A Robust Autoregressive Gaussian Process Motion Model Usingl1-Norm Based Low-Rank Kernel Matrix Approximation

Eunwoo Kim, Sungjoon Choi, and Songhwai Oh

Abstract— This paper considers the problem of modelingcomplex motions of pedestrians in a crowded environment. Anumber of methods have been proposed to predict the motionof a pedestrian or an object. However, it is still difficult to makea good prediction due to challenges, such as the complexity ofpedestrian motions and outliers in a training set. This paperaddresses these issues by proposing a robust autoregressivemotion model based on Gaussian process regression usingl1-norm based low-rank kernel matrix approximation, calledPCGP-l1. The proposed method approximates a kernel matrixassuming that the kernel matrix can be well represented using asmall number of dominating principal components, eliminatingerroneous data. The proposed motion model is robust againstoutliers present in a training set and can reliably predict themotion of a pedestrian, such that it can be used by a robotfor safe navigation in a crowded environment. The proposedmethod is applied to a number of regression and motionprediction problems to demonstrate its robustness and effi-ciency. The experimental results show that the proposed methodconsiderably improves the motion prediction rate compared toother Gaussian process regression methods.

I. INTRODUCTION

For service mobile robots assisting humans, adapting tocomplex and real-world environments is a challenging task[1]. The environments can be highly dynamic and it is dif-ficult to predict behaviors of humans. In such environments,a service robot can easily get stuck or collide with movinghumans or obstacles and it can lead to an unsafe situationfor humans nearby. Since a safe operation is an importantrequirement for the success of service robots, an abilityto predict motions of pedestrians and moving objects is ofparamount importance.

In this paper, we consider the problem of safe navigationof a mobile robot under a dynamic and crowded environment[2]. A related topic is autonomous robot navigation and it hasbeen studied extensively in recent years [3]–[10]. For safenavigation, it is required to predict the trajectories of pedes-trians and moving objects in order to schedule a collision-free path for a robot. In [3], a probabilistic framework basedon a partially observable Markov decision process is used topredict trajectories of humans and obstacles for autonomousrobot navigation. Henry et al. [5] proposed a learning methodfor human-like navigation in a crowded environment using

This research was supported in part by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) funded by theMinistry of Education (NRF-2013R1A1A2009348) and by the ICT R&Dprogram of MSIP/IITP (14-824-09-014, Basic Software Research in Human-level Lifelong Machine Learning, Machine Learning Center).

E. Kim, S. Choi, and S. Oh are with the Department of Elec-trical and Computer Engineering, ASRI, Seoul National University,Seoul 151-744, Korea (e-mail: {eunwoo.kim, sungjoon.choi, songh-wai.oh}@cpslab.snu.ac.kr).

inverse reinforcement learning. Asaula et al. [6] proposeda stochastic abstraction of human motion using a discrete-time Markov chain to predict trajectories of moving objectsand estimate the probability of an accident. Fulgenzi et al.[4] proposed a motion pattern model of pedestrians using aGaussian process (GP). In many approaches, it is assumedthat the current positions of a robot and dynamic obstaclesare available [9] or positions can be estimated from anexternal device or infrastructure, such as an overhead cameranetwork system [8]. However, such assumption makes anapproach impractical in many real-world environments sincecollecting exact locations using an external device can beexpensive and available only in a laboratory setting.

An autonomous navigation method can be grouped intofour categories based on the perspective of a view (referenceor egocentric) and the method used to predict future positionsof obstacles (model-based or data-driven) [10]. A complexmotion of a pedestrian is modeled using autoregressiveGaussian processes (AR-GP) in [10]. The AR-GP basedmethod is data-driven and egocentric, hence, no externaldevice is required for collecting location information. In[10], the authors have shown that the AR-GP based methodgives a dramatic improvement over existing reactive controlmethods, such as the reactive planner [7] and vector fieldhistogram [11]. However, in order for an AR-GP motionmodel to perform well, the training set has to be carefullycollected such that no outliers are included. But it is anontrivial task to collect an outlier-free training set due tonoises in sensors and object detection algorithms used in arobot sensing system. In this paper, we address the limitationof [10] by introducing a method which can robustly learnfrom a training set with outliers.

