# Near-Optimal Budgeted Data Exchange for Distributed Loop Closure Detection

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3 metadata reveals a number of potential matches. Each potential match comes with a similarity score [10] or an occurrence probability (estimated directly or obtained by normalizing the similarity scores) that estimates the likelihood of that potential match corresponding to a true loop closure. The result can be naturally represented as an exchange graph [12]. Definition 2 (Exchange Graph). An exchange graph [12] is a simple undirected graph G x = (V x,e x ) where each vertex v V x corresponds to an observation, and each edge {u,v} E x corresponds to a potential inter-robot loop closure between the corresponding observations. Furthermore, G x is endowed with w : V x R >0 and p : E x (0,1] that quantify the size of each observation, 1 and the occurrence probability of an edge, respectively. We make the simplifying assumption that edges occur independently. Note that in an n-rendezvous, the exchange graph will be an n-partite graph. Figure 1a illustrates a simple exchange graph with n = 3. For simplicity, we assume that a single robot participating the rendezvous (known as the broker in [12]) is responsible for forming the exchange graph and solving the subsequent optimization problem introduced in Section III-C. A fully decentralized approach is left for future work. B. Optimal Lossless Data Exchange After identifying potential matches, robots must exchange full observations for geometric verification and relative pose estimation [12]. This stage is substantially more communication-intensive than the metadata exchange phase as it involves the transmission of actual observations (e.g., the complete set of keypoints in an image). In [6,7], each robot sends its full query only to the owner of the most promising potential match. This heuristic obviously comes at the risk of losing some loop closures. Our recent work [12] presents an efficient algorithm for finding the optimal data exchange policy that minimizes data transmission among all lossless (i.e., allows robots to verify every potential match) exchange policies in 2-rendezvous. In [12] we demonstrate that by forming the exchange graph and exploiting its unique structure one can come up with intelligent resource-efficient data exchange policies which were not possible in [6,7]. In particular, for 2-rendezvous we prove that in the optimal exchange policy, agents share the observations that correspond to the minimum weighted vertex cover of the exchange graph, C w (E x ) V x [12]. Thus, the minimum amount of data transmission required for verifying all potential matches is given by c w (E x ) v C w(e x) w(v). Here we note that this result can be trivially generalized to the case of n-rendezvous. Figure 1b illustrates an example. The key difference between 2-rendezvous and n-rendezvous is that finding the optimal lossless exchange policy (i.e., the minimum weighted vertex cover) is NP-hard for general n-rendezvous (i.e., general n-partite graphs) when n 3. Nevertheless, by rounding the solution of linear programming (LP) relaxation 1 For instance, this can be the number of keypoints in an image. we obtain a lossless exchange policy with at most 2 c w (E x ) data transmission [40]. C. Data Exchange under Budgeted Communication Intrinsic limitations of the communication channel (e.g., bandwidth in underwater acoustic communication [31]), operational constraints, and/or resource scheduling policies (e.g., enforced to regulate resource consumption) necessitate and lead to budgeted communication. We are specifically interested in situations where the data transmission budget b is strictly less than c w (E x ). Based on our earlier remarks, one can immediately conclude that in this regime, verifying all potential matches without exceeding the budget b becomes impossible. Therefore, resource-adaptation and task-oriented prioritization are inevitable. In what follows, we approach this problem from two perspectives. It will become clear shortly that these two seemingly different viewpoints lead to two problem statements that are intimately connected. 