Approximate Distributed Spatiotemporal Topic Modelsfor Multi-Robot Terrain Characterization

Kevin Doherty1,2, Genevieve Flaspohler1,2, Nicholas Roy1, and Yogesh Girdhar2

Abstract— Unsupervised learning techniques, such asBayesian topic models, are capable of discovering latentstructure directly from raw data. These unsupervised modelscan endow robots with the ability to learn from theirobservations without human supervision, and then use thelearned models for tasks such as autonomous exploration,adaptive sampling, or surveillance. This paper extendssingle-robot topic models to the domain of multiple robots.The main difficulty of this extension lies in achieving andmaintaining global consensus among the unsupervised modelslearned locally by each robot. This is especially challengingfor multi-robot teams operating in communication-constrainedenvironments, such as marine robots.

We present a novel approach for multi-robot distributedlearning in which each robot maintains a local topic model tocategorize its observations and model parameters are shared toachieve global consensus. We apply a combinatorial optimiza-tion procedure that combines local robot topic distributionsinto a globally consistent model based on topic similarity,which we find mitigates topic drift when compared to abaseline approach that matches topics naı̈vely. We evaluateour methods experimentally by demonstrating multi-robot un-derwater terrain characterization using simulated missions onreal seabed imagery. Our proposed method achieves similarmodel quality under bandwidth-constraints to that achieved bymodels that continuously communicate, despite requiring lessthan one percent of the data transmission needed for continuouscommunication.

I. INTRODUCTION

Unsupervised machine learning techniques can enableadaptive robotic systems that are robust to unexpectedchanges in their environment. Furthermore, the operationof robots in environments like the deep benthic sea, wherethey may encounter species and terrains not previouslydocumented, demands the flexibility to novel observationsafforded by unsupervised learning. A team of underwaterrobots capable of unsupervised scene categorization wouldenable more efficient ocean surveying and exploration. How-ever, because unsupervised learning methods do not have apriori defined labels, combining the local models discov-ered by individual robots into a globally cohesive modelis a hard combinatorial optimization problem. Achievinga globally cohesive scene model is critical for both post-mission analysis and so that multi-robot teams can makebetter global planning and exploration decisions in real-time.This paper presents a novel approach to multi-robot learning

1Computer Science and Artificial Intelligence Laboratory (CSAIL), Mas-sachusetts Institute of Technology, 77 Massachusetts Ave, MA 02139, USA{kjd, geflaspo, nickroy}@csail.mit.edu

2Department of Applied Ocean Physics and Engineering, Woods HoleOceanographic Institution, 86 Water St, Woods Hole, MA 02543, USAyogi@whoi.edu

Fig. 1: Illustration of the multi-robot terrain charac-terization problem. Each individual AUV builds its ownlocal topic model. Model parameters from all vehicles aretransmitted top-side via acoustic modem, where the modelsmust be combined into a consistent global model. However,there is no guarantee that the individual robots will agreeon which visual category corresponds to each topic ID. Thiscorrespondence issue must be addressed to achieve globalconsensus.

that obtains a globally consistent scene categorization undercommunication constraints, endowing a team of exploratoryrobots with the ability to reach consensus in their semanticdescription of the world.

In this work, we consider the problem of underwater ter-rain characterization. Formally, given a sequence of images,we would like to predict for each image the distribution overa set of latent categories that generated the data. Previousapproaches to this problem for single robots [1] make use ofspatiotemporal variants of topic models like latent Dirichletallocation (LDA) [2] and the nonparametric hierarchicalDirichlet process (HDP) [3]. The extension of these methodsto the multi-robot setting presents several challenges. We aimto have every robot build a model that discovers thematicstructure within an image stream that coincides well withhuman semantics, but also for each robot’s model to beconsistent with the other robots in a multi-robot exploration

team. We define consistent topic models to be those withsimilar topic-word distributions.

