NeuroMLR: Robust & Reliable Route Recommendation on Road Networks 
Authors: Jayant Jain
Vrittika Bagadia
Sahil Manchanda
Sayan Ranu
Venue: NeurIPS 2021 Poster (2021)
Keywords: Route recommendation
Spatio-temporal
Deep Learning
Road networks
Graph Convolutional Networks
Inductive Learning

Abstract: Predicting the most likely route from a source location to a destination is a core functionality in mapping services. Although the problem has been studied in the literature, two key limitations remain to be addressed. First, our study reveals that a significant portion of the routes recommended by existing methods fail to reach the destination. Second, existing techniques are transductive in nature; hence, they fail to recommend routes if unseen roads are encountered at inference time. In this paper, we address these limitations through an inductive algorithm called NeuroMLR. NeuroMLR learns a generative model from historical trajectories by conditioning on three explanatory factors: the current location, the destination, and real-time traffic conditions. The conditional distributions are learned through a novel combination of Lipschitz embedding with Graph Convolutional Networks (GCN) using historical trajectory data. Through in-depth experiments on real-world datasets, we establish that NeuroMLR imparts significant improvement in accuracy over the state of the art. More importantly, NeuroMLR generalizes dramatically better to unseen data and the recommended routes reach the destination with much higher likelihood than existing techniques.

NEUROMLR: Robust & Reliable Route Recommendation on Road Networks
Jayant Jain', Vrittika Bagadia", Sahil Manchanda, Sayan Ranu Indian Institute of Technology Delhi (jayant vrittttt aaaakaaaaadaaa@@ggaaaa com {sahil..
(sahil.manchanda manchanda, (ayanramu)@@aa iitd. ac. in
Abstract
Predicting the most likely route from source location to a destination is core functionality in mapping services. Although the problem has been studied in the literature, two key limitations remain to be addressed. First, our study reveals that significant portion the routes recommended by existing methods fail reach the destination. Second, existing techniques transductive in nature; hence, they fail to recommend routes if unseen roads are encountered at inference time, In this paper, we address these limitations through an inductive algorithm called NEUROMLR. NEUROMLR learns generative model from historical trajectories by conditioning three explanatory factors: the current location, the destination, and real-time traffic conditions. The conditional distributions are learned through a novel combination of Lipschitz embeddings with Graph Convolutional Networks (GCN) using historical trajectory data. Through in-depth experiments on real- world datasets, establish that NEUROML. imparts significant improvement in accuracy over the state of the art. More importantly, NEUROMI. generalizes dramatically better unseen data and the recommended routes reach the destination with much higher likelihood than existing techniques.
1 Introduction and Related Work
Given historical trajectory data, we study the problem of predicting the most likely route from source node to a destination node in road network. This problem has two prominent applications: route recommendation and route recovery. Route recommendation is one of the core functionalities in GPS-aided mapping applications. They are routinely used in the cab and food-delivery industry [26] 34 6) as well as by common people through navigation systems when they are unfamiliar with their surroundings [3). Route recovery, as the name suggests, focuses on recovering the actual traversed route from a partially observed GPS trajectory ( Due various reasons, for instance, limiting the power consumption of GPS devices, trajectories are often recorded at low sampling rate.
1.1 Existing work
The simplest approach is to predict the shortest or the quickest path (route) between the source and destination. However, several studies have shown that human beings rarely travel in shortest paths [21]. Rather, the probability of path being taken is complex mixture of several latent factors such road quality [5 12 road scenery [4]. pollution levels [2]] presence of road tolls, etc. Furthermore, it has also been shown that human beings systematically misperceive travel time 20) Modeling these complex factors is challenging. Consequently, large body of work exists
on predicting the most likely route 28 33 4 IO 27], with DEEPST [10] and CSSRNN (27 being the best performing algorithms. CSSRNN models the trajectory patterns through a Recurrent Neural Network and exploits the topological constraints presented by the road network. DEEPST learns representations for the trajectories, destination and traffic conditions using variational autoencoders.
