] 3

4 Gaussian Process Input data points x1, x2,... xn ℜQ Stochastic process {y(x) ℜ x ℜQ} Gaussian Process is a stochastic process such that: y(x1), y(x2),... y(xn) are jointly Gaussian P[y] = N(y 0, K) μ(x) = E[y(x)] Cov[y(xn),y(xm)] = E[(y(xn)-μ(x))(y(xm)-μ(x))T] Cov[y(xn),y(xm)] = k(xn, xm) 4

5 Gaussian Processes = kernel methods Input data points x1, x2,... xn ℜQ For each xn, associated yn = y(xn) ℜ Linear combination of basis functions φ(.) yn = wtφ(xn) On all datapoints, using design matrix Φ y=φw Gaussian prior on weights: P[w] = N(w 0, 1/α I) E[y] = Φ E[w] = 0 Cov[y,y] = Φ E[wwT] ΦT = 1/α ΦΦT = K Gram matrix K: k(xn, xm) = 1/α φ(xn)t φ(xm) [Bishop, 2006] 5

6 Gaussian kernel parameters [Bishop, 2006] 6

7 Gaussian Process yn=y(xn) tn=yn+ηn Sample function from Gaussian process prior (specific kernel) Joint Gaussian distribution on y(x1), y(x2),... y(xn) Gaussian kernel K k(xn, xm) Gaussian additive observation noise tn=yn+ηn P[T Y] = N(T Y, σ2noise I) P[T]=N(T 0, K + σ2noise I) 2 C = K + σ I noise [Bishop, 2006] 7

8 Gaussian Process Regression Conditional Gaussians: Prediction of y* at new point x*: C [Rasmussen, 2006] 8

9 Gaussian Process Regression Sampled (xn, tn) with additive noise Target function Mean of predictive distribution ±2 std of predictive distribution [Bishop, 2006; Williams and Rasmussen, 1996] 9

10 Relevance of individual inputs w1 = w2 Scaling matrix x1 Different variances along different input dimensions x2 w1 >> w2 x1 x2 [Rasmussen, 2006; Wang et al, 2005] 10

11 Learning a Gaussian Process [Rasmussen, 2006] 11

12 Learning a Gaussian Process Need to invert N x N matrix K: O(N3) at each step of the algorithm Problems with large datasets Usual limitations (and advantages) of kernel methods 12

13 Gaussian Processes Introduction to Gaussian Processes Interlude Gaussian Process Latent Variable Models Gaussian Process Dynamical Models Application to motion capture data 13

14 Gaussian Processes and Brownian Motion Ornstein-Uhlenbeck process Non-gaussian kernel: exp(-θ x-x' ) Stationary, mean reverting Conversely, Wiener Process (Brownian Motion) is not stationary [Bishop, 2006] 14

15 Gaussian Processes Introduction to Gaussian Processes Interlude Gaussian Process Latent Variable Models Gaussian Process Dynamical Models Application to motion capture data 15

16 Gaussian Process Latent Variable Models D Probabilistic PCA: Observations (output): Q W Y N D Q Latent variables (input): iid observation noise: [Lawrence, 2004; 2005] X N 16

17 GPLVM (PCA interpretation) Integrating parameters W Optimizing latent variables X D Q Q N [Lawrence, 2004; 2005] Y D WT X 17 N

18 GPLVM learning Replace linear kernel by Gaussian kernel Learn both kernel K and hyperparameters α, β Conjugate gradient (each step requires O(N3) operations: impractical) Sparsification + Informative Vector Machine [Lawrence 2004] [Lawrence, 2004; 2005] 18

19 Gaussian Processes Introduction to Gaussian Processes Interlude Gaussian Process Latent Variable Models Gaussian Process Dynamical Models Application to motion capture data 19

20 Gaussian Process Dynamical Models Observation model: GPLVM for the mapping X -> Y Dynamical model: GPLVM for the mapping X -> Xout Q Y D Observation 1 model Q 1 N 2 Dynamical model 1 N [Wang et al, 2005] N X Q X Xout N-1 20

21 Two types of kernels Observation model: RBF kernel Dynamical model: RBF+linear kernel [Wang et al, 2005] 21

22 Reconstruction of walk sequence using GPDM with 3 latent variables Uncertainty of reconstruction Latent positions xt [Wang et al, 2005] 22

23 Naïve iterated prediction in GPDM Gaussian distribution of 1-step prediction: Mean: targets Variance: Kernel (training data) Kernel evaluated on test data Most likely (closed-loop) prediction: Long sequences generated by naïve prediction can diverge Need to propagate uncertainty [Girard et al, 2003] [Wang et al, 2005] 23

24 Naïve iterated prediction in GPDM 1st order Markov (RBF + linear) kernel 1st order Markov Linear kernel Walk sequence: 2nd order Markov RBF kernel Golf sequence: [Wang et al, 2005] 24

25 Optimization = inference of latent variables Latent variables from training data (fixed) Inferred Hyperparameters latent (fixed) variables Naïve prediction Naïve prediction after optimization [Wang et al, 2005] 25

26 Comparison of {linear, RBF} kernels All models are GPDM 3-dimensional latent variables xt Observation model g has RBF kernel Dynamical model f has {linear, RBF, RBF+linear} kernel Walk sequence 8-step iterated naïve prediction Training dataset size: 130 frames Testing dataset size: 23 frames [Wang et al, 2005] 26

27 Inference of missing data 1st order Markov Linear kernel 1st order Markov (RBF + linear) kernel 157-frame walk sequence 50 missing frames Latent coordinates xt initialized using cubic splines in 3D latent space [Wang et al, 2005] Need little hack: start learning on subsampled sequence 27

28 References Christopher Williams, Carl E Rasmussen, Gaussian Processes for Regression, NIPS, Agathe Girard, Carl E Rasmussen, Joaquin Q Candela, Roderick Murray-Smith, Gaussian Process Priors with Uncertain Inputs: Application to Multiple Step Ahead Time Series Forecasting, NIPS, Neil D Lawrence, Gaussian Process Latent Variable Models for Visualization of High Dimensional Data, NIPS, Jack M Wang, David J Fleet, Aaron Hertzmann, Gaussian Process Dynamical Models, NIPS, Christopher Bishop, Pattern Recognition and Machine Learning, Springer, 2006 (Chapter 6.4) Carl E Rasmussen, Advances in Gaussian Processes, Tutorial at NIPS,