Rank Priors for Continuous Non-Linear Dimensionality Reduction
Andreas Geiger
Department of Measurement and Control
Karlsruhe Institute of Technology

Raquel Urtasun and Trevor Darrell
UC Berkeley EECS & ICSIAbstract

been shown to be very effective when dealing with large
datasets that are homogeniously sampled. However,as demonstrated here, they suffer in the presence ofnoisy and sparse data. Unfortunately, a large set of realworld computer vision datasets are sparse. Human motion datasets comprise small numbers of examples fromdifferent subjects performing different activities [2, 11].
While these databases are typically densely sampled intime, they are sparse in the motion style and activity type. Object recognition databases [1], also sufferfrom sparsity: only a few examples may be labeled for
categories with large variation in appearence.

Discovering the underlying low-dimensional latent
structure in high-dimensional perceptual observations(e.g., images, video) can, in many cases, greately improve performance in recognition and tracking. However, non-linear dimensionality reduction methods areoften susceptible to local minima and perform poorly
when initialized far from the global optimum, evenwhen the intrinsic dimensionality is known a priori.
In this work we introduce a prior over the dimensionality of the latent space that penalizes high dimensional spaces, and simultaneously optimize both the latent space and its intrinsic dimensionality in a continuous fashion. Ad-hoc initialization schemes are unnecessary with our approach; we initialize the latent space
to the observation space and automatically infer thelatent dimensionality. We report results applying ourprior to various probabilistic non-linear dimensionality
reduction tasks, and show that our method can outperform graph-based dimensionality reduction techniques
as well as previously suggested initialization strategies.
We demonstrate the effectiveness of our approach whentracking and classifying human motion.

Non-linear probabilistic models, such as the GPLVM[13], can recover complex manifolds. They have received considerable attention in recent years, having
been applied to human motion tracking [25, 23, 14, 10],detection [3, 18, 7], 3D shape estimation [20] and character animation [6, 24]. However, they have only been
applied to small databases typically composed of very
few examples of a single activity [25]. Moreover, thelatent dimensionality was either chosen by the user
[25, 23, 14] or optimized by cross-validation [20], whichis computationally expensive.
While their representation power is desirable, suchmethods suffer from local minima, since they rely on
optimization of complex non-linear functions that are
generally non-convex. Even with the right dimensionality, they can result in poor representations if initialized far from the optimum [24]. Factors which contribute to this include the distortion introduced by the
initialization and the non-convexity of the optimization. This is aggravated when optimizing extremely
low-dimensional latent spaces, which is typically thecase in applications such as tracking [25].

1. Introduction
Many computer vision problems involve high dimensional datasets that are computationally challenging toanalyze. In such cases it is desirable to reduce the dimensionality of the data while preserving the original
information in the data distribution, allowing for more
efficient learning and inference. Linear dimensionality reduction techniques (e.g., PCA) have been very
popular in the past, due to their simplicity and efficiency. However in practice, as shown below, they can
result in poor approximations when dealing with complex datasets.
Graph-based methods, e.g., LLE [19] and Isomap[21] exploit local neighborhood distances to approximate the geodesic distance in the manifold. They have

In this paper we present a new learning paradigm
that reduces the problem of local minima by performing continuous dimensionality reduction. In contrast to
previous GPLVM-based approaches, no distortion is introduced by an initialization step in our approach, since
the latent coordinates are initialized to be the original
observations. By introducing a prior over the dimen1

sionality of the latent space that encourages sparsity ofthe singular values, our method is able to simultaneously estimate the latent space and its dimensionality.
Regularization via sparsity has recently become afocus of attention in the machine learning and visioncommunities, and has been applied to feature selection
for linear dimensionality reduction, e.g. Sparse PCA[28], or transfer learning [17]. Here we are interested
in learning non-linear low-dimensional latent spaces; toour knowledge, ours is the first non-linear dimensionality reduction technique that penalizes the latent space
rank and simultaneously optimizes the structure of anon-linear latent space as well as its intrinsic dimensionality.
In the remainder of the paper we first briefly review latent variable models. We then show how toincorporate a rank prior in the optimization of a nonlinear probabilistic model, and demonstrate our approach when tracking and classifying complex articulated human body motions.