To handle outliers, methods based on the l1-norm arewidely used to robustly solve problems such as backgroundmodeling, image denoising, dimension reduction, and datareconstruction [12], [13]. These techniques assume a Lapla-cian noise model instead of a Gaussian noise model andare used to obtain a robust low-rank approximation of theoriginal data.

In this paper, we propose a robust autoregressive Gaussianprocess motion model (AR-GP) using the l1-norm basedlow-rank kernel matrix approximation. The proposed method(PCGP-l1) approximates a kernel matrix used in AR-GP byits low-rank approximation, assuming that the kernel matrixcan be represented using a small number of dominating prin-cipal components, eliminating outliers and erroneous data inthe training set. By extracting orthogonal basis vectors froml1-norm based low-rank kernel matrix approximation, we

obtain the robust solution of AR-GP. The proposed methodis applied to regression and motion prediction problems insimulation to demonstrate its robustness against outliers. Themethod is also applied in a physical experiment using aPioneer 3DX mobile robot and a Microsoft Kinect camerafor motion prediction. We also applied the proposed motionmodel to a motion control problem in a school cafeteria.The experimental results show that our proposal is robustcompared to existing Gaussian process regression methodsunder the existence of outliers or unwanted data.

The remainder of this paper is organized as follows: InSection II, we briefly review the Gaussian process motionmodel. In Section III, we proposed a robust autoregressiveGaussian process motion model using l1-norm based low-rank kernel matrix approximation. In Section IV, the perfor-mance of our proposal is evaluated in experiments.

II. AUTOREGRESSIVE GAUSSIAN PROCESS MOTIONMODEL

A Gaussian process (GP) is a collection of random vari-ables which has a joint Gaussian distribution and is specifiedby its mean function m(x) and covariance function k(x,x′)[14]. A Gaussian process f(x) is expressed as:

f(x) ∼ GP (m(x), k(x,x′)). (1)

Suppose that xi ∈ Rni is an input and yi ∈ R is an output.For a noisy observation set D = {(xi, yi)|i = 1, · · · , n}, wecan consider the following observation model:

yi = f(xi) + wi, (2)

where wi ∈ R is a zero-mean white Gaussian noise withvariance σ2

w. Then the covariance of yi and yj can beexpressed as

cov(yi, yj) = k(xi,xj) + σ2wδij , (3)

where δij is the Kronecker delta function which is 1 if i = jand 0 otherwise. k(xi,xj) = 〈φ(xi), φ(xj)〉 is a covariancefunction based on some nonlinear mapping function φ. Thefunction k is also known as a kernel function.

We can represent (3) in a matrix form as follows:

cov(y) = K + σ2wI, (4)

where y = [y1 · · · yn]T and K is a kernel matrix such that[K]ij = k(xi, xj).

The conditional distribution of a new output y∗ at a newinput x∗ given D becomes

y∗|D,x∗ ∼ N (y∗,V(y∗)), (5)

wherey∗ = kT∗ (K + σ2

wI)−1y (6)

and the covariance of y∗, V(y∗), is

V(y∗) = k(x∗,x∗)− kT∗ (K + σ2wI)−1k∗. (7)

Here, k∗ ∈ Rn is a covariance vector between the new datax∗ and the existing data, such that [k∗]i = k(x∗, xi).

In [10], Choi et al. proposed an autoregressive Gaussianprocess (AR-GP) for motion modeling and prediction of apedestrian trajectory for real-time robot navigation. Let Dt =(xt, yt) ∈ R2 be the position of a moving object at timet, they modeled the current velocity, ∆Dt = Dt − Dt−1,with an appropriate time scaling using the AR-GP model asfollows [10]:

∆Dt = f(Dt−1, Dt−2, ..., Dt−p)

∼ GPf (Dt−1, Dt−2, ..., Dt−p).(8)

Hence, the AR-GP model approximates the current motionof a moving object from p recent positions of the object.It can be considered as a nonlinear generalization of anautoregressive model under the Gaussian process framework.They have shown that an AR-GP based method gives adramatic improvement compared with other reactive controlalgorithms, such as the reactive planner [7] and vector fieldhistogram [11].

