Relational Object Maps for Mobile Robots

Benson Limketkai and Lin Liao and Dieter FoxDepartment of Computer Science and Engineering

University of WashingtonSeattle, WA 98195

In Proc. of the International Joint Conference on Artificial Intelligence (IJCAI), 2005.

AbstractMobile robot map building is the task of generat-ing a model of an environment from sensor data.Most existing approaches to mobile robot mappingeither build topological representations or generateaccurate, metric maps of an environment. In thispaper we introduce relational object maps, a novelapproach to building metric maps that represent in-dividual objects such as doors or walls. We showhow to extend relational Markov networks in or-der to reason about a hierarchy of objects and thespatial relationships between them. Markov chainMonte Carlo is used for efficient inference and tolearn the parameters of the model. We show that thespatial constraints modeled by our mapping tech-nique yield drastic improvements for labeling linesegments extracted from laser range-finders.

1 IntroductionBuilding maps of indoor spaces is an extremely importanttask in mobile robotics. Over the last years, the SLAM (si-multaneous localization and mapping) community has madetremendous progress in the development of efficient andhighly accurate map building techniques. Most of thesetechniques focus either on capturing the metric layout ofan environment with high accuracy[Eliazar and Parr, 2003;Konolige et al., 2005] or on representing the topologicalstructure of an environment[Kuipers, 2000]. While metricmaps have the advantage of being well suited for robot nav-igation tasks, they are typically unstructured and contain noinformation about the different types of places or objects inan environment. Topological approaches, on the other hand,are more compact and expressive in that they describe signif-icant places in an environment. However, these techniquesoften ignore valuable metric information and they still lackthe ability to describe individual objects in an environment.

Our goal is to build on the progress made in metric map-ping and to enhance metric maps with semantic informa-tion about different objects in an environment. Object-basedrepresentations have much higher expressive power in thatthey combine metric accuracy with a semantic description ofan environment. In addition to allowing more natural inter-faces between humans and robots (“take the third door on theright”), they enable robots to perform better reasoning, espe-cially about dynamic objects such as doors.

Recently, Anguelov and colleagues made important stepstoward building such maps by developing techniques for de-tecting and labeling objects in metric maps. In[Anguelovet al., 2002], they showed how to learn models of non-stationary, free-standing objects such as rectangular or roundboxes. To overcome the limitations of this technique, theythen developed a generative Bayesian approach for detect-ing doors and walls in a hallway environment[Anguelovetal., 2004]. While this approach yields good results for la-beling line segments extracted from laser range-scans, it haslimited capabilities in modeling context information. Eventhough the approach can express information such as all doorsshare the same width and color, it can not model more com-plex, spatial relationships between objects. As[Murphy etal., 2003] showed for scene analysis in computer vision, suchcontext information can be extremely useful for distinguish-ing between different types of objects. In mobile robot mapbuilding, for example, an object standing in the middle of ahallway cannot be a door, even if it has the same width andcolor as other doors in the hallway.

In this paper, we show how to reason about the appear-ance of and spatial relationships between objects so as tobuild metric maps with object representations. We denotethe resulting mapping paradigmRelational Object Maps, RO-Maps for short. RO-Maps build on Relational Markov Net-works (RMN) to represent hierarchical structure and the re-lationships between objects. RMNs are undirected graphicalmodels that are learned discriminatively[Taskaret al., 2002].They provide an extremely flexible framework for describingand reasoning about relations. Our RO-Maps detect objectssuch as doors and walls by a combination of features, includ-ing appearance features such as the width of a door, neigh-borhood features describing what type of objects are next toeach other, and spatial features such as the indentation of adoor relative to the wall. In our experiments we show that theparameters of RO-Maps can be learned from a set of hallwaysand then be successfully applied to labeling the objects in ahallway of a different environment.

This paper is organized as follows. After providing back-ground on RMNs, we will show how RO-Maps model objectappearance and the spatial relationships between them. Infer-ence and parameter learning will be discussed in Section 3.Experimental results are presented in Section 4, followed byconclusions and a discussion of future work.

2 Relational Map BuildingThis section describes RMNs and several extensions thatmake them applicable to the problem of mobile robot mapbuilding. We denote the resulting framework RO-Maps, shortfor relational object maps.