Semantically meaningful, consistent topic models can beachieved by allowing the robots to share raw sensor obser-vations directly with one another continuously; however, inmany practical scenarios this approach is infeasible due tocommunication constraints. Additionally, it does not allowthe individual robots to build their own models for localcontrol. We opt instead to build a local topic model on eachrobot and provide an approach allowing the correspondencesbetween topic models to be identified and fused withoutsharing the raw sensor observations. An illustration of ourproblem scenario is depicted in Figure 1, where we showseafloor exploration with multiple autonomous underwatervehicles (AUVs) communicating acoustically with a centraltop-side modem. Topic models learned by each robot aretransmitted top-side, where the models are combined tobuild a consistent global model that is then sent back toall vehicles.

The primary contributions of this work are as follows:• We present a novel algorithm for multi-robot real-time

online spatiotemporal topic modeling.• We experimentally validate our approach by simulating

a multi-robot underwater mission where we seek tocategorize terrains and fauna observed on the seafloorwithout human supervision or a priori defined labels.

• We examine the performance of the proposed approachunder different communication constraints, and showthat our approach produces consistent topic modelseven under substantial communication delays, as wellas when one robot observes topics that go completelyunobserved by another.

• We additionally show that the proposed model achievessimilar model quality under bandwidth-constraints tothat achieved by models that continuously communi-cate, despite requiring < 1% of the data transmissionneeded for continuous communication.

To our knowledge, this work is the first application ofdistributed topic models to streaming data from a multi-robotmission. Consequently, this work provides new avenues forintelligent, coordinated multi-robot exploration.

The remainder of the paper is organized as follows: inSection II we discuss related work in machine learning androbotics. In Section III we briefly describe the latent Dirichletallocation topic model and real-time online spatiotemporaltopic modeling (ROST) [1], and then present our approach,approximate distributed ROST (AD-ROST). In Section IVwe present both qualitative and quantitative results from topicmodeling experiments on a simulated multi-robot missionusing data collected at the Hannibal Bank Seamount off thecoast of Panama. Finally, in Section V we discuss severalcompelling directions for future work in this area.

II. RELATED WORK

Unsupervised machine learning models have been of in-terest in the robotics community for several years. In the area

of visual topic models, both parametric and nonparametricmodels have been used for analysis of seabed imagery [4],[5], underwater exploration [6], [1], and navigation [7]. Ourapproach is based on the realtime online spatiotemporal topicmodeling (ROST) framework developed by Girdhar et al. in[1]. More recently, the seabed imagery analysis and anomalydetection work in [5] was adapted to use features learned bya deep convolutional auto-encoder [8]. The parametric visualtopic models developed in the aforementioned works couldall be directly extended to the multi-robot setting using ourproposed methods.

Within the machine learning community, there are sev-eral approaches for distributing topic models. Newman etal. [9] provide methods for distributing LDA and HDPon multiple processors in an offline setting. Our baselineapproach is an adaptation of their approximate distributedLDA algorithm to the online setting, while inheriting thespatial and temporal modeling capabilities of ROST. Thebipartite matching approach we take toward addressing thecomponent correspondence issue is grounded in methods formulti-processor learning, and has been applied in distributedlearning literature to LDA and HDP in centralized, sampling-based [9] and variational inference [10], [11], as well as inthe decentralized case [12].

Our problem differs from the traditional distributed learn-ing setting fundamentally in that our data is partitionednaturally by virtue of being collected on independent robots.In contrast, in a traditional learning setting, we would haveaccess to the full dataset and partition the data into mini-batches across multiple processors, for example by samplingrandomly from the dataset. Our data also contains signifi-cant spatial and temporal correlations, since they consist ofconsecutive observations from a sensor stream taken as arobot navigates through the environment. Beyond that, weare subject to the practical constraints imposed by acousticcommunication on our ability to send updates to a globalmodel. The consequence of this is not only that topic drift ismore likely, but also that some robots may make observationsof visual categories that have not been observed by others.Our work focuses on addressing these topic correspondenceissues as they apply to learning on multi-robot systems.