1) Edge Selection: From the perspective of measurement selection [3,18,19], we seek to select a subset of potential loop closures such that the minimum amount of data transmission needed to verify them is at most b. We need a task-oriented objective f e : 2 Ex R 0 that quantifies the value of all subsets of potential loop closures. Note that f e must take into account the stochastic nature of potential matches, captured by the edge weights p : E x (0,1] in G x. Later in this section we discuss several suitable choices of f e for CSLAM and place recognition. This perspective leads to the following problem statement: maximize f e (E) s.t. c w (E) b, (P e ) E E x where c w (E) gives the minimum amount of data transmission needed to verify E (see Section III-B) and c w (E) b imposes a global communication budget on the rendezvousing robots. Remark 1. Merely deciding whether a given E E x is P e - feasible, (i.e., there exists a vertex cover for E with a size of at most b), is an instance of the weighted vertex cover problem which is NP-complete [16] in n-rendezvous with n 3. Put differently, deciding whether a set of potential loop closures can be verified by exchanging at most b units of data is NPcomplete when n 3. 2) Vertex Selection: From the perspective of data exchange polices, one may naturally search for an optimal subset of vertices (observations) that need to be broadcasted without exceeding the budget b. Recall that once an observation is broadcasted, other robots can collectively verify all potential loop closures involving that observation (i.e., incident to the selected vertex); see Figure 1 and [12]. This motivates the following objective function, f v : 2 Vx R 0 : V f e ( edges(v) ). (1) where edges : 2 Vx 2 Ex gives the set of all edges such that for each edge at least one end is in the given subset of vertices. This intuitively means that the value of any subset of vertices

4 is equal to the f e -value of all edges incident to those vertices. This leads to the following optimization problem: maximize V V x f v (V) s.t. v V w(v) b, (P v ) Note that the budget constraint in P v is simpler than what we encountered in P e. Needless to say, P e and P v are closely related. The connection between these two problems is discussed and exploited in Section IV. D. Objective Function Definition 3. A set function f : 2 W R 0 for a finite W is normalized, monotone, and submodular (NMS) if it satisfies the following properties: Normalized: f( ) = 0. Monotone: for any A B, f(a) f(b). Submodular: for any A B and u W\B, f(b {u}) f(b) f(a {u}) f(a). We focus only on cases where f e is NMS. In what follows, we briefly review two examples of such functions used recently for measurement selection in the context of SLAM and VIN [3,18,19]. In addition, we consider also a third objective that is suitable for general place recognition scenarios. Note that our framework is compatible with any NMS objectives, and is not limited to the instances considered below. 1) D-optimality Criterion The D-optimality design criterion (D-criterion), defined as the log-determinant of the Fisher information matrix (FIM), is one of the most popular design criteria in the theory of optimal experimental design with well-known geometrical and information-theoretic interpretations; see, e.g., [15,33]. D- criterion has been widely adopted in many problems including sensor selection [15,35] and measurement selection in SLAM [3,18]. Let I init 0 denote the information matrix of the joint CSLAM problem before incorporating the potential loop closures. Moreover, let I e = J e Σ 1 e J e 0 be the information matrix associated to the candidate loop closure e E x in which J e and Σ denote the measurement Jacobian matrix and the covariance of Gaussian noise, respectively. Following [3], one can approximate the expected gain in the D-criterion as: ( f FIM (E) log det I init + ) p(e) I e log det I init. (2) e E f FIM is NMS [3,35]. 2) Tree-Connectivity The D-criterion in 2D pose-graph SLAM can be closely approximated by the weighted number of spanning trees (WST) hereafter, tree-connectivity in the graphical representation of SLAM [17]. In [18,19] tree-connectivity is used as a graphical surrogate for the D-criterion for (potential) loop closure selection. Evaluating tree-connectivity is computationally cheaper than evaluating the D-criterion and, furthermore, does not require any metric knowledge of robots trajectories. Let t wp (E) and t wθ (E) denote the weighted number of spanning trees in a pose-graph specified by the edge set E whose edges are weighted by the precision of the translational and rotational measurements, respectively [19]. Furthermore, let E init be the set of edges that exist in the CSLAM pose-graph prior to the rendezvous. Define Φ(E) 2 log E [ t wp (E init E) ] + log E [ t wθ (E init E) ] where expectation is with respect to the anisotropic random graph model defined in Definition 2; see [19]. Khosoussi et al. [19] then seek to maximize the following objective: f WST (E) Φ(E) Φ( ). (3) It is shown in [18,19] that f WST is NMS if the underlying pose-graph is connected prior to the rendezvous. 