While Gaussian process regression is a nonparametricmethod which enjoys the expressive power of the fullBayesian framework, it suffers from a major drawback whichis its high computational cost for evaluating the inverseof a kernel matrix whose size grows with the number oftraining data [14]. To deal with the computation issue, sparseGaussian process methods have been proposed [15], [16].These methods give results comparable to the standard GPbut with lower computational complexity. However, note thatwhen it comes to make a prediction from a fixed trainingset, the computational load of calculating (6) and (7) can bereduced by pre-computing the inverse of the kernel matrixas follows [10]:

y∗ = kT∗ Λy = kT∗ Γ, (9)

where Λ = (K + σ2wI)−1 and Γ = Λy.

III. ROBUST AUTOREGRESSIVE GAUSSIAN PROCESSMOTION MODEL

A. Principal Component Gaussian Process (PCGP)

To reduce the computational cost of inverting the ker-nel matrix Λ in (9), a number of approximation methodshave have been proposed, including Incomplete CholeskyFactorization (ICF) [16] and the Nystrom method [17]. Inthis paper, we consider another kernel matrix approximationmethod, low-rank kernel matrix approximation, which is alsoknown as kernel principal component analysis (KPCA) [18].It has been attracted much attention for a wide range ofproblems in order to efficiently process a large quantityof data and to discover a hidden low-dimensional structurebased on the Euclidean distance (l2-norm).

The main idea behind the kernel-based approximationmethod is that by using a kernel function the original linearoperations of PCA are performed in a high-dimensionalHilbert space [18]. Performing linear PCA in a high-dimensional space has an effect of performing nonlinear PCAin the original input space [18]. Hence, we can apply low-rank kernel matrix approximation to reduce the computa-tional load of Λ in (9) to speed up the kernel machine.

Suppose that a nonlinear function φ : Rni → X is amapping from the input space to a high-dimensional featurespace. Then, for centered data x1, . . . ,xn, the covariancematrix in X is

C =1

n

n∑i=1

φ(xi)φ(xi)T

and the eigenvector v with nonzero eigenvalue of C canbe represented as v =

∑ni=1 βiφ(xi). The coefficients

β = [β1 · · ·βn]T can be found by solving the followingeigenvalue problem [18]:

Kβ = nλβ, (10)

where K is a kernel matrix such that [K]ij = 〈φ(xi), φ(xj)〉.It follows that principle components in X can be extractedusing eigenvectors vk for k = 1, . . . , r with r largest eigen-values of K using coefficents found from (10) with a propernormalization [18]. Hence, we can effectively represent akernel matrix using a subset of eigenvectors with r largesteigenvalues.

Given the eigenvalue decomposition of K = RΣRT ,where Σ = diag(λ1, · · · , λn) is a diagonal matrix of eigen-values of K, such that λ1 ≥ · · · ≥ λn, we can approximatethe inverse of K as follows:

K−1 = (RΣRT )−1 = RΣ−1RT ≈ RRT , (11)

where R = RrΣ− 1

2r . Here, Rr ∈ Rn×r collects the first

r vectors from R and Σr = diag(λ1, · · · , λr) ∈ Rr×r isa diagonal matrix of r largest eigenvalues. Hence, we canreformulate (9) by treating Λ as K in (11) as

y∗ = kT∗ Λy ≈ kT∗ RRTy = kT∗ y, (12)

where k∗ = RT k∗ is a kernel vector which is projectedinto the orthogonal feature space R and y = RTy is aprojected output vector into R. This means that the low-dimensional approximation of a kernel matrix can be appliedto Gaussian process regression problems by using k∗ andy which are projected on R, and the inverse of kernelmatrix becomes an identity matrix which represents theindependent relationship between basis vectors. Hence, (12)can be another representation of y∗ in the dimensionallyreduced orthogonal feature space R. Figure 1 shows theconcept of the proposed method using low-rank kernel matrixapproximation.