III. METHODS

In the following sections, we review the latent Dirichletallocation (LDA) topic model and its extension to spa-tiotemporal domains, highlighting the permutation symmetryproblem as it pertains to these two models. We then providetwo methods for multi-robot topic modeling. The first is anaı̈ve adaptation of parallel LDA for multiple processors [9]to the multi-robot setting, in which we merge topics directlyby their ID, without considering topic consistency. In thesecond approach, we explicitly account for the permutationsymmetry problem during inference in order to improveglobal consistency between models.

Fig. 2: Graphical model representation of the ROST frame-work. ROST incorporates both spatial and temporal relation-ships into the traditional LDA formulation.

A. Latent Dirichlet Allocation

The latent Dirichlet allocation (LDA) topic model [2] isused to extract latent thematic structure from a corpus ofdocuments. In the LDA model, a document d consists of aset of discrete words {w1, w2, . . . , wN} from a vocabularyof size V . In LDA, we infer the distribution over a set oflatent “topics” for each document and the likelihood of allwords given these topics by computing a topic assignmentzi for each word wi in a document. With this factorizationinto the distributions p(zi|d) and p(wi|zi), the likelihood ofan observed word wi given its document d is computed bymarginalizing over the topics as follows:

p(wi | d) =

K∑k=1

p(wi | zi = k)p(zi = k | d). (1)

We will often abbreviate the topic mixing proportions fora document p(z|d) as θd and the per-topic word distributionp(w|z = k) as φkw. Dirichlet priors are placed over bothdistributions, i.e. θd ∼ Dir(α) and φkw ∼ Dir(β) where αand β are hyperparameters.

B. Real-time Online Spatiotemporal Topic Models

The traditional formulation of LDA does not considercorrelations between observed words in a document. SpatialLDA [13] incorporates spatial correlations between visualwords within an image. The real-time online spatiotemporaltopic model (ROST) [1] further extends this method to imagestreams by additionally incorporating correlations betweenvisual word observations within the same temporal neigh-borhood. In the ROST framework, depicted graphically inFigure 2, we consider a document, here denoted Gi, as theset of all observed words in a spatiotemporal neighborhood.Neighborhoods are determined by cells of fixed-size in spaceand time. Similar to LDA, ROST places Dirichlet priors overφkw and the topic mixing proportions, which we denote θGi

to make explicit the spatiotemporal context. The likelihoodof a word wi can then be written

p(wi | x, t) =

K∑k=1

p(wi | zi = k)p(zi = k | x, t) (2)

Algorithm 1 AD-ROST-IDrepeat

// Local model updatesfor each robot r in parallel do

// Receive global counts NkwNkwr ← Nkwfor t from tcurr to tcurr + T do

w,x, t← ExtractWords(It)z, Nkwr ← RefineTopics(z,w,x, t)

end forend for// Global model updatesSynchronize // Retrieve each NkwrNkw ← Nkw +

∑r(Nkwr −Nkw)

until no new observations

where x and t denote the location of wi in an image and thetime of its observation, respectively.

Exact inference in the ROST model and LDA more gener-ally is intractable. Instead, we perform approximate inferenceusing Gibbs sampling [14]. Gibbs sampling is performed bysampling a topic zi for every word wi = v conditioned onthe set of all words w and the current set of topics assignedto all other words, denoted z−i. This sampling distributionis computed as follows:

p(zi = k | z−i,w) ∝

[n(v)k,−i + β

n(·)k,−i + V β

][n(Gi)k + α

n(Gi)−i +Kα

], (3)

where n(v)k,−i is the count of assignments of topic k to everyother observation of word v, n(·)k,−i is the count of all currentassignments of topic k, n(Gi)

k is the count of all assignmentsof topic k in spatiotemporal neighborhood Gi, and n(Gi)

−i isthe corresponding total count of topic assignments to all otherwords in Gi. Gibbs sampling is performed continuously asnew images are observed by the robot.