2. Background: Latent Variable Models
Latent Variable Models (LVMs), e.g., Probabilistic
PCA [22], assume that the data has been generated
by some latent (unobserved) random variables that lie
on or close to a low-dimensional manifold. Let Y =[y1 , · · · , yN ]T be the set of observations yi ∈ ℜD , and
let X = [x1 , · · · , xN ]T be the set of latent variables
xi ∈ ℜQ , with Q ≪ D. Probabilistic LVMs relate
the latent variables to a set of observed variables via aprobabilistic mapping, y (d) = f (x) + η, with y (d) thed-th coordinate of y, and η ∼ N (0, θ3 ) iid Gaussian
noise.
The Gaussian Process Latent Variable Model(GPLVM) [13] places a Gaussian process prior over the
space of mapping functions f . Marginalizing over f
and assuming conditional independence of the output
dimensions given the latent variables results inp(Y|X) =DY

d=1

N (Y(d) |0, K)

where Y(d) is the d-th column in Y, and K is the covariance matrix, typically defined in terms of a kernel function. Here RBF + noise kernel,
 we use ankxi −xj k22+ θ3 δij , since it allows
k(xi , xj ) = θ1 exp − 2θ22
for a variety of smooth non-linear mappings using only
a limited number of hyperparameters, Θ = {θ1 , θ2 , θ3 }.
Learning is performed by maximizing the posteriorp(X|Y) ∝ p(Y|X)p(X) with respect to the latent variables X and the kernel hyperparameters Θ. p(X) encodes prior knowledge about the latent space X.

PCA and graph-based techniques are commonlyused to initialize the latent space in GPLVM-based
dimensionality reduction; both offer closed-form solutions. However, as shown below, PCA [16] cannot capture non-linear dependencies, LLE [19] gives a goodinitialization only if the data points are uniformly sampled along the manifold, and Isomap [21] has difficultywith non-convex datasets [9]. Generally, when initialized far from the global minimum, the GPLVM optimization can get stuck in local minima [13, 24].
To avoid this problem different priors over the latent space have been developed. In [27] a prior was
introduced in the form of a Gaussian process over the
dynamics in the latent space. This results in smoothermodels but performs poorly when learning stylistic
variations of a motion or multiple motions. In [24] aprior over the latent space, inspired by the LLE costfunction, that encourages smoothness and allows theintroduction of prior knowledge, e.g., topological information about the manifold, was proposed. However,such prior knowledge is not commonly available, reducing the applicability of this technique. In contrast,in this paper we introduce a generic prior that requires
no specific prior knowledge, directly penalizing the dimensionality of the latent space to learn effective lowdimensional representations.

3. Continuous Dimensionality Reduction
via Rank Priors
In this section we introduce a continuous dimensionality reduction technique that initializes the latent
space to the observation space to avoid initial distorsions, and learns the latent space and its dimensionality
by introducing a prior that penalizes latent spaces withhigh dimensionality. The dimensionality of the latent
space can be described by the rank of the Gram matrixof the latent coordinates, which can be computed as thenumber of non-zero singular values of X. However, it
is difficult to enforce directly a prior on the rank since
it is a discrete quantity.
Instead, we propose a relaxation that results in a
penalty function which gradients are continuous andcan be easily computed. In particular, we introduce aprior of the form!
DX
1p(X) = exp −αφ(si )
Zi=1(1)

where si are the normalized singular values of themean-subtracted matrix of latent coordinates, with D
the dimensionality of the latent space, and Z a normalization constant.

Figure 1. Comparison of our Rank Prior with a GPLVM initilized to PCA: The goal is to recover the 1D manifoldthat is embedded in a 2D space. The GPLVM gets stuck in local minima very early (upper row) since the PCA initialization
does not capture non-linear dependencies, whereas our method decreases dimensionality gradually and recovers the correct
structure (lower row).