2.1 Relational Markov NetworksWe will now briefly review the basic ideas of relationalMarkov networks, more detailed information can be foundin [Taskaret al., 2002]. RMNs are an extension of Condi-tional Random Fields (CRFs), which are undirected graphicalmodels that were developed for labeling sequence data[Laf-ferty et al., 2001]. CRFs are discriminative models that havebeen shown to out-perform generative approaches such asHMMs and Markov random fields in areas such as naturallanguage processing[Lafferty et al., 2001] and computer vi-sion [Kumar and Hebert, 2003]. RMNs extend CRFs byproviding a relational language for describing clique struc-tures and enforcing parameter sharing at the template level.Thereby RMNs are an extremely flexible and concise frame-work for defining features and relationships that can be usedin our map building context.

Specifically, an RMN consists of three parts: aschemaEfor the domain, a set ofrelational clique templatesC, andcorrespondingpotentialsΦ. The schemaE specifies the setof classes(i.e., entity types) and the attributes in each class.In the map building context, the classes are types of objects,such as walls or doors, and the attributes describe their phys-ical properties. AninstantiationI of a schema specifies theset of entities for each class and the values of all attributesfor each entity. In our case,I is an RO-Map consisting ofline segments extracted from laser range-scans. Arelationalclique templateC ∈ C is similar to a query in a relationaldatabase. It selects tuples from an instantiationI; the queryresult is denoted asC(I). In an RO-Map, for example, aclique template can select all wall or door objects on one sideof a hallway.

Each clique templateC is associated with a potential func-tion φC(vC) that maps valuesvC of variables in the clique tonon-negative real numbers. Using a log-linear combinationof feature functions to define these potentials, we get the fol-lowing representation:φC(vC) = exp{wT

C · fC(vC)}, wherefC() defines a feature vector forC andwT

C is the transpose ofthe corresponding weight vector. For instance, a feature couldbe the indentation of a door, defined on a clique template se-lecting a door and wall objects on one side of a hallway.

For a specific instantiationI, an RMN defines a condi-tional distributionp(y|x) over labelsy given observationsx. To compute the conditional distribution of a label vec-tor y, the RMN generates anunrolled Markov network, inwhich the nodes correspond to the entities. The cliques of theunrolled network are built by applying each clique templateC ∈ C to the instantiation, which can result in the genera-tion of several cliques per template. All cliques that originatefrom the same template must share the same weightswC .The resulting cliques factorize the conditional distribution as

p(y | x) =1

Z(x)

∏C∈C

∏vC∈C

φC(vC) (1)

=1

Z(x)

∏C∈C

∏vC∈C

exp{wTC · fC(vC)} (2)

=1

Z(x)exp{wT · f}, (3)

where the normalizing partition functionZ(x) =∑y′

∏C∈C

∏v′

C∈C φC(v′C). (3) follows by moving the

products into the exponent and combining all summationsinto w andf .

For efficient inference in RMNs,[Taskaret al., 2002] sug-gest using (loopy) belief propagation. However, because ofthe extensions described below, we cannot apply belief prop-agation directly. Instead, we will show how to perform effi-cient inference and learning using MCMC.

2.2 Relational Object MapsIn the most general form, RO-Maps represent an environmentby a hierarchical collection of objects and relations betweenthem. RO-Maps estimate the types of objects under the as-sumption that their locations are known. This assumptionis justified by the progress made by the SLAM community,which focuses on the problem of concurrently estimating thepose of a robot and the locations of sensor measurements.Such techniques generate well aligned laser range-scans evenfor large scale environments (e.g., [Eliazar and Parr, 2003;Konoligeet al., 2005]).

We introduce an extension of the relational framework ofRMNs to reason about physical objects in RO-Maps. TheRO-Map of a specific environment is modeled by an unrolledMarkov network. Each node in such a network corresponds toan object or geometric primitive, and the links between nodesrepresent relations between these objects.

Objects: The basic building blocks of RO-Maps are geo-metric primitives such as line segments extracted fromsensor data. These primitives typically describe objectssuch as doors or wall segments. More complex objectscan be generated from existing ones by a process we callphysical aggregation. This process first selects all ob-jects with a certain property (just like relational cliquetemplates do in RMNs), and then computes the param-eters of the aggregated object. For example, physicalaggregation generates a wall object by selecting all wallsegments on one side of a hallway, followed by the com-putation of the line parameters and color distribution de-scribing the wall. The left and right wall of a hallwaycan be aggregated into a hallway object, which has pa-rameters such as orientation and width.