B. PCGP-l1: PCGP via l1-norm based kernel matrix ap-proximation for robust AR-GP

As mentioned earlier, Λ in (9) can be approximated bya conventional low-rank approximation method which trans-forms data into a low-dimensional subspace and maximizesthe variance of the given data based on the Euclidean dis-tance (l2-norm). However, the method is sensitive to outliersbecause the l2-norm can sometimes amplify the negativeeffects of such data. When outliers are present in the data, thecovariance matrix of the data can be corrupted by outliers.Therefore, l2-norm based low-rank approximation methods

Fig. 1. A graphical illustration of principal component Gaussian processregression using low-rank kernel matrix approximation. We can performthe prediction step of Gaussian process regression in the dimension-reducedfeature space.

may find projections which are far from the desired solution.As an alternative, various methods based on the l1-normhave been proposed recently, which are more robust againstoutliers [13]. Hence, we approximate the kernel matrix Λusing l1-norm based kernel matrix approximation for robustautoregressive Gaussian process motion model learning.

A minimization problem based on the l1-norm can beregarded as a maximum likelihood estimation problem underthe Laplacian noise distribution [13]. We first consider anapproximation problem for vector g = (g1, g2, · · · , gm)T bya multiplication of vector x ∈ Rm and scalar α, i.e.,

g = αx+ δ, (13)

where δ is a noise vector whose elements are independentlyand identically distributed from a Laplacian distribution. Theprobability model for (13) can be written as

p(g|x) ∼ exp

(−‖g − αx‖1

s

), (14)

where ‖ · ‖1 denotes the l1-norm and s > 0 is a scalingconstant. We assume for a moment that x is given. In thiscase, maximizing the log likelihood of the observed data isequivalent to minimizing the following cost function:

J(α) = ‖g − αx‖1. (15)

Let us rewrite the cost function for matrix approximationusing (15):

minU,V

J(U, V ) = ‖G− UV ‖1. (16)

where G ∈ Rm×n, U ∈ Rm×r, and V ∈ Rr×n are the obser-vation, projection, and coefficient matrices, respectively. Gis a given data matrix and it is Λ from (12) for the low-rankkernel matrix approximation problem, i.e., m = n. Here, ris a predefined parameter less than min(m,n) and || · ||1 isthe entry-wise l1-norm, i.e., ‖G‖1 =

∑ij |gij | where gij is

the (i, j)-th element of G. In general, (16) is a non-convexproblem, because both U and V are unknown variables.

We can efficiently perform this optimization process byminimizing the cost function over one matrix while keepingthe other fixed and then alternately exchanging roles ofmatrices. Such minimization technique based on alternatingiterations has been widely used in subspace analysis [13] andcan be written as

U (j) = arg minU‖G− UV (j−1)‖1

V (j) = arg minV‖G− U (j)V ‖1,

(17)

where the superscript j denotes the iteration number. How-ever, conventional alternating minimization based approachesare still computationally very expensive and require a largememory when the matrix dimension is large.

In order to speed up the optimization step (17), [13] pro-posed two l1-norm based low-rank approximation methodsusing alternating rectified gradient. In this paper, we usethe second method (l1-ARGD) which solves the problemby optimizing its dual problem which is convex [13]. Inaddition, l1-ARGD enjoys the convergence guarantee [13].The problem of finding the gradient of V for a fixed U inl1-ARGD is formulated as

min∆V

J(∆V |U, V ) = ‖G− U(V + ∆V )‖1

s.t. ||U∆V ||2F = ε2,(18)

where ∆V is the variation of V that we are seeking. Inthis formulation, l1-ARGD solves the problem based onthe primal-dual formulation and uses the proximal gradientand QR factorization for exact and fast computation. Thecomputational complexity of l1-ARGD at each iteration isO(rmn) for a matrix of size m×n and is O(rn2) for a kernelmatrix of size n×n. l1-ARGD typically requires between 10and 20 iterations before convergence [13]. It has shown thesuperiority of l1-ARGD for large-scale problems with respectto the running time and reconstruction errors compared withother l1-norm based methods [13].

