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Dynamical Binary Latent Variable Models for 3D Human Pose Tracking Supplementary Material

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Dynamical Binary Latent Variable Models for 3D Human Pose TrackingSupplementary MaterialGraham W. TaylorNew York UniversityNew York, [email protected] SigalDisney ResearchPittsburgh, [email protected] J. Fleet and Geoffrey E. HintonUniversity of TorontoToronto, Canada{fleet,hinton}@cs.toronto.edu1. Details of imCRBM weight updatesIn this section we describe the positive and negativephase statistics that were referenced in the Appendix of thesubmitted paper, and derive the weight updates for each setof parameters in the imCRBM.Note that Wk, Akand Bkcan be thought of as either (1)the kthslice of weight tensors W ,A and B, respectively; or(2) the weights of the kthcomponent CRBM in the mixture.Similarly, ck, and dkcan be thought of as either (1) the kthcolumn of weight matrices C and D respectively; or (2) thebiases of the kthcomponent CRBM in the mixture. Bothviews are equivalent.Contrastive divergence learning consists of (1) a posi-tive phase in which we hold the visible variables fixed to atraining vector, x+t, and sample the latent variables qt, zt;and (2) a negative phase, in which we alternate between re-constructing (by sampling) the visible variables given latentvariables and sampling the latent variables given the recon-structed visible variables. The positive phase yields k+(i.e.q+kt= 1) and z+twhich are our sampled latent variablesconditional on the training data x+t. The negative phaseyields x−twhich are the values of the visible variables afterM steps of alternating sampling and k−and z−twhich areour sampled latent variables conditional on x+t. Note thatall distributions which we draw from are conditional on thehistory xhtwhich remains constant throughout the entirelearning algorithm.The imCRBM has two types of learnable parameters:weights and offsets (biases). The weights capture pairwiseinteractions between variables and therefore the statisticsthat comprise the updates are outer products. The statisticsfor the offset parameters are simply average activities. Re-flecting the pairwise connectivity of the model, the positivephase statistics are:W+k= x+tz+tT(1)A+k= x+txTht(2)B+k= z+txTht(3)c+k= x+t(4)d+k= z+t(5)The negative phase statistics have the same form:W−k= x−tz−tT(6)A−k= x−txTht(7)B−k= z−txTht(8)c−k= x−t(9)d−k= z−t(10)Repeating the positive and negative phase for a mini-batch of S training pairs of the form {xht, x+t} resultsin ten sets of statistics for each component k. The firstfive sets correspond to the Uktimes k was chosen inthe positive phase: {W+k1, . . . , W+kUk},{A+k1, . . . , A+kUk},{B+k1, . . . , B+kUk}, {c+k1, . . . , c+kUk}, {d+k1, . . . , d+kUk}. Thenext five sets correspond to the Vktimes k was chosenin the negative phase: {W−k1, . . . , W−kVk},{A−k1, . . . , A−kVk},{B−k1, . . . , B−kVk}, {c−k1, . . . , c−kVk}, {d−k1, . . . , d−kVk}.The weight updates for the kthcomponent CRBM areaverages over the mini-batch:∆Wk=λS UkXu=1W+ku−VkXv =1W−kv!, (11)∆Ak=λS UkXu=1A+ku−VkXv =1A−kv!, (12)∆Bk=λS UkXu=1B+ku−VkXv =1B−kv!, (13)∆ck=λS UkXu=1c+ku−VkXv =1c−kv!, (14)∆dk=λS UkXu=1d+ku−VkXv =1d−kv!. (15)10 50 100 1500102030405060708090Test: S5 Walking Validation; Order 1 priorError (mm)Frame Train:S5Train:S1Train:S1S2S3Figure 1. Multi-view tracking. Tracking of subject S5 using a1st order prior model learned from S5, S1, and S1+S2+S3 train-ing data. Results are averaged over 10 runs (per-frame standarddeviation is shaded).where λ is a learning rate which could be made parameter-specific.Note that the Ukand Vkneed not be the same for a givencomponent, k, since k+and k−are sampled. For the caseof K = 1, U1= V1= S and these weight updates reduceto the updates of a standard CRBM [2].2. Multi-view trackingFigure 1 shows the result of tracking subject S5 usingthree different first-order CRBM priors, each trained on adifferent dataset. The plot is the illustration of the con-densed results reported in Table 1 of the submitted paper.It was not included in the main document due to space con-straints. We see that the prior is able to generalize to sub-jects who are not included in the training set.3. Monocular trackingThe analysis of performance of imCRBM-2L prior onthe monocular ”combo” sequence for subject S3 is illus-trated in Table 1. Similarly to other experiments in the paperwe use 1000 particles (though similar results were achievedeven with 200 particles). We run imCRBM-2L on each ofthe 3 views independently and report the performance av-eraged over 5 runs in each case. A single, representative,run with Camera 2 data was illustrated in Figure 5 of themain submission. Since with monocular observations wedo not expect to resolve the depth reliably (due to depthambiguity), as is standard in the literature, we report therelative average marker error. The average relative markererror corresponds to the marker distance with respect to thepelvis (see [1] for details). The performance is very encour-aging, particularly considering that only one camera view isused and the motion contains transitions. Consequently, thebaseline algorithm is not able to track the sequence resultingin a quick failure.imCRBM-2LCamera 1 118.87 ± 33.12Camera 2 84.26 ± 6.85Camera 3 90.44 ± 7.64Table 1. Monocular tracking with transitions: quantitativeperformance. Tracking of subject S3 “combo” sequence using afirst-order imCRBM-2L prior model with 1000 particles. Resultsare averaged over 5 runs.References[1] L. Sigal, A. Balan, and M. Black. HumanEva: Synchronizedvideo and motion capture dataset and baseline algorithm forevaluation of articulated human motion. IJCV, 2010.[2] G. Taylor, G. Hinton, and S. Roweis. Modeling human mo-tion using binary latent variables. NIPS, 19,


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