• We propose a general-purpose Bayesian network prior of human pose. • Fully non-parametric: Estimation of both optimal information-theoretic topology and local conditional distributions from data. • Compositional: E↵ective handling of the combinatorial explosion of articulated objects, thereby improving generalization. • Superior performance: Better data representation than traditional global models and parametric networks on the large Human 3.6M dataset. • Real-time: Fast and accurate computation of approximate likelihoods on datasets with up to 100k training poses. • We compute expected log-likelihoods for our Chow-Liu/CKDE model and several baselines on the Human 3.6M dataset. Figure 1: Samples drawn from a single Chow-Liu/CKDE model. training data test data cluster centers exact evaluations approx. evaluations 4 25 50 10 −15 10 −10 10 −5 10 −2 10 0 Exact LL Mean absolute error (nats) Core clusters 4 25 50 0 1.5 5 10 Mean runtime (ms) −82% Approximate LL Approximation with 4 core clusters • Our formulation allows to freely combine substructures, but only if they do not share a lot of information. = ) Compositionality exactly where needed and only where appropriate. Perceiving Systems – ps.is.tue.mpg.de 2 Non-parametric Networks “Standing” “Sitting” “Kneeing / Lying” Training samples Samples from model “wave both” ... ... “neutral” “wave left” ... “wave right” ... ... “wave right”: 50% “wave left”: 50% ... Inferred network Table 1: Expected log-likelihoods. Method Graph structure Training Testing Gaussian Global -266.84 -271.15 KDE Global -239.61 -263.77 GPLVM * Global -327.85 -341.89 Independent -352.80 -345.94 Gaussian linear Kinematic chain (order 1) -311.54 -310.98 network Kinematic chain (order 2) -305.54 -307.88 Chow-Liu tree -283.82 -284.03 CKDE network Independent -322.64 -322.25 Kinematic chain (order 1) -260.04 -270.52 Kinematic chain (order 2) -247.35 -263.83 Chow-Liu tree (ours) -242.24 - 254.98 * 25% subsampling; FITC Kinematic chain Mutual information Chow-Liu tree • Learning the conditional distributions: We use a conditional kernel density estimate (CKDE) to learn the local models of the inferred tree, p ( X j X pa(j ) ) = p ( X j ,X pa(j ) ) R X j p ( X j ,X pa(j ) ) dX j = P i N ⇣ (X j ,X pa(j ) ) (X (i) j ,X (i) pa(j ) ),BB > ⌘ P i N ⇣ X pa(j ) X (i) pa(j ) , (BB > )| X pa(j ) ⌘ , where p ( X j ,X pa(j ) ) is an unconditional KDE with isotropic Gaussian kernel and bandwidth B proportional to the square root of the covariance. • Important operations are efficient: – Computation of a log-likelihood requires O (|V |) KDE evaluations. – Ancestral sampling requires O (|V |) samples from the local models. [Gaussian mixture models with non-uniform weight distribution] A Non-parametric Bayesian Network Prior of Human Pose Andreas M. Lehrmann¹, Peter V. Gehler¹, Sebastian Nowozin² MPI for Intelligent Systems¹, Microsoft Research Cambridge² 3D pose dataset Learn topology / local models Non-parametric Bayesian Network Prior of Human Pose Score Test 3D skeletal data e.g., pose estimation 1 Overview 3 Compositionality & Generalization 4 Live Scoring References [1] C. Chow and C. Liu. Approximating discrete probability distributions with dependence trees. IEEE Transactions on Information Theory, 1968. [2] A. Gray and A. Moore. Nonparametric density estimation: Toward computational tractability. SIAM International Conference on Data Mining, 2003. [3] C. Ionescu, D. Papava, V. Olaru, and C. Sminchisescu. Human3.6M: Large Scale Datasets and Predictive Methods for 3D Human Sensing in Natural Environments. Technical report, University of Bonn, 2012. Visit Us Learn a sparse and non-parametric Bayesian network B =(p, G (V,E )). • Learning the graph structure: Minimize KL-divergence between the high-dimensional pose distribution q (X) and the tree-structured network p(X)= Q |V | j =1 p ( X j X pa(j ) ) , G := argmin pa KL (q (X) k p(X)) = MST(G 0 ), where G 0 is the complete graph with edge weights e jk = c MI(X j ,X k ). • Applications in real-time environments require additional speed. • Training: Cluster the training points into clusters {C (i) } i using k -means and build a kd-tree for their centres. • Testing: Given a test pose x, use the kd-tree to compute a k -NN partitioning {C (i) } i = C e (x) U C a (x) and approximate the likelihood as p (x) ⇡ (S e + S a ) . (N · det(B )), with S e = X C 2C e X j 2C ⇣ B -1 ⇣ x - x (j ) ⌘⌘ , [exact] S a = X C 2C a |C | ( B -1 ( x - C )) , [approx.] where C and |C | denote the centre and size of cluster C , respectively.
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• We propose a general-purpose Bayesian network prior of human pose.
• Fully non-parametric: Estimation of both optimal information-theoretictopology and local conditional distributions from data.