Different penalty functions φ can be considered.
Common choices for sparsity are the power family andthe (generalized) elastic net [4]. In the power family
φ(si , p) = |si |p(2)

sparsity is achieved for p ≤ 1. The L2 norm (i.e., p = 2)
is a well studied penalty, but does not encourage sparsity. It is equivalent to a Gaussian prior over the singular values. The most commonly used penalty thatencourage sparsity is the L1 norm (i.e., p = 1), that results in a Laplace prior over the singular values in Eq.
(1). This case is in general attractive since the penalty
function is linear, and when the objective function is
also linear the optimization can be effectively solved
with a Linear Program, even with large number of variables [17].
However here we are interested in learning nonlinear latent spaces; our objective function is non-linear
even when φ is linear. In particular, we minimize thenegative log posterior
L=

D
X
D
D
φ(si ) , (3)ln |K| + tr(K−1 YYT ) + α22i=1

where α controls the influence of the penalty in the optimization. Of particular interest to us are functions φthat drive small singular values faster towards 0 thanlarger ones. Examples of such functions are the powerfamily with p < 1, the logarithmic and sigmoid functions. In practice we use a logarithmic prior for allof our experiments since converges faster to a sparse
solutionφ(s) = ln(1 + βs2 )with β a constant.

The derivatives of our rank prior with respect to thelatent coordinates X are
!
D
D
X
X
∂φ(sm )
1∂
φ(sm ) = √Uim Vjm ,∂Xij m=1
N − 1 m=1 ∂sm(4)where we have used the fact that the derivatives of
the normalized singular values with respect to the la∂sm=tent coordinates can be easily computed as ∂x
ij
1
√
U
V
[15],
with
U
the
matrixof
left-singular
N −1 im jm
vectors, and V the matrix of right-singular vectors ofm)
X. The value of ∂φ(sdepends on the sparsity func∂smtion and the derivatives of the first two terms in Eq.
(3) with respect to the latent coordinates X and thekernel hyperparameters Θ are given in [13].
Minimizing Eq. (3) results in a reduction of theenergy of the spectrum (since the singular values areminimized). To prevent this from happening, one caninstead solve a constrained optimization problem suchthat the energy of the singular values remains constant,
min L

s.t. ∀i si ≥ 0, ∆E = 0,(5)where ∆E = E(Y) − E(X) is the difference of energiesof the observation space and the
Platent space, and theenergy is computed as E(X) = i s2i . We use SNOPT[5], a non-linear constraint optimizer to minimize Eq.
(5).
Finally, we choose the dimensionality of the latent
space to besi
Q = argmaxi(6)
si+1 + ǫ
where ǫ ≪ 1, and s1 ≥ s2 ≥ · · · ≥ sD .1 Once Q is com1 This strategy is commonly used in statistics to compute theamount of signal sources in noisy data.

2.5
2observation space
linear sparsity function
logarithmic sparsity function
sigmoid sparsity function

2
1.5

σ

m

1.5
1

0.5

0.5
0
0

1

20

40
60
iteration number

80

100

Figure 2. Spectrum of a 30D motion database: Evolution of the first ten singular values as a function of the
optimization iteration number for a logarithmic sparsity
penalty function.

0

2

4

6

8

10

m

Figure 3. Comparison of different penalty functions
on a 30D motion database. Note that the logarithmic
penalty imposes sparsity more strongly.