Relations between objects can be spatial, such as “distancefrom each other”, or appearance based, such as “similarcolor”. Relations can be defined on basic objects and ob-jects generated by (multiple) physical aggregations. Re-lational clique templates of RMNs can be used to de-fine RO-Map relations. For example, to consider thesimilarity between the widths of the doors in a hallway,RO-Maps use a clique templateC that first selects allobjects that are labeled as doors (such selections are de-fined via SQL queries on the objects[Taskaret al., 2002;

Wall object

Indent Indent

Width

WidthWidth

Figure 1: Upper graph: Endpoints of laser range-scans observed in a hallway. The points line up very accurately due to the position correctionperformed by a scan matching procedure. Middle graph: Line segments extracted from the scans. Wall segments are shown as dashed lines,doors as solid lines, others as thin black lines. Lower figure: Markov network with example cliques generated for this specific labeling. Thinsolid lines define neighborhoods. The doors in the upper part of the hallway are linked by the door variance clique template. The wall objectis generated by physical aggregation (dashed lines). It is linked to doors by the door indentation clique template (solid lines).

Liao et al., 2005]). The widths of the selected doors de-fine the value vectorvC of this clique, which is thenfed into the feature functionfC() that computes the vari-ance. The potential of the clique,φC(vC), is given byexp{wC · fC(vC)}, wherewC is the feature weight.

RO-Maps are illustrated in Figure 1. The upper plot showslaser range-scans observed by a robot traveling down a hall-way; the middle plot shows the labeled line segments ex-tracted from these scans; and the lower plot shows the nodesin the corresponding RO-Map along with some of the undi-rected links generated by the clique templates for this label-ing. Among others, the figure shows two cliques generatedby the template that measures the indentation of doors. Thetemplate generates pairs of doors and the corresponding wallobject. The potential of the clique is then based on the dis-tance of the door from the line representing the wall. Moredetails on features for detecting doors and walls will be givenin the experimental results section.

3 Inference and Learning3.1 Inference: Labeling objects in RO-MapsThe goal of inference is to estimate the labels (types) of theobjects represented in an RO-Map. Inference in RO-Mapsis more complicated than in regular RMNs. This is due tothe fact that the clique structure of the Markov networks gen-erated by RO-Maps depends on the unknown labels of ob-jects, and thus changes during inference. Since it is not clearhow (loopy) belief propagation can be applied to suchlabel-

specific cliques, we use Gibbs sampling, an MCMC tech-nique, for inference[Gilks et al., 1996]. Whenever the la-bel of an object is changed during sampling, we determineall (aggregated) objects and cliques that could be affected bythis change and re-compute their parameters and potentials.To initialize a Markov chain, we randomly assign labels tothe basic objects in the RO-Map. Based on these labels, wegenerate all aggregate objects and their parameters. Finally,the relational clique templates generate the cliques of the un-rolled, fully instantiated Markov network.

At each MCMC step, we update the label of a randomlyselected object by sampling from the conditional distribution

p(yk | y−k,x,w) ∝ exp{wT · f(x,y−k ∪ yk)} (4)

wherek is the index of the object,yk is one of its possiblelabels, andy−k are the labels for the other objects. To com-pute the probabilities of the different labelsyk, we update thecliques and the parameters of aggregated objects involvingthe objectk. Even though this process can become inefficientfor highly complex RO-Maps, inference on our current testhallways is very efficient, as we will describe in Section 4.We expect further efficiency gains by more elaborate sam-pling techniques.

3.2 Learning: Determining feature weightsThe parameters of RO-Maps are the weightsw of the fea-tures that define the clique potentials. The key idea of ourtechnique is to learn these parameters from a set of labeledenvironments, and then apply the learned weights when therobot explores a new environment.

Figure 2: Examples of hallways used for learning and testing. Line segments were labeled manually as Door, Wall, or Other.

The details of our learning technique are beyond the scopeof this paper, more information is given in[Liao et al., 2005].In a nutshell, the weights are learned by minimizing theneg-ative log-likelihood of the labeled training data collected inN environments:

L(w) ≡N∑

j=1

− log p(yj | xj ,w) +wT w2σ2

(5)

Here,j is the index of mapmj , andyj are the labeled ob-jects in thej-th map. The rightmost term avoids overfittingby imposing a zero-mean, Gaussian shrinkage prior on eachcomponent of the weight vector[Taskaret al., 2002]. Since(5) is convex, we can determine the global minimum using aquasi-Newton method, where the value and gradient ofL(w)are determined by MCMC sampling[Liao et al., 2005].