The overall procedure of the proposed robust Gaussianprocess regression is described in Algorithm 1. We call thealgorithm as PCGP-l1 because the algorithm uses principalcomponents obtained from l1-norm based low-rank kernelmatrix approximation. In Algorithm 1, we perform the stan-dard PCA to L, which is a low-rank kernel matrix, because itis difficult to find an exact solution with the same orthogonalU and V using the l1-norm based method. We precomputethe kernel matrix and its principal components in line 4-8,and test a new input x∗ given the principal components Rin line 10-11.

IV. EXPERIMENTAL RESULTS

In this section, we have evaluated the performance of theproposed method (PCGP-l1) by experimenting with variousdata and compared with well-known sparse Gaussian processmethods, such as SPGP1 [15] and GPLasso2 [16], along withthe standard full GP. In our experiments, we used the RBF

1Available at http://www.gatsby.ucl.ac.uk/˜snelson/.2Available at https://www.cs.purdue.edu/homes/alanqi/

softwares/softwares.htm.

Algorithm 1 PCGP-l11: Input: X,y, and x∗2: Output: y∗3: // Training4: Compute Λ = K + σ2

wI5: Perform low-rank kernel matrix approximation [13]:6: [U, V ] = arg minU,V ||Λ− UV ||17: L← UV8: Compute R and Σ by performing PCA to L9: // Testing

10: Compute k∗ = k(x∗, X)11: Compute y∗ by (12)

(a) (b)

Fig. 2. (a) A Pioneer 3DX mobile robot with a Kinect camera. (b) Collectedtrajectories from Kinect. We show a few trajectories in bold for bettervisualization.

Gaussian kernel for all methods and hyperparameters learnedusing a conjugate gradient method [14].

For an actual experiment, we collected trajectories ofmoving pedestrians using a Pioneer 3DX differential drivemobile robot and a Microsoft Kinect camera, which ismounted on top of the robot as shown in Figure 2(a). Allalgorithms are written in MATLAB using the mex-compiledARIA package3 on a notebook with a 2.1 GHz quad-coreCPU and 8GB RAM. The position of a pedestrian is detectedusing the skeleton grab API for Kinect.

A. Simulation

First, we tested on a synthetic example to compare theproposed l1-norm based PCGP to the l2-norm based PCGPin a regression problem when there are some outliers. Sincewe are interested in how the proposed PCGP-l1 performsin the presence of outliers, we only compare PCGP-l1 tol2-norm based PCGP and the full GP.

Figure 3 shows the simulation results of a regressionproblem according to outlier levels. Two cases with outliersare constructed by adding additional outliers (10%) to the nooutlier case with two different magnitude levels. As shownin Figure 3(a), all methods give the exact results whenthere are no outliers, so the proposed method shows itscompetitiveness compared with the full GP. If we add someoutliers as shown in Figure 3(b), the full GP and PCGP-l2try to fit outliers so they show fluctuations when there isan outlier, but PCGP-l1 is less affected by outliers, showing

3Available at http://robots.mobilerobots.com/wiki/ARIA.

0 2 4 6 8 10−2

−1

0

1

2

3

4

5

6

Reference FieldGP train dataFullGPPCGP−L2PCGP−L1

(a)

0 2 4 6 8 10−10

−5

0

5

10

15

Reference FieldGP train dataFullGPPCGP−L2PCGP−L1

(b)

0 2 4 6 8 10−30

−20

−10

0

10

20

30

Reference FieldGP train dataFullGPPCGP−L2PCGP−L1

(c)

Fig. 3. Simulation results on a synthetic example. (a) No outlier case. (b) Mid-level outlier case. (c) High-level outlier case.

its robustness against outliers. When there are outliers withlarger magnitudes (Figure 3(c)), l2-norm based PCGP tries tofit the outliers which results in a bad regression result. UnlikePCGP-l2, PCGP-l1 gives the best regression result. From thisexperiment, we see the clear benefit of the proposed methodto a regression problem when the train set contains outliers.