4. Examples and Experimental results
puted, we apply PCA in the optimized low-dimensional
space. Note that the mapping is still non-linear since
PCA is performed in the latent space, not in the observation space, and simply rotates the latent coodinates
to produce the most compact Q-dimensional representation. The last step consists of refining the kernel
hyperparamters by optimizing p(Y|X, Θ) with respectto Θ keeping X fix.
Fig. 1 compares the GPLVM (initialized via PCA)
with the result of optimizing Eq. (5) on a toy example,
where a 1D manifold is embedded in a 2D space. PCA
provides a non-optimal initialization, and the GPLVM
gets trapped in local minima, whereas our method recovers the correct structure. Note that our final PCA
projection rotates the latent space and results in a
1D manifold. In this example, using spectral methods
could lead to a successful initialization for the GPLVM.
However, for more complex datasets this is not necessarily the case in general, as shown in Figs. 4 and 8.
Fig. 2 depicts the evolution of the first ten singular values when optimizing Eq. (5) with a logarithmic penalty function for a 30D motion database. Note
how our method drops dimensions as the optimization
evolves (i.e., the smallest singular values drop to zero
within the first few iterations). The behavior of different penalty functions is shown in Fig. 3.
When the observations are high-dimensional, one
can reduce the complexity by whitening the observations using PCA, removing negligible singular values.
The complexity of our approach is of the same order as
the complexity of the GPLVM, i.e., O(N 3 ), since thecomplexity of computing the singular values of X ismin(O(P 3 ), O(N 3 )), with P the dimensionality of thewhitened space, and N the number of examples. Whenusing sparsification the complexity can be reduced.
We initialize the kernel width using neighborhood
distances, and we set the noise variance to 0.0001. A
value of α ∈ (1, 10) is used in practice.

In this section we demonstrate the effectiveness of
our approach to discover the latent structure and its
dimensionality in a variety of artificial datasets. We
then illustrate the application of our method to the
problem of tracking and classifying 3D articulated motion.

4.1. Dimensionality and latent structure estimation
Graph-based methods rely on local neighborhoods
to unravel the data and discover the underlying latent
structure, but often suffer in the presence of sparseand noisy data. We illustrate this problem on a sparseswiss role, which is a 2D manifold embedded in a 3D
space. We simulate sparsity by providing as training
data only the black points in Fig. 4 (a). The first row
in Fig. 4 (b) shows the result of applying PCA, Isomap,
Laplacian Eigenmaps, LLE, LTSA and MVU (see [26]for a review on these techniques). The second row depicts our technique and the result of optimizing the
GPLVM with these algorithms as initialization. The
last two rows of Fig. 4 (b) show the test data (i.e.,colored samples) reconstructed in the 2D latent space
and in the original 3D space. Note that our method,
unlike PCA, graph-based techniques and the GPLVM
with any of the initializations, is able to recover thecorrect structure.
We evaluate quantitatively the manifolds estimatedby the different algorithms computing both a global
and a local measure of accuracy. The reconstruction
error is a global measure of the ability to generalize,
and was obtained by first finding the latent coordinates
x∗ of the test data y∗ by maximizing p(x∗ |y∗ , X, Y),
and then
P computing the average mean prediction error N1t i kµ(x∗i ) − y∗i k2 , with Nt the number of test
data. The relationship error, Rerror , measures how
well local neighborhoods are preserved and is defined
2
PNt P
Γi,j − Γ̄i,j , where ηi is theas Rerror = i=1j∈ηiFigure 4. Finding a 2D manifold in 3D space on a sparsely sampled swiss roll. Only a sparse, noisy subset(depicted in black) of the full manifold is assumed to be known (a). (b) shows the initialization (with neighborhood sizek=6), GPLVM result and 2D/3D reconstruction of the full manifold (from top to bottom).
kx −x kset of neighbors of the i-th test data, Γi,j = kyii −yjj k22
is the ratio between the distances in the latent space
and in the observation space for two neighbors, andΓ̄i,j is the mean ratio in the local neighborhood. Fig.
5 depicts the two error measures when performing theexperiment in Fig. 4 averaged over 20 random partitions of the data. We use a local neighborhood of size
4 to compute the relationship error in all experiments,
and a logarithmic sparsity function with α = β = 10,
and Θ = {0.5, 1.5, 0.01} for our approach. We reportresults over a wide range of parameters for the different techniques. The hyperparameters were optimized
for the GPLVM baselines. Note that our method outperforms the baselines independent of the initialization
used for the GPLVM.
We further illustrate our method on 5 complex syn-thetic examples. In Fig. 6 (a) a spiral with a wide
separation between rings is reduced to a 1D manifold. When the distance between the different rings
decreases, the manifold dimensionality changes from
1D to 2D (see Fig. 6 (b)), since relationships between
points that have the same phase are considered. In
Fig. 6 (c) a 2D manifold from a cut-off sphere sampled
along longitudinal lines is recovered. The manifold in
Fig. 6 (d) is determined to be 3D, while its truncatedversion in Fig. 6 (e) is estimated to be 2D.