3) Expected Number of True Loop Closures The previous two estimation-theoretic objective functions are well suited for CSLAM. However, in the context of distributed place recognition, one may simply wish to maximize the expected number of true loop closures (NLC) between the agents. From Definition 2 recall that the probability associated to a potential loop closure e E x is p(e). Consequently, the expected number of true loop closures in a subset of edges E can be expressed as: { e E p(e) E, f NLC (E) 0 E =.. (4) f NLC is clearly NMS. IV. ALGORITHM AND THEORETICAL GUARANTEES This section presents approximation algorithms for our two perspectives P v and P e with performance guarantees that hold for any NMS f e and the corresponding f v. Theorem 1. For any NMS f e, the corresponding f v is NMS. This theorem implies that P v is an instance of the classical problem of maximizing an NMS function subject to a knapsack constraint. Although this class of problems generalizes the maximum coverage problem [13] and thus are NP-hard in general, they enjoy a rich body of results on constant-factor approximation algorithms [20]. 2 We discuss these algorithms in more detail and show how they can be applied to P v in Section IV-A. Now, we show that by establishing a simple approximation factor preserving reduction from P e to P v, we can also obtain constant-factor approximation schemes for P e. Let OPT e and OPT v be the optimal values of P e and P v, respectively. The following lemmas shed more light on the connection between our two perspectives P e and P v. Lemma 1. 1) For any P v -feasible V, edges(v) is P e -feasible. 2) OPT e = OPT v. 2 Recall that an α-approximation algorithm for a maximization problem is an efficient algorithm that produces solutions with a value of at least α OPT for a constant α (0,1).

5 Using the above lemma, we establish the following approximation factor preserving reduction from P e to P v. It is worth mentioning this reduction belongs to a strong class known as S-reductions [2,9]. Theorem 2. Given an α-approximation algorithm for P v, the following is an α-approximation algorithm for P e : 1) Run the α-approximation algorithm on P v to produce Ṽ. 2) Return edges(ṽ). Theorem 2 illustrates how constant-factor approximation algorithms for P v can be used as a proxy to obtain constantfactor approximation algorithms for the dual perspective in P e. In Section IV-A, we show how the interplay between P v and P e can be exploited in certain situations to further improve the approximation guarantees of our algorithms. A. Approximation Algorithms So far, we have shown that near-optimal solutions to P v and by virtue of Theorem 2 P e can be obtained using constant-factor approximation algorithms for maximizing NMS functions under a knapsack constraint. Some of these algorithms are listed in Table I; see [20] for a comprehensive survey. The approximation factors obtained by these algorithms hold for any NMS objectives. In the special case of maximizing the expected number of loop closures f NLC (Section III-D) under a cardinality constraint, P v reduces to the well-studied maximum coverage problem over a graph [13]. In this case, a simple procedure based on pipage rounding can improve the approximation factor to 3/4 [1]. Furthermore, if the graph is bipartite, a specialized algorithm can improve the approximation factor to 8/9 [4]. Nonetheless, in this work we focus on greedy algorithms described in Table I due to their generality, computational efficiency, and incremental nature (see Remark 2). In many real-world scenarios, the number of primitives (e.g., keypoints in an image) is roughly the same across all measurements. By ignoring insignificant variations in observations sizes, one can assume that each vertex (i.e., observation) has unit weight or size. In this case, the knapsack constraint in P v reduces to a cardinality constraint V b. We first discuss this case and then revisit the more general case of knapsack constraints. 1) Uniform Observation Size: The standard greedy algorithm gives the optimal approximation factor for maximizing general NMS functions under a cardinality constraint (Table I). This algorithm, when applied on P v under a cardinality constraint, is as follows: for b rounds, greedily pick (without replacement) a vertex v with the highest marginal gain f v (V {v}) f v (V), where V denotes the current set of selected vertices. Refer to Algorithm 1 in the extended version of this paper for the complete pseudocode [37]. The solution V grd produced by the greedy algorithm satisfies f v (V grd ) (1 1/e) OPT v [28]. Furthermore, by our approximation-factor-preserving reduction (Theorem 2), V grd can in turn be mapped to a P e -feasible subset of edges E grd = edges(v grd ), for which we also have f e (E grd ) (1 1/e) OPT e. Remarkably, in some cases the interplay between P e and P v reveals a pathway to further improve the solution even after b rounds of standard greedy selection. We already know that E