The maximum-likelihood estimates for the per-topic worddistribution and topic mixing proportions can be calculatedusing the counts recorded during Gibbs sampling as follows:

φ̂kw =n(w)k + β

n(·)k + V β

(4)

θ̂Gik =n(Gi)k + α

n(Gi) +Kα. (5)

C. Approximate Distributed ROST with Topic IDs

The marginalization over topics in the generative modelsof LDA and ROST (Equations 1 and 2 respectively) causesthe likelihood of a word to be invariant to the permutationof topics in these models. For a single topic model, thispermutation symmetry does not warrant consideration. Whenwe seek to distribute the model, however, this permutationsymmetry becomes vital: two models can be equivalent interms of the likelihood, but be inconsistent in terms of topic

correspondence. This problem of resolving correspondencebetween latent variables in unsupervised models is a generalproblem in distributed unsupervised learning.

The baseline approach that we consider is an adaptationof the parallel LDA formulation of Newman et al. [9] tothe ROST model. We make the naı̈ve assumption that thetopic IDs assigned by each robot directly correspond, i.etopic 1 on robot 1, topic 1 on robot 2, through topic 1 onrobot R are topics corresponding to the same visual category,where R is the number of robots. Under this assumption, itis straightforward to derive a global update rule based onthe per-topic word counts for each agent. Letting Nkw bethe global K × V matrix of per-topic word counts obtainedfrom Gibbs sampling across all agents, and letting Nkwr bethe per-topic word count matrix on robot r, we combine thelocal counts by summation, taking care to avoid duplicatesof the global count matrix:

Nkw ← Nkw +R∑r=1

(Nkwr −Nkw), (6)

where R is the number of robots. The complete procedure fordistributing ROST with ID-based topic matching, AD-ROST-ID, is described in Algorithm 1. Each robot r receives theglobal count matrix Nkw, then begins processing the next Timages, where T is a fixed number of observations betweenglobal synchronizations of all robot models. The procedureExtractWords produces a set of visual words consisting ofextracted feature descriptors. RefineTopics then performsGibbs sampling over the data given the new observations.After we have added T new observations, we synchronizemodels by communicating local counts to a central node,which combines the counts according to the update in Equa-tion 6. On the next cycle, each robot retrieves the new globalcount matrix, replacing its local model with the global model.We repeat this process until every agent ceases to recordnew observations. If one agent finishes before the others, itsfinal updates are made and the remaining agents proceed asusual, relying on the central node to handle synchronizationappropriately.

It may seem counter-intuitive to assume that modelscombined by ID in this manner would converge to a globallyconsistent set of topics describing the same visual categories.Consistency between models is achieved by the repeatedmixing of model parameters into an averaged global modeland the resetting of the local models. When mixing hap-pens frequently enough, ID-based matching performs Gibbssampling from an approximation of the true posterior forthe combined dataset [9]. As a consequence, successfullyarriving at the appropriate topic correspondences in the caseof ID-based topic matching rests on the frequency of globalmodel updates. In a traditional distributed learning setting,it is not unreasonable to expect that we can synchronizeprocessors after only a few iterations of Gibbs sampling.However, on a team of robots, communication delays maybe substantial, and models may converge to different sets oftopics locally between synchronizations.

Algorithm 2 AD-ROST-SIMrepeat

// Local model updatesfor each robot r in parallel do

// Receive global counts NkwNkwr ← Nkwfor t from tcurr to tcurr + T do

w,x, t← ExtractWords(It)z, Nkwr ← RefineTopics(z,w,x, t)

end forend for// Global model updatesSynchronize // Retrieve each Nkwrfor each robot r do