4.2. Tracking and classifying human motion
We conducted experiments tracking and classifying
complex motions in synthetic and real data. We created semi-synthetic databases using motion capture
data, where the task is, given 2D joint locations inour method
PCA init
ISOMAP init (k=6)
ISOMAP init (k=9)
ISOMAP init (k=12)
ISOMAP init (k=15)
Laplacian init (k=6)
Laplacian init (k=9)
Laplacian init (k=12)
Laplacian init (k=15)
LLE init (k=6)
LLE init (k=9)
LLE init (k=12)
LLE init (k=15)
LTSA init (k=6)
LTSA init (k=9)
LTSA init (k=12)
LTSA init (k=15)
MVU init (k=6)
MVU init (k=9)
MVU init (k=12)
MVU init (k=15)

Reconstruction Errormeanstddev
0.00410.01070.08450.18600.38130.17940.17200.84530.35830.42210.23500.23263.70272.65532.31361.31501.60380.28780.55612.03213.55142.18712.61752.51411.6235
1.13862.73004.54882.63770.72733.51493.88352.17343.20994.09503.66810.37830.5381
0.33830.36650.32280.34770.07060.2560Relationship Errormeanstddev
0.00080.00130.42320.19150.00380.01200.00110.00760.01300.00350.07310.03149.27352.55400.6426
8.1716
7.97841.50292.24000.05087.55911.48078.54851.48810.10580.1064
1.78208.92151.26361.64982.05752.40382.37003.07182.18194.51500.12380.00550.0491
0.04650.06720.01640.58562.8057

Figure 5. Quantitative performance on a synthetic sparse swiss roll example Reconstruction and Relationship
Error for the experiment in Fig. 4 averaged over 20 random partitions of the data. (Left) 8 best dimensionality reduction
techniques. (Right) More detailed results, including PCA and graph-based methods with different neighborhood sizes.

Figure 6. Dimensionality estimation. (Top) Five 2D manifolds embedded in 3D. (Bottom) Latent spaces and dimensionalities Q learned using our continous dimensionality reduction method.

monocular images, infer the 3D pose. The databases
were composed of running examples from 3 different
subjects, and walking examples from a single subject,
all 62D. We split the data into training and testing byrandomly sampling 3 running and 2 walking motions
for training and 10 running and 5 walking motions fortesting. Note that most of the splits contain training motions from only 2 or fewer subjects, requiring
generalization to unseen styles. We use the Condensation algorithm [12] with a second-order Markov modeland the reprojection error as image likelihood [25, 23].
We compare our approach to two baselines: tracking
in the original space, and tracking in a latent spacelearned using GPLVM with PCA initialization. Fig. 7(right) depicts tracking accuracy in cm averaged over
10 splits as a function of the number of particles. Notethat our approach (green) outperforms significantly thebaselines. Tracking in the original space (red) resultsin the worst performance. For all splits, our approach
discovered a 2D latent space. GPLVM with PCA ini-

tialization performes worse than our approach since itlearns non-smooth latent spaces, as illusted in Fig. 7(left).
In the second experiment we track and classify human motion from real images in a kitchen domain [8].
The dataset consists of multiple instances of rolling,milling and brooming motions performed in front ofa multi-camera array and synchronized mocap. Our
method found a 3D latent space from 30D joint angle
observations of 2 trials for each of the 3 different motions (N = 1120 training examples). Note that in previous work, without the inclusion of hand-tuned priorknowledge, GPLVM-based approaches were not able tolearn latent spaces with multiple motions [24]. We useda simple sparsification technique in order to speed up
learning: Instead of optimizing one single GP, we subdivided the training data into overlapping local andglobal sets and calculate the product of those smallsized GPs. This reduces the computational cost fromO(N 3 ) to O(2N 2 ). Fig. 8 shows the result of learning

Running: Tracking Error (3000 experiments)