B. Motion Prediction

We performed experiments in our laboratory to predictthe future position of a person. From training, we collecteda diverse set of trajectories of pedestrians, which are in thefield of view of a robot as shown in Figure 2(b). To makea training set from the collected trajectories, we uniformlysampled positions to have about ten samples in a trajectorywhen a trajectory has many detected positions.

We evaluated the proposed method under this experimentalsetting compared with other Gaussian process methods (fullGP [10], SPGP [15], and GPLasso [16]). To test the proposedmethod, we predicted the next location of a pedestriangiven the detected locations of the moving pedestrian andthis process is continuously performed until the pedestriandisappears from the field of view of a robot. We tested about100 trajectories of pedestrians from a mobile robot in ourlaboratory and measured the prediction accuracy using theroot mean squared error.

Figure 4 shows the motion prediction results in centimetersunder various low rank conditions, r/n = {5%, 10%, 15%},using Kinect-based pedestrian trajectories. From the figure,we can see that the proposed method gives a lower errorthan other methods regardless of the rank condition, whereasthe sparsity-based GP methods shows a higher root meansquare error (RMSE) compared to the propose method andfull GP. Some snapshots from our laboratory experiments areshown in Figure 5. For this example, the reduced rank wasr = 0.15n. As shown in the figure, the proposed methodpredicted the future trajectories better than the full GP [10].

C. Motion Control

We evaluated the performance of the proposed method,PCGP-l1, for a motion control problem. A non-parametricBayesian motion controller can navigate through crowdeddynamic environments using an autoregressive Gaussian pro-cess model (AR-GP). Thus, we applied the proposed method

5 10 150.1

0.15

0.2

0.25

0.3

0.35

Rank percentage (Rank/FullRank*100)

RM

SE

[cm

]

Motion Prediction with Microsoft Kinect

FullGPSPGPGPLassoPCGP−L1

Fig. 4. Motion prediction results using Kinect-based human trajectories.

to the autoregressive Gaussian process motion controller[10]. The number of pedestrians varies from one to eight tovalidate the safety of the the proposed method under differentconditions. Similar to [10], the robot tries to reach the goalusing the pure pursuit method [19] if there is no pedestrian.

We tested the motion controller based on the proposedmotion model in a crowded school cafeteria. The goal ofthe experiment was to follow a given path while avoidingdynamic or static pedestrians. Figure 6 shows some snapshotsfrom the experiment. The robot successfully avoided movingpedestrians without any collisions and arrived at the goalposition.

V. CONCLUSIONS

In this paper, we have proposed a robust Gaussian pro-cess regression method using l1-norm based low-rank ker-nel matrix approximation, PCGP-l1, for robustly modelinga complex pedestrian motion pattern in the presence ofoutliers. Using the relationship between Gaussian processregression and low-rank kernel matrix approximation, weobtain orthogonal basis vectors under the l1-norm cost func-tion. The proposed method is applied to various problemsincluding experiments using a Pioneer 3DX mobile robotand a Microsoft Kinect. The experimental results show theperformance and robustness of the proposed method againstoutliers and sensor errors compared to existing methods.

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Fig. 5. Snapshots from laboratory experiments. The predicted positions by the proposed algorithm and the full GP motion model [10] are also shown.Each snapshot shows a photo taken by a camera, a photo taken by the robot, and the robot’s internal state.

Fig. 6. Snapshots from the school cafeteria experiment using the autoregressive Gaussian process motion controller based on PCGP-l1. Each snapshotshows a photo taken by a camera, a photo taken by the robot, and the robot’s internal state.

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