Cr1 ← ComputeCost(φ̂kwr, φ̂kw1)π∗r ← Hungarian(Cr1)

end forNkw ← Nkw +

∑r π∗r (Nkwr −Nkw)

until no new observations

D. Approximate Distributed ROST with Topic Similarity

In order to mitigate the potential topic drift issues causedby naı̈ve ID-based matching in a direct adaptation of parallelLDA [9], we now propose an alternative approach. We wouldlike to obtain local topic models on each robot that areconsistent in the sense that their topics are appropriatelyaligned and the same topic ID on multiple robots correspondsto the same semantic visual category. Accounting for thisconsistency should result in a visual topic model that bettercorresponds with human semantics. If the probability distri-bution over the vocabulary is similar for two topics, we saythat the topics are similar. We adopt the heuristic in [11],measuring the similarity between two topic models in termsof the sum of squared distances between these distributions1

D(φ̂kw, φ̂′kw) =

K∑k=1

‖φ̂kw − φ̂′kw‖2. (7)

Given this measure of model similarity, we pose modelconsistency as an explicit optimization objective during theinference procedure. Each time the global model is updated,we permute each robot’s topic-word distribution to optimizemodel consistency. There are K! possible permutations oftopics each time we perform model synchronization. If wesynchronize S times during a mission, there are K!S totalpossible sequences of permutations over the entire mission.Consequently, it is intractable to maintain a full distributionover all possible permutations of topics for every model syn-chronization. Instead, we optimize the permutations greedilyat every model update.

Let φ̂kwr denote the empirical topic-word distribution onrobot r and let π(φ̂kwr) denote a permutation of topics in

1Although we use the sum of squared Euclidean distance heuristic, othermeasures of topic similarity for related models have been presented, suchas symmetric Kullback-Leibler divergence [9].

φ̂kwr. We aim to solve the following optimization:

π∗r = arg minπ(φ̂kwr)

f(π(φ̂kwr)) (8)

f(π(φ̂kwr)) =

K∑k=1

‖π(φ̂kwr)− φ̂kw1‖2, (9)

where π∗r is the optimal permutation of the topic-worddistribution on robot r. In our case, the optimal permutationminimizes the sum of squared Euclidean distances betweenthe topic-word distributions on robot r and those on the firstrobot. Since we optimize permutations with respect to thefirst robot, we take π∗1 to be the identity.

We minimize this objective for R robots using the Hungar-ian algorithm [15], which provides the optimal assignmentin O(K3), giving an overall complexity of O(RK3) for theoptimization procedure at each model synchronization step.

After finding the optimal permutation for each robot, π∗r ,we update the global topic-word counts according to thefollowing rule:

Nkw ← Nkw +

R∑r=1

π∗r (Nkwr −Nkw), (10)

where, as before, Nkw and Nkwr are the global and localcounts of topic k for word w, respectively. After subtractingthe set of previous global topic-word counts, we permute themodel updates from each robot using its respective optimalpermutation from Equation 8. The new update is exactly theupdate from Equation 6 when π∗r is identity for all r, whichis the only scenario where ID-based matching is optimal withrespect to the objective posed in Equation 9.

This approach, which we call AD-ROST-SIM, is summa-rized in pseudocode in Algorithm 2. The individual vehiclesperform the same operations as in the case of AD-ROST-ID.The primary distinction is that the central node, after receiv-ing the model parameters of the robots, then computes foreach robot the pairwise distances between word distributionsfor each topic on that robot and that of the first robot (i.e. theprocedure ComputeCost). Optimization of costs producesthe optimal permutation of topics on robot r to match thetopics on the first robot. Differences in counts are permuted,then incorporated into the global model by summation.

IV. EXPERIMENTAL RESULTS

A. Experimental Setup

We compare the proposed distributed learning approachesusing real video data collected by a single AUV equippedwith a downward-facing camera at the Hannibal BankSeamount, off the coast of Panama [16]. To simulate a multi-robot mission, we treat data from multiple deployments ofa single robot as though it were retrieved simultaneouslyby multiple vehicles. Each simulated “robot” communicateswith a central node to update a globally synchronized modelin order to produce a consistent scene categorization modelshared between all agents in the system. To aid in evaluation,each mission has been annotated using a set of thirteen

(a) Topic distributions predicted by AUV 1 and AUV 2 usingthe AD-ROST-ID method. AD-ROST-ID results in different visualcategories being assigned to the same topic.