• Compositional: E↵ective handling of the combinatorial explosion ofarticulated objects, thereby improving generalization.
• Superior performance: Better data representation than traditionalglobal models and parametric networks on the large Human 3.6M dataset.
• Real-time: Fast and accurate computation of approximate likelihoods ondatasets with up to 100k training poses.
• We compute expected log-likelihoods for our Chow-Liu/CKDE model andseveral baselines on the Human 3.6M dataset.
Figure 1: Samples drawn from a single Chow-Liu/CKDE model.
training datatest datacluster centersexact evaluationsapprox. evaluations
4 25 5010
−15
10−10
10−5
10−2
100
Exact LL
Me
an
ab
solu
te e
rro
r (n
ats
)
Core clusters4 25 50
0
1.5
5
10
Me
an
ru
ntim
e (
ms)
−82%
Approximate LL
Approximation with 4 core clusters
• Our formulation allows to freely combine substructures, but only if theydo not share a lot of information.
=) Compositionality exactly where needed and only where appropriate.
Perceiving Systems – ps.is.tue.mpg.de
2 Non-parametric Networks
“Standing”! “Sitting”!
“Kneeing / Lying”!
Training samples! Samples from model!
“wave both”!
...!
...!“neutral” !
“wave left” !
...!“wave right”!
...!...!“wave right”: 50% !
“wave left”: 50% !
...!
Inferred network!
Table 1: Expected log-likelihoods.
Method Graph structure Training Testing
Gaussian Global �266.84 �271.15KDE Global �239.61 �263.77GPLVM
*Global �327.85 �341.89
Independent �352.80 �345.94Gaussian linear Kinematic chain (order 1) �311.54 �310.98network Kinematic chain (order 2) �305.54 �307.88
Chow-Liu tree �283.82 �284.03
CKDE network
Independent �322.64 �322.25Kinematic chain (order 1) �260.04 �270.52Kinematic chain (order 2) �247.35 �263.83Chow-Liu tree (ours) �242.24 � 254.98
*25% subsampling; FITC
Kinematic chain Mutual information Chow-Liu tree
• Learning the conditional distributions:
We use a conditional kernel density estimate (CKDE) to learn the local
models of the inferred tree,
p�Xj
��Xpa(j)
�=
p�Xj , Xpa(j)
�RXj
p�Xj , Xpa(j)
�dXj
=
Pi N
⇣(Xj , Xpa(j))
��� (X(i)j , X(i)
pa(j)), BB>⌘
Pi N
⇣Xpa(j)
��� X(i)pa(j), (BB>
)|Xpa(j)
⌘ ,
where p�Xj , Xpa(j)
�is an unconditional KDE with isotropic Gaussian
kernel and bandwidth B proportional to the square root of the covariance.
• Important operations are e�cient:
– Computation of a log-likelihood requires O(|V |) KDE evaluations.
– Ancestral sampling requires O(|V |) samples from the local models.
[Gaussian mixture models with non-uniform weight distribution]
A Non-parametric Bayesian Network Prior of Human Pose Andreas M. Lehrmann¹, Peter V. Gehler¹, Sebastian Nowozin²
MPI for Intelligent Systems¹, Microsoft Research Cambridge²
3D pose dataset
Learn topology / local models
Non-parametric Bayesian Network
Prior of Human Pose
Score Test
3D skeletal data
e.g., pose estimation
1 Overview
3 Compositionality & Generalization
4 Live Scoring
References [1] C. Chow and C. Liu. Approximating discrete probability distributions with dependence trees.
IEEE Transactions on Information Theory, 1968.[2] A. Gray and A. Moore. Nonparametric density estimation: Toward computational tractability.
SIAM International Conference on Data Mining, 2003.[3] C. Ionescu, D. Papava, V. Olaru, and C. Sminchisescu. Human3.6M: Large Scale Datasets and
Predictive Methods for 3D Human Sensing in Natural Environments.Technical report, University of Bonn, 2012.
Visit Us
Learn a sparse and non-parametric Bayesian network B = (p,G(V,E)).
• Learning the graph structure:Minimize KL-divergence between the high-dimensional pose distribution
q(X) and the tree-structured network p(X) =
Q|V |j=1 p
�Xj
��Xpa(j)
�,
G := argmin
paKL (q(X) k p(X)) = MST(G0
),
where G0is the complete graph with edge weights ejk =
cMI(Xj , Xk).
• Applications in real-time environments require additional speed.
• Training: Cluster the training points into clusters {C(i)}i usingk-means and build a kd-tree for their centres.
• Testing: Given a test pose x, use the kd-tree to compute a k-NN
partitioning {C(i)}i = Ce(x)U
Ca(x) and approximate the likelihood as
p (x) ⇡ (Se + Sa)
.(N · det(B)),
with
Se =
X
C2Ce
X
j2C
⇣B�1
⇣x� x
(j)⌘⌘
, [exact]
Sa =
X
C2Ca
|C|�B�1
�x� C
��, [approx.]
where C and |C| denote the centre and size of cluster C, respectively.
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