The outcome of the learning procedure is a set of featureweights that best represent the appearance and spatial con-straints of the training maps by maximizing the classificationrate. RO-Maps inherit the important benefit from RMNs ofautomatically enforcing parameter sharing between (aggre-gated) objects and relations of the same type. This “smooth-ing” over instances of classes provides a very powerful toolfor dealing with sparse training data.

4 Implementational Details and ExperimentsWe evaluated RO-Maps on the task of labeling line segmentsextracted from laser range-finder scans collected by a robottraversing a hallway. The data sets were taken from theRadish robotics data set repository[Howard and Roy, 2003].We took two to three hallways from five different environ-ments each and manually labeled the extracted and alignedline segments into the three categories Door, Wall, and Other.Line segments were extracted using the technique describedin [Gutmann and Konolige, 1999]. Some of the test hallwaysare shown in Figure 2. To perform the labeling, we reliedon our experience in typical layouts of hallways. In severalcases, we were not sure about the labeling or were only ableto label doors based on the fact that we knew the real build-ings represented by the maps.

4.1 FeaturesTo label the line segments in the hallways, we implementedvarious features. The features were represented by the log-likelihood of the measured values under a generative model.This approach allows our technique to automatically takevariability between different features into account.

Physical aggregationWall objects are the only physicallyaggregated objects in our current system. Wall objectsare lines that are generated from line segments that arelabeled Walls and are on the same side of a hallway(see Figure 1). The Wall object parameters can be com-puted efficiently from the parameters of the line seg-ments.

Local features describe appearance properties of objects. InRO-Maps, such features are modeled by generating aclique for each object and the corresponding attribute.We use the length of line segments as a feature. The cor-responding feature functions return the log-likelihood ofa specific length given a Gaussian representing the distri-bution over lengths, conditioned on the object type. Themeans and variances of the Gaussians for Doors, Walls,and Others were estimated from the labeled maps. Inour framework it is straightforward to add other features,such as the color of objects[Anguelovet al., 2004]. Un-fortunately, the Radish data sets do not contain camerainformation.Aggregated Wall objects have a feature that measuresthe alignment of the line segments that make up the wallobject. Alignment is measured by the average distanceof the line segments from the aggregated Wall line. Thisfeature helps RO-Maps to label line segments as Wallsonly when they are well aligned with other Wall seg-ments.

Neighborhood features describe which object types are lo-cated next to each other. For example, it is common toobserve a Wall segment next to a Door, but uncommonto find many Doors in close proximity to one another.We compute a line segment’s neighborhood by deter-mining all other segments with endpoints within 40 cm.Neighborhood features are then modeled by binary indi-cator functions on the possible label pairs (e.g., “WW”,“WD”, “DO”).

Spatial features describe the relative position of objects.The indentation of a Door object is computed relative tothe aggregated Wall object on the same side of the hall-way. Indentation is given by the minimum distance be-tween the Door line endpoints and the Wall line, therebyestimating indentation at the hinge point of a door. Aswith line width, the indentation feature function returnsthe log-likelihood under a Gaussian indentation modelestimated from the training maps.Unless an object is a Wall segment, it is highly unlikely

Doors labeled as Other

Figure 3: Hallway with the highest classification error. The doors in the lower half of the hallway are wide double doors with very largeindentations. Since none of the other training environments contained such doors, our approach labeled them as Other. Some of the opendouble doors were detected as doors. All doors in the upper half and all walls (dashed lines) are labeled correctly.

to be both very close and well-aligned with a Wall ob-ject. RO-Maps detect Other objects by considering thedistance and angle alignment of such objects relative tothe closest Wall object. To compute the likelihood ofspecific distances and angles, we estimate discrete dis-tributions from the training maps (these distributions aretypically non-Gaussian).

Global features depend on multiple objects that are poten-tially far apart. In our experiments, we measure the sim-ilarity of the indentations of all doors in a hallway bycomputing their variance.