grd can be covered by a subset of vertices of size b, namely V grd. However, there may be an even cheaper (i.e., of size b < b) subset of vertices that covers the entire E grd. Such a subset can be found by computing the minimum vertex cover of the graph induced by E grd. This is in general NP-hard. We can, however, find a vertex cover by rounding a solution of the LP relaxation of this problem; see [40]. For 2-rendezvous (bipartite exchange graphs), this gives the minimum vertex cover as noted in [12]. As mentioned earlier, for general n-rendezvous (n 3), size of the resulting vertex cover is guaranteed to be at most twice the size of the minimum vertex cover; see, e.g., [40]. If such a subset can be found, we can continue running the greedy algorithm for b b additional rounds while still ensuring that the final solution is budget-feasible. Suppose repeating the process of recomputing the (approximate) minimum vertex cover leads to b + k rounds of greedy decisions in total and produces V grd. Using [20, Theorem 1.5], it can be shown that f v (V grd ) (1 1/e 1+ k b ) OPT v. However, we must note that in practice, recomputing the vertex cover only tends to improve the solution when the input exchange graph is sufficiently dense, and in 2-rendezvous where the actual minimum vertex cover can be computed. Remark 2. According to [30], an algorithm is any-com if it finds a suboptimal solution quickly and refines it as communication permits. The greedy algorithm described above has a similar trait: (i) for any budget b, it finds a nearoptimal solution; (ii) let (E b grd,vb grd ) be the pair of near-optimal solutions produced for budget b. Then, V b grd must be sent as a priority queue to robots to initiate the data exchange process by following the original ordering prescribed by the greedy algorithm (i.e., first round, second round, etc). Now imagine the exchange process is interrupted after exchanging b < b observations. Due to the incremental nature of the greedy algorithm, at this point robots have already exchanged V b grd, which is the solution that would have been produced by the greedy algorithm if the budget was b. But note that we know that this solution and its corresponding subset of loop closures E b grd are near-optimal for budget b. 2) Non-uniform Observation Size: A modified version of the greedy algorithm is guaranteed to provide a solution with an approximation factor of 1/2 (1 1/e) [20,24] (Table I) for the knapsack constraint. The algorithm is very intuitive: first, we run the standard greedy algorithm described above (stopping condition in this case will be the knapsack constraint). Then, we rerun the greedy algorithm with a minor modification: instead of picking the vertex with the highest marginal gain, we select the one with the highest normalized marginal gain, i.e., ( f v (V {v}) f v (V) ) /w(v). Finally, we return the better solution. Note that the normalized marginal gain encodes the marginal gain achieved in our task-oriented objective per one bit of data transmission. For example, for f NLC (4) this term quantifies the expected number of realized loop closures

7 (a) KITTI 00 (b) Simulation Fig. 2: Left: KITTI 00; Right: 2D simulation. Each base graph shows trajectories of five robots. Before inter-robot data exchange, trajectories are estimated purely using prior beliefs and odometry measurements, hence the drift displayed in the KITTI trajectories. The simulation trajectories shown are the exact ground truth. as it requires (approximately) solving the minimum vertex cover problem multiple times during the greedy loop. In our implementation, we further augment Edge Greedy by allowing it to select extra communication free edges, i.e., edges that are incident to the current vertex cover. Our second baseline is a simple random algorithm, which picks a random budgetfeasible subset of observations and selects all potential loop closures that can be verified by sending these observations. When comparing different algorithms, performance is evaluated in terms of the normalized objective value, i.e. achieved objective value divided by the maximum achievable value given infinite budget (i.e., selecting all candidates in E x ). A crucial step is to demonstrate and verify the near-optimal performance of the proposed algorithm. Since finding the optimal value of Problem P e is impractical, we rely on the proposed convex relaxation approach (Section IV-C) as an a posteriori certificate of near-optimality. In addition, to quantify how much accuracy we lose by using only a subset of loop closures, we compute the absolute trajectory error (ATE) between our estimated trajectory and the full maximum likelihood estimate (obtained by selecting all candidate loop closures). B. KITTI Odometry Sequence 00 