'eeotees equal contribution
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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(00
100
Percentage nodes
30
Reachability (%)
V10
70
30
Routes obtained from Historical
60
Trajectory database
NeuroM R-G
1 1/2 + 1/3 - 1/4
CSSRNN
Tz: - V/3 - 1/4
T3:0; + F/2 + 8y + Ugg
20
50
(CHG)
(BJG)
(CTI)
2814
T:: 1
Dy VB P10
Percent of training data kept
Number of road segments
(a)
(b)
(c) Reachability
Figure 1: (a) A sample road network and historical database D. For simplicity, we ignore edge weights. (b) Percentage of unseen nodes in the road network of Beijing against the size of the training data (c) Impact of road network size on performance of CSSRNN and NEUROML.RGG.
1.2 Limitations of existing work
Reachability: Existing techniques have primarily used recall and preciston measure accuracy of predictions [TO] 27 Critically, these ignore whether the predicted trajectory actually reaches the destination. The deployability of any route recommendation algorithm is severely compromised if the recommended route does not reach the destination. Our experiments reveal that for both DEEPST and CssRNN, significant portion of recommended routes fail to reach the destination (See Table. Further, the state of the art does not scale well to large road networks (see Fig. [c]. The performance gap between CssRNN and NEUROMLR becomes more prominent with increasing network size.
Inductive Learning: good prediction model should be capable making predictions on parts of the road network which unseen rarely seen during training. From Fig lb we observe that there is a significant percentage of nodes in the road network that remain unseen even we the entire training data. Both CsSRNN and DEEPST learn node embeddings in transductive manner, limiting knowledge sharing across nodes. Hence, quality on unseen/lesser seen nodes suffers. DEEPST employs similar destinations. However, this introduces a limitation that different destinations get mapped to identical representations, impairing reachability.
approach for sharing statistical strength across trips having
Prediction Accuracy:
The accuracy of even the best performing techniques is often below Consequently, there is scope for improvement.
50%
[00.
1.3
Contributions
Decoupled route prediction: Our proposed problem formulation ($ 2 allows decoupling the problem into two independent subproblems of route search ($3) and predicting transition probabilities ( This allows to reduce the problem of finding the most likely route that of identifying the shortest path in a modified road network, thus guaranteeing destination reachability ($ 3
Generalization capability: We propose an inductive learning method using novel combination of Lipschitz embeddings with Graph Convolutional Networks. Lipschitz embeddings serve as a rich initialisation derived from the global road network structure, independent of historical data. GCNs, via message passing, further propagate information learnt during training, to unseen and rarely seen nodes.
Empirical Evaluation: Extensive experiments on five large, real datasets establish: (1) NEU- ROMLR is up to 100% more accurate and 4 times faster, (2) the recommended route reaches destination with > 0.96 probability, which is up to 90% better than the state of the art, and (3) NEUROMLR is dramatically more effective in generalizing to unseen data.
Problem Formulation
Definition (Road Network). road network directed graph G (V, 5, Tt), where V is the set of nodes representing road intersections, E C V V is the set of edges representing road

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segments, distance function ò: 3 - IR representing the length (weight) of each road segment, and function T4 & E. representing the average time taken to traverse each road segment time t.
We use the notation e = (u, v) to denote a road segment (edge) from node u to u and its length is denoted by 5(e).
Definition 2 (Route). A route (a.k.a path) R(s, d) (vi..... fe corresponds to a simple path from the source node vn to destination d = in the road network G, i.e, path without cycles. Analogously, a route can also expressed in terms of sequence of edges R(8, (E1,'". " ck-1 1). where E (14, 0(41).
To denote generic route between any arbitrary source and destination nodes, we use instead of R(s, d). We use the notation [R] to denote the number of edges in R and R.e, to refer to the 7th edge in R. Furthermore, E denotes that R goes through edge e. The above notations are analogously extended from edges to nodes.
We assume we have access a dataset of historical trajectories. A trajectory is sequence of GPS pings, made by a vehichle, which corresponds to a path in the road network. Each GPS ping is tuple of the form (lalitude, longilude, lime), which records vehicle's location Definition 3 (Query). In a route recommendation query, the input is a tuple q : {s, d, f), where a.d. E V are the source and destination nodes respectively, and is the time at which the journey is to be taken.
particular time.