4.2 Experiments and ResultsTo test the performance of our learning method, we used 5-fold cross-validation on the five environments extracted fromthe Radish repository. To do so, the model parameters weretrained on labeled hallways from four environments and thentested on the remaining one. Training on a set of 10 hall-ways took 2-3 hours. To generate the statistics needed foreach iteration of the quasi-Newton optimization technique,we ran a Gibbs sampler for 1,000 iterations on each trainingmap. The final 200 samples from each chain were used tocompute differences between the sampled feature values andthe feature values extracted from the labeled maps. Thesedifferences were then used to compute the relative, negativelog-likelihood (5) and the gradient.

Testing, that is, labeling of the objects in an unknown map,was very efficient and took around one minute or less on av-erage. For each environment, the labeling error rate was de-termined by comparing the most probable assignment to themanually generated label. Most probable assignments wereextracted from the Markov chains by counting the differentlabels for each object. The label with the highest count wascompared to the ground truth. To gauge the importance ofdifferent features for labeling the objects in these hallways,we varied the set of features we used.

Environm. Lengths Lengths + Neighbors All1 62.6% 88.5% 90.7%2 58.7% 63.0% 93.5%3 59.0% 79.2% 89.7%4 51.8% 96.5% 97.7%5 60.0% 68.5% 77.9%

Table 1: Average accuracy rates on maps using different fea-tures.

Table 1 summarizes the results. The numbers give the ac-

curacy rates of the labels, averaged over the two hallways ofeach test environment. The leftmost result column gives theaccuracy rates achieved when using only the lengths of linesegments to distinguish between the different object types.The next column summarizes results obtained when consid-ering length and neighborhood features. The resulting modelsare similar to standard HMMs or MRFs in that they considerlocal appearance features (length) and connect neighboringobjects. As can be seen, object neighborhood improves theresults significantly. Furthermore, these results indicate thatthe labeling task is by far not trivial. The last column presentsthe accuracy rates achieved when using all features discussedin the previous section. The results show that the additionalconsideration of spatial constraints such as door indentationimproves the classification performance by a large margin.

Figure 3 shows the hallway with the highest classificationerror (environment 5). As can be seen, most of the doorsalong the lower wall of the hallway are mislabeled as Other.However, this is not surprising given that the indentation ofthese doors is far larger than in the other training maps. Fur-thermore, most of these double doors are closed, which re-sults in very long line segments. Thus, the labels are veryreasonable, given the training data available to the approach.

Inferred labelsTruth Wall Door OtherWall 221 5 8Door 1 122 21Other 10 12 93

Table 2: Confusion matrix (counts).

Table 2 shows the confusion matrix summarized over alltest environments using all features. The table shows thatmost errors are due to confusion between Others and Walls,and Others and Doors. Walls and Doors are rarely confused.The overall accuracy rate is 88.4%. This is a very encourag-ing result, especially since several maps contained objects forwhich even humans were uncertain about the correct labeling.

It should be noted that[Anguelovet al., 2004] achieved rel-atively high accuracy using a hierarchical Bayesian approachwithout spatial constraints. This is due to the fact that theytrained model parameters in the same environment and tookvisual color information into account. Adding such appear-ance information to RO-Maps is straightforward and we ex-pect strong improvements from such additional information.

5 Conclusions and Future WorkWe introduced relational object maps for mobile robot map-ping. RO-Maps use relational, discriminative techniques togenerate metric maps with semantic object representations.RO-Maps build upon and extend Relational Markov Net-works so as to reason about the appearance of and the spatialconstraints between objects in an environment. In order todistinguish between different types of objects, RO-Maps canincorporate a variety of features, ranging from global, suchas the width of all doors in a hallway, to spatial, such as theindentation of a door. To handle the changing structure of theMarkov networks instantiated by RO-Maps, we showed howto perform inference and parameter learning using MCMC.During learning, RO-Maps benefit from automatic parametersharing provided by relational clique templates. This allowsthe models to be trained from relatively sparse training sets.

Experiments in five different environments demonstratethat RO-Maps are able to achieve high accuracy in label-ing objects in previously unknown hallways. We show thatthe accuracy is improved drastically by taking spatial con-straints between objects into account. These results are par-ticularly encouraging given that objects are labeled based onlaser range-data only.

RO-Maps provide an extremely flexible framework forcombining metric accuracy with the representation of individ-ual objects and structures in an environment. We believe thatRO-Maps are an important step toward more expressive, se-mantic representations of environments. However, the workpresented here is only the first step in this direction, layingthe foundation for future research projects.