In order to show that the proposed algorithms are useful in real-world scenarios, we performed a number of offline experiments on data from the KITTI odometry benchmark [11]. Similar to [8], we divide the odometry sequence 00 into sub-trajectories representing individual robots paths (Figure 2a). This particular sequence contains many realistic path intersections and re-traversals, providing a large number of inter-robot loop closures that make the measurement selection problem interesting. For the KITTI experiments, we use a modified version of ORB-SLAM2 [27] to generate stereo visual odometry and potential loop closures. As in [12], we define the probability associated with each potential loop closure by normalizing the corresponding visual similarity score outputted by DBoW2 [10]. Since the weighted treeconnectivity (WST) objective only supports 2D data, we project the KITTI data to the x-y plane and work with the projected 2D trajectories. The same projection is done for the other two objective functions to ensure a fair comparison. robot 1 robot 2 robot 3 robot 4 robot 5 In practice, the number of keypoints is roughly the same (around 2000) for all observations obtained by ORB-SLAM2. In our experiments, we ignore this insignificant variation in observation sizes and assign each vertex a unit weight. In this way, our communication budget reduces to a cardinality constraint on the total number of observations the team can exchange. Assuming each keypoint (consisting of a descriptor and coordinates) uses approximately 40 bytes of data, a communication budget of 50, for example, translates to 4MB of total data exchange bandwidth. Figures 3a-3c illustrate the performance of the proposed algorithm under different objective functions and varying communication budget. Under all three objectives, the proposed algorithm outperforms both baselines by a significant margin. Intuitively, the Edge Greedy baseline blindly maximizes the objective function without controlling resource consumption, while the random baseline only aims at achieving a budgetfeasible solution. In contrast, by having foresight over resource consumption while maximizing the objective, the proposed method spends the budget more wisely, hence always achieving a higher score. Furthermore, Figures 3b (WST) and 3c (NLC) show upper bounds on the value of the optimal solution obtained using convex relaxation (Section IV-C). We do not include the upper bound for the FIM objective because solving the convex relaxation in this case is too time consuming. 3 For both WST and NLC objectives, we observe that the performance of the proposed algorithm is close to the convex relaxation upper bound. This confirms our intuition that in practice, the proposed algorithm performs much better than the theoretical lower bound of (1 1/e) OPT e. It is particularly interesting that for the NLC objective, the value achieved by the proposed approach matches exactly with the upper bound, indicating that in this case the solution is actually optimal. This happens because the specific problem instance is sparse, i.e., candidate edges form several distinct connected components. In this case, the entire problem breaks down into multiple easier sub-problems, for which it is more likely for the greedy algorithm to obtain an optimal solution. Due to the same sparse structure, we do not observe improvement caused by recomputing the minimum vertex cover, as discussed in Section IV-A. Figure 3d illustrates the cross-objective performance of the proposed method evaluated on the absolute trajectory error (ATE). We note that the overall decrease in error is small as communication budget increases. This suggests that our initial estimate is relatively accurate. Nevertheless, as the communication budget increases, the estimates of all three objectives eventually approach the full maximum likelihood estimate (obtained by selecting all potential loop closures). Interestingly, the ATE metric is not monotonically decreasing. Furthermore, the performance of the NLC objective is more stable compared to the FIM and WST objectives. Both observations are due to the fact that the ATE is highly 3 The KITTI dataset we use contains more than 2000 poses in total. Consequently, the Fisher information matrix with three degrees of freedom has dimension larger than