Problem: Most Likely Route: Given a road network 9, a historical database DD of trajectories, and query : (s, t). we would like infer the mostly likely route R (s, d) based the traffic patterns embodied in D. Formally,
RJ R* d) = arg max Pr(R(s, d) 1))
(I)
rR(g.d)Eg
2.1 Problem Characterization
The route recommendation problem can be framed as path search problem on the graph correspond- ing the road network. Mathematically, the probability of route can be expressed in terms of its constituent edges.
[
Pr(R () = II Pr(R.e: | R.co + R.
Here, Pr(R.e; R.eg R. 8, d, represents the probability that route R. goes through edge R.e, given the path taken so far and the query parameters q (s. d, () Past studies have shown that human mobility patterns conform to the Markovian assumption [I 25). Thus, Eq. 2 reduces to:
[R]
Pr(R 4) = II
Pr(R.e: 0i d,
(3)
1=1
With these simplifications, Eq.
reduces to:
[R]
[
ds. arg max
II
Pr(R.e, d,,, arg min YREG
log(Pr(R.e) Us d)t))
(4)
YREG
To summarize the above observations, there are two key challenges that we need to tackle:
Route Search: Given the conditional transition probability distribution, searching the route (path)
minimizing Eq. 4
Modeling: Learning the conditional distribution that governs transitions in the road network, from the historical trajectory dataset 'D.
Henceforth, while talking about transitions in the road network, we will use the notation curr refer to the current location(node), curr E V.
3 Route Search
In this section, we assume that the transition probability Pr(elcurr, d, t) for any edge E E is known. Our algorithm to learn this distribution is discussed in $
The Optimal Search: We first note that the negative log likelihood of transition probability (see Eq. 4 would be non-negative. Thus, we have an edge weight for each edge in the road network and
3

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Neural Model
( (curr, 1) cur,, d, 6)
GCN
Iterate TeeT
2 curF
vT E
Historical trajectories
MLP
D
TT
Road Network
Raw Traffic
Traffic Representation
Lipschilz
Vector
Embedding
Data Pre-processing
Figure 2: Architecture of NEUROMLR.
our goal is to identify the path from the source to the destination that has the minimum cumulative weight. This computational task maps the problem of finding the shortest path in graph and can be solved using Dijkstra's 's Algorithm. More importantly, the optimal path is guaranteed to reach the destination. The pseudocode for this search algorithm is provided in App- A
Computational Complexity: The complexity of the Dijkstra's Algorithm is O([VI + [&] log(
This may be prohibitively expensive on large road networks where real-time predictions are desired.
Greedy Approach: We start from the source node and greedily choose the transition with the highest probability till the destination is reached. Inaccurate estimation transition probabilities may however divert towards the wrong direction and may never reach the destination. This may result in [ iterations in the worst case. To handle cases of this nature, we terminate when either the destination is reached or the Haversine distance from the current node to the destination is significantly higher than the closest point in the current route to the destination. This idea is motivated from the fact that, in general, vehicle progressively moves closer to the destination with each transition [31] The pseudocode of the greedy approach can be found in Alg. 2 [Bp.
Computational Complexity: At each node in R', we evaluate each neighbor and select the one with highest likelihood. Hence, the complexity is Og RR where g the average degree G.
NEUROMLR: The Neural Approach to the Most Likely Route Problem
Revisiting Eq. 4 the key requirement is to accurately model the conditional transition probability function Pr(e curr, d, t) where e = (curr, v). We want to estimate the true distribution that governs transitions in the road network. However, this distribution is hidden from us and we only have access D. which a sample drawn from this distribution.
Mathematically. we wish to estimate the underlying transition probability distribution from D using surrogate function (nlcc d, t; θ), such that, Qeeccur,, t; θ) Pr(e|curr, t).
We learn θ using the neural network depicted Fig. Our core idea is to learn useful representations of road intersections (nodes) and real-time traffic conditions, and use them infer the transition probabilities. To learn inductive node representations, we use a novel combination GCN g with Lipschitz embeddings (2). In addition, low-dimensional traffic representation of the road network