(b) Topic distributions predicted by AUV 1 and AUV 2 usingthe AD-ROST-SIM method. Matching by topic similarity helps toensure that distinct visual categories are appropriately assigned todifferent topics.

Fig. 3: Comparison of topic distributions inferred by AD-ROST-ID (a) and AD-ROST-SIM (b) with global updatesevery 100 observations. Neglecting to consider permutationsymmetry in this limited communication setting results indifferent visual categories being placed in the same topic.Accounting for permutation symmetry with AD-ROST-SIMcauses allocation of separate topics for these categories.

possible ground-truth terrain labels, including “crab congre-gation,” “water column,” and “sparse rocks.”

In the first simulated mission, the first deployment, “AUV1,” contains 2,296 image frames, each taken four secondsapart over approximately 2.5 hours. In this mission, the robotdescends through the water column until it arrives near theseafloor, which primarily consists of smooth, sandy terrains.Notably, however, these visually nondescript sandy terrainsare punctuated by several dense crab swarms [16]. Finally,the vehicle ascends through the water column. The seconddeployment, “AUV 2,” contains 1,117 image frames takenfour seconds apart, with a total duration of approximately1.25 hours. In the second mission, the robot observes severaldistinct seafloor terrains. It initially descends through thewater column, then observes a rocky terrain, followed byporous, sandy seafloor, before ascending back through thewater column. Since the second AUV mission ends beforethe first AUV mission, the vehicle stops sending updates tothe global model and the first vehicle continues its missionalone until it receives no new observations.

In the second simulated mission, the first deployment,

Fig. 4: AD-ROST-SIM achieves higher mutual informationthan AD-ROST-ID as we increase the period between modelsynchronizations. Despite requiring a fraction of the datatransmission, the mutual information of AD-ROST-SIM withT = 250 is comparable to that of either method with T = 1.

“AUV 1,” contains 1,244 frames over about 1.25 hours inwhich the robot descends through the water column, travelsover sandy and rocky seafloor terrains, and then ascends backthrough the water column. The second deployment, “AUV2,” contains 2,733 image frames with a total duration ofapproximately 3 hours. In this deployment, after the vehicle’sdescent, it simply observes a variety of nondescript sandyterrains over the course of the mission, before ascending backthrough the water column. In this mission, the first AUVdeployment finishes before the second, so the first vehiclestops sending updates to the global model while the secondvehicle continues.

Since our focus is on the relative performance of eachmethod under a variety of communication constraints, wefixed the hyperparameters K = 7, α = 0.1, and β = 10 in allexperiments. In practice, hyperparameters may be optimizedin cross-validation by comparing mutual information be-tween maximum-likelihood topics and a set of ground-truthlabels, if annotations are available [8]. We evaluated modelswith T = 1, 20, 40, 60, 80, 100, and 250 observationsbetween global model updates. Because the Gibbs samplingprocess may produce different estimates when run multipletimes on the same data, we trained each model three timesand show average performance and standard deviation for allquantitative results.

The topic models used in our experiment make use ofvisual words consisting of ORB features [17]. Our approachis also compatible with other methods of feature extraction,including features learned by a deep neural network [8]. Thevisual vocabulary we use has size V = 6500 and was builtusing k-means clustering of ORB features extracted fromunrelated video. At run time, ORB features extracted from

Fig. 5: Sum of squared differences in topic-word distributionson each model evaluated on the first mission. As we increasethe number of observations between model synchronizationsteps, the topic distributions begin to diverge in the caseof AD-ROST-ID. AD-ROST-SIM is comparatively robust todecreases in communication rate.

images count as instances of their nearest visual vocabularyword, quantified by Euclidean distance.

B. Evaluation Metrics

We tested the correspondence between the discoveredtopics and human annotation by measuring the mutual in-formation between the most likely topic labels for eachmission and ground-truth annotations. Mutual information isthe amount of entropy reduced in a random variable Y afterthe observation of a random variable X , and is formallydefined as

I(X;Y ) =∑x

∑y

p(x, y) logp(x, y)

p(x)p(y). (11)

Mutual information here measures the degree to whichthe maximum-likelihood topic labels correlate with anno-tations. Specifically, normalized mutual information equalto 1 signifies that maximum-likelihood topics are in one-to-one correspondence with annotated labels. A normalizedmutual information score of zero indicates the opposite:that the maximum-likelihood topics and annotated labels arestatistically independent.

We quantify topic divergence by examining the similaritybetween the topic-word distributions for the two vehicles atthe end of the mission. Specifically, we measure the sum ofsquared distances between topic-word distributions for eachtopic, consistent with the heuristic distance D(φ̂kw, φ̂

′kw)

posed in Equation 7. A large value of D(φ̂kw, φ̂′kw) indicates

that the pair of models has drifted from one another, resultingin dissimilar topic-word distributions.

C. Results and Discussion

In Figure 3 we show, for both vehicles in the first mission,the topic distributions produced by AD-ROST-ID and AD-

Fig. 6: Sum of squared differences in topic-word distri-butions on each model evaluated for the second mission.As communication becomes less frequent, AD-ROST-IDquickly begins to show topic model divergence that persiststo 250 iterations between synchronizations. AD-ROST-SIMperformance degrades slightly with 250 iteration gaps incommunication, but consistently performs much better thanAD-ROST-ID.

ROST-SIM with synchronizations every 100 observations.The portion of the mission highlighted in Figure 3 demon-strates that when updates to the global model are infrequent(e.g. T = 100), merging topics naı̈vely based on ID (AD-ROST-ID) causes topic distributions to drift. This problemis most apparent when comparing the maximum-likelihoodtopics near t = 950 on the first AUV and t = 400 on thesecond AUV. The second AUV observes a rocky terrain att = 400, but at t = 950, the same maximum-likelihoodtopic corresponds to a crab colony on AUV 1. On theother hand, merging topics based on similarity (AD-ROST-SIM) appropriately places these distinct visual categories intodifferent topics, despite also synchronizing the models onlyevery 100 observations.

In Figure 4, we show the mutual information between themaximum-likelihood topic predictions and a set of humanannotations of the data from the first mission. We observethat as we increase the period between synchronizations,the performance of AD-ROST-ID degrades. With AD-ROST-SIM, we achieve nearly the same model quality when wesynchronize every 250 observations as when we synchronizeafter every observation, despite the fact that we transmit lessthan one percent of the data required to synchronize afterevery observation.

In Figure 5, we show the sum of squared differencesbetween the per-topic word distributions on each robot atthe end of the first mission as we increase the number ofobservations between global updates. We observe that thesum of squared model differences begins to grow in the caseof AD-ROST-ID. This quantifies the phenomenon that weobserved in Figure 3, when the same topic was assigned to

(a) Topic exemplars, grouped by topic ID, from the model inferredusing AD-ROST-ID with T = 100 show several instances wherethe robots assign the same topic to visually different categories.

(b) Exemplars from the model inferred using AD-ROST-SIM withT = 100 show that the model successfully assigns differenttopics to different visual categories. Additionally, when a particularcategory is observed by one AUV but not the other, such as in thecase of the “crab congregation,” there is no image from AUV 2where the corresponding topic occurs with maximum likelihood.

Fig. 7: Annotated exemplar images for each topic. Topicsthat did not occur with maximum-likelihood on any imageare denoted “No ML Image.” This occurs when a visual cat-egory is only observed by one of the two AUVs. Mismatcheswhere the individual robot topic models assign the same topiclabel to images of visually distinct categories arise with AD-ROST-ID, but not when using AD-ROST-SIM.

both images of a crab colony and the rocky seafloor. Consis-tent with our previous observations, AD-ROST-SIM is robustto infrequent model communication up to intervals of 250observations between synchronizations, which correspondsto synchronization roughly every 15 minutes.