1

20

0.8

full space
rank
gplvm 2d

1
18

0.4

Average joint error (cm)

0.6
0.5

0.2
0

0

−0.2
−0.4

−0.5

16
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−0.6
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−1

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1000
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1500

Walking: Tracking Error (1500 experiments)

2

20

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full space
rank
gplvm 2d

18

1

Average joint error (cm)

0.5

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2.5

(GPLVM initialized with PCA)

−2

−1

0

1

(our method)

2

500
1000
Number of particles

1500(Error comparison)

Figure 7. Tracking running (top) and walking (bottom) motions from 2D mocap data. (Left) Example of a
2D space learned with GPLVM initialized to PCA. (Middle) Latent space learned with our rank priors. (Right) Tracking
performance in cm for our approach vs. tracking in the original space and GPLVM initialized with PCA, when avaraged
over 10 splits. Note that our method outperforms the baselines for all number of particles.

FscoreRerror

GPLVM initialized LLE
0.73122.4105

Our approach
3.97780.0180

Table 1. Fisher score and reconstruction error for GPLVM
initialized with LLE and our approach.

such motions using our method. Note that the latent
space is smooth (i.e., consecutive frames in time areclose in latent space), and separates well the different
classes. To quantify the latter, we computed
the Fisher−1score defined as Fscore = tr Sw
Sb , where Sw is thewithin class matrix and Sb is the between class matrix. Smoothness implies lower relationship error, as
depicted by Table 1. Note that our method performs
significantly better than traditional GPLVM in termsof the relationship error and the fisher score.
Fig. 9 depicts tracking and classification performance for the milling and rolling motions. The 3D
latent space learned by our approach is depicted by
Fig. 8. For tracking we used a particle filter in the
low dimensional space with second-order Markov dynamics. Our image likelihood is based on low-level silhouette features. We labeled the data using 7 classes(rest, grasp pin, rolling, grasp broom, brooming, grasp
mill, milling) and used Nearest Neighbors for classification. Our method significantly outperforms the
GPLVM with LLE initialization in both tracking ac-

Figure 8. Learning different types of motion into one
single 3D latent space using our rank prior. The varianceof the mapping is depicted in transparency (red = rolling,green = brooming, blue = milling).

curacy and classification performance.

5. Conclusions and Future WorkIn this paper we have presented a new method
for non-linear dimensionality reduction that penalizes
high dimensional spaces and results in an optimization problem that continuously reduces dimensionlity
while solving for the latent coordinates. Our approach
can discover the structure of the latent space and its
intrinsic dimensionality, without an ad-hoc initialization step. Our approach has proven superior to PCA,
graph-based and non-linear dimensionality reduction

Tracking rolling sequences

8
6
4
2

GPLVM+LLE, 1 camera
GPLVM+LLE, 2 cameras
our method, 1 camera
our method, 2 cameras

12
10
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80
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100

0
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Classifying rolling sequences

100
Classification Accuracy (%)

10

Tracking Error (cm)Tracking Error (cm)

12

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14
GPLVM+LLE, 1 camera
GPLVM+LLE, 2 cameras
our method, 1 camera
our method, 2 cameras

100

80
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GPLVM+LLE, 2 cameras
our method, 1 camera
our method, 2 cameras

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Classification Accuracy (%)

Tracking milling sequences
14

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GPLVM+LLE, 1 camera
GPLVM+LLE, 2 cameras
our method, 1 camera
our method, 2 cameras

20
0
0

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100

Figure 9. Tracking and classification performance for milling and rolling motions using our method (red) and GPLVM
initialized to LLE (blue) as a function of the number of particles used in the particle filter.

techniques in a variety of tasks involving synthetic andreal-world databases, including tracking and classifying human motion. While our approach avoids most of
the local minima present in the GPLVM, the objectivefunction remains non-linear; we plan as future work toinvestigate simulated annealing and stochastic gradientdescend to further mitigate this problem.

Acknowledgements
The first author would like to thank Rainer Stiefelhagen for his support and advice. The last author
would like to acknowledge support from NSF #IIS0704479.

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