8 1 0.8 Normalized FIM Score Proposed Alg. Edge Greedy Random Normalized WST Score Proposed Alg. Upper bound on OPT Edge Greedy Random Normalized NLC Score Proposed Alg. Upper bound on OPT Edge Greedy Random Absolute Trajectory Error (ATE) 10 Proposed Alg. on FIM Proposed Alg. on WST 8 Proposed Alg. on NLC (a) FIM (Eq. 2) (b) WST (Eq. 3) (c) NLC (Eq. 4) (d) ATE Fig. 3: (a) to (c) shows performance of the proposed algorithm in KITTI 00 under different objective functions and varying communication budget. Objective value is normalized by the maximum achievable value of each objective given infinite budget (selecting all potential loop closures). (d): cross-objective performance evaluated on ATE, compared against the maximum likelihood estimate given infinite budget. In the KITTI experiments, budget b is defined as the total number of observations the team can exchange. In this dataset, the minimum budget to cover all potential loop closures is 250, which translate to roughly 20MB of total data exchange bandwidth. dependent on the random realization of the selected loop closures. Intuitively, although the FIM and WST objectives tend to select more informative loop closure candidates, these candidates may turn out to be false positives and hence do not contribute to the final localization accuracy. From this perspective, the NLC objective is advantageous as it seeks to maximize the expected number of true loop closures. However, the other two objectives can be easily augmented to better cope with the stochastic nature of candidate loop closures. This can be done by, e.g., prefiltering the candidates and removing those with low occurrence probabilities WST Score (Budget = 50) Proposed Alg. Upper bound on OPT Edge Greedy Random C. Simulation Experiments Synthetic data was produced using the 2D simulator functionality of g2o [22]. The simulator produced noisy odometry and loop closure constraints for robots moving in a random square grid pattern (Figure 2b). The trajectory was divided into multiple robots in the same manner as the KITTI data, but loop closures probabilities p were generated from a uniform random distribution. We use a simulated graph to study the behavior of the proposed algorithm under varying densities of the input exchange graph. In our simulation, we control this density by enforcing different maximum degrees on the vertices and randomly pruning excess candidate edges. Figure 4 displays the effect of increasing density (in terms of the maximum vertex degree) on the WST objective. The algorithms were applied with a fixed communication budget of 50 over the same exchange graph. As expected, the proposed algorithm outperforms the baselines by a significant margin. VI. CONCLUSION & FUTURE WORK Inter-robot loop closures constitute the essence of collaborative localization and mapping; without them collaboration is impossible. Detecting inter-robot loop closures is a resourceintensive process, during which robots must exchange substantial amounts of sensory data. Recent works have made great strides in making CSLAM front-ends more data efficient [8,12]. While such approaches play a crucial role in saving scarce mission-critical resources, in many scenarios the resources available onboard (mainly, battery and bandwidth) may be insufficient for verifying every potential loop closure. Fig. 4: Performance of the proposed algorithm maximizing the WST objective in simulation under fixed communication budget (50 in this case) and varying maximum degree of the exchange graph. Consequently, robots must be able to solve the inter-robot loop closure detection problem under budgeted communication. This paper addressed this critical challenge by adopting a resource-adaptive approach. In particular, we presented anycom constant-factor approximation algorithms for the problem of selecting a budget-feasible subset of potential loop closures while maximizing a task-oriented objective. Performance guarantees presented in this work hold for any monotone submodular objective. Extensive experimental results using the KITTI benchmark dataset and realistic synthetic data validated our theoretical results. We plan to extend our results by considering new forms of communications constraints. In particular, in heterogeneous multirobot systems, one may wish to partition the team into several disjoint blocks and impose different communication budgets on each block. Our preliminary results indicate that our approach can be extended to cover this case. While beyond the scope of this paper, analyzing more complex communication protocols and network topologies is another important future direction. ACKNOWLEDGEMENT This work was supported in part by the NASA Convergent Aeronautics Solutions project Design Environment for Novel Vertical Lift Vehicles (DELIVER), by the Northrop Grumman Corporation, and by ONR under BRC award N

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### arxiv: v1 [cs.ro] 17 Jan 2019

Resource-Aware Algorithms for Distributed Loop Closure Detection with Provable Performance Guarantees arxiv:9.5925v [cs.ro] 7 Jan 29 Yulun Tian, Kasra Khosoussi, and Jonathan P. How Laboratory for Information

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