Our current implementation relies on line segments ex-tracted from laser range-scans as the building blocks of RO-Maps. As shown by[Anguelov et al., 2004], color infor-mation extracted from cameras strongly improves the perfor-mance of object classification. The flexibility of our frame-work allows us to integrate such additional appearance in-formation without modifications to the underlying RO-Mapconcept. We observed significant improvements in initial ex-periments with a camera pointed toward the walls. We addi-tionally plan to improve line segmentation by incorporatingsegmentation into the RO-Map inference process. Thereby,high-level information provided by the RO-Map can guiderange-scan segmentation.

The RO-Maps presented here model objects in individualhallways only. A key next step will be the extension to moreobject classes and multiple, connected hallways, rooms, andopen spaces. The RO-Map framework enables the definitionof high-level structures such as hallways or rooms using thenotion of physical aggregation. We are currently developinghierarchical representations and inference techniques that la-bel both high-level structures and individual objects withinthese structures.

6 AcknowledgmentsThe authors would like to thank Jeff Bilmes for fruitful dis-cussions during the early stages of this project. This researchis sponsored in part by the National Science Foundation (CA-REER grant number 0093406).

References[Anguelovet al., 2002] D. Anguelov, R. Biswas, D. Koller,

B. Limketkai, S. Sanner, and S. Thrun. Learning hier-archical object maps of non-stationary environments withmobile robots. InProc. of the Conference on Uncertaintyin Artificial Intelligence (UAI), 2002.

[Anguelovet al., 2004] D. Anguelov, D. Koller, E. Parker,and S. Thrun. Detecting and modeling doors with mobilerobots. InProc. of the IEEE International Conference onRobotics & Automation (ICRA), 2004.

[Eliazar and Parr, 2003] A. Eliazar and R. Parr. DP-SLAM:Fast, robust simultaneous localization and mapping with-out predetermined landmarks. InProc. of the InternationalJoint Conference on Artificial Intelligence (IJCAI), 2003.

[Gilks et al., 1996] W.R. Gilks, S. Richardson, and D.J.Spiegelhalter, editors. Markov Chain Monte Carlo inPractice. Chapman and Hall/CRC, 1996.

[Gutmann and Konolige, 1999] J.S. Gutmann and K. Kono-lige. Incremental mapping of large cyclic environments.In Proc. of the IEEE International Symposium on Compu-tational Intelligence in Robotics and Automation, 1999.

[Howard and Roy, 2003] A. Howard and N. Roy.The robotics data set repository (radish), 2003.radish.sourceforge.net .

[Konoligeet al., 2005] K. Konolige, D Fox, C. Ortiz,A. Agno, M. Eriksen, B. Limketkai, J. Ko, B. Moris-set, D. Schulz, B. Stewart, and R. Vincent. Centibots:Very large scale distributed robotic teams. In M. Angand O. Khatib, editors,Experimental Robotics: The 9thInternational Symposium, Springer Tracts in AdvancedRobotics (STAR). Springer Verlag, 2005.

[Kuipers, 2000] B. Kuipers. The spatial semantic hierarchy.Artificial Intelligence, 119:191–233, 2000.

[Kumar and Hebert, 2003] S. Kumar and M. Hebert. Dis-criminative random fields: A discriminative framework forcontextual interaction in classification. InProc. of the In-ternational Conference on Computer Vision (ICCV), 2003.

[Lafferty et al., 2001] J. Lafferty, A. McCallum, andF. Pereira. Conditional random fields: Probabilisticmodels for segmenting and labeling sequence data.In Proc. of the International Conference on MachineLearning (ICML), 2001.

[Liao et al., 2005] L. Liao, D. Fox, and H. Kautz. Location-based activity recognition using relational Markov net-works. InProc. of the International Joint Conference onArtificial Intelligence (IJCAI), 2005.

[Murphyet al., 2003] K. Murphy, A. Torralba, and W. Free-man. Using the forest to see the trees: A graphical modelrelating features, objects and scenes. InAdvances in Neu-ral Information Processing Systems (NIPS), 2003.

[Taskaret al., 2002] Ben Taskar, Pieter Abbeel, and DaphneKoller. Discriminative probabilistic models for relationaldata. InProc. of the Conference on Uncertainty in Artifi-cial Intelligence (UAI), 2002.