For the second mission, we show the sum of squareddifferences in per-topic word distributions on each of the twovehicles in Figure 6. We similarly observe topic divergence inthe case of AD-ROST-ID, which happens much more quicklythan in the first mission and persists to 250 iterations betweenglobal model synchronizations. Distance in the topic modelsgrows in the case of AD-ROST-SIM as well around 250iterations, but the resulting model similarity is comparableto AD-ROST-ID with synchronization every iteration.

Another way to examine the topics learned by thesemodels is through “exemplar images” of each topic. Tofurther compare the effects of model drift on the topiccorrespondences between the two vehicles, we provide high-likelihood exemplars from four of the discovered topics inthe first mission in Figure 7. Topics that never occur withmaximum-likelihood are not displayed. We show that AD-

ROST-ID incorrectly matches several topics between the twovehicles. Most notably, we observe again the phenomenonwhere AUV 1’s topic representing a congregation of crabsis matched to a rocky terrain image on AUV 2. Using AD-ROST-SIM, we see that the topic exemplified by an image ofcrabs has no maximum-likelihood match on AUV 2. Sincewe know that the crab congregation never appears throughoutthe second AUV’s mission, this is appropriate. Furthermore,the remaining topic exemplars from the similarity-basedmatching procedure are in direct correspondence with oneanother, in contrast to AD-ROST-ID.

V. FUTURE WORK

The multi-robot topic modeling framework presented hereenables many interesting directions for future research. Weprovide a multi-agent topic modeling framework, enablingdistributed variants of unsupervised exploration algorithms[1], [5], or perplexity-based navigation methods, as in [7].In general, leveraging our globally consistent, unsupervisedmodels for multi-agent robotic exploration is a compellingarea for future research.

We have not considered the situation where robots cancommunicate directly with one another absent a global,centralized node. Application of decentralized variants oftopic models, either using variational inference [12] or Gibbssampling [18], to the multi-robot domain would greatlyimprove the robustness of these methods to unpredictablecommunication delays or node failures.

Another avenue for future work is the extension of thesemethods to the case where the number of topics is notchosen a priori and consistently across robots, such as in thecase of the hierarchical Dirichlet process (HDP) model [3].A topic matching procedure similar to Algorithm 2 couldbe used with HDP models in communication-constrainedsettings, for example by allocating new global topics usinga threshold on distance between topic-word distributions asin [9]. The notion of category discovery is vital for the sortof lifelong learning required of systems operating in novelenvironments, such as in underwater exploration. Extensionof our methods to nonparametric unsupervised models is apriority for further work on this problem.

VI. CONCLUSION

We have presented a novel algorithm for multi-robot real-time online spatiotemporal topic modeling. We applied ouralgorithm in the context of underwater terrain characteriza-tion. By explicitly incorporating an optimization procedureto match topics during the distributed inference process,we achieve more globally consistent models despite therobots’ having never shared their raw sensor observationsdue to communication constraints. Beyond resolving issuesdue to topic drift, we also found that our method is robustto situations where certain visual categories were observedby one robot but not by the other. Finally, enforcing topicconsistency can lead to improved correspondence betweenmaximum-likelihood topic labels and human annotations,as measured by mutual information. The proposed method,

AD-ROST-SIM, infers semantically meaningful, consistenttopics comparable to those of models requiring constantcommunication, while transmitting substantially less data.

ACKNOWLEDGMENTS

We thank James Kinsey, Brian Claus, and Trevor Campbellfor helpful discussions of ideas presented in this paper.This project was supported in part by National ScienceFoundation (NSF) NRI Award #1734400, and NOAA Award#NA17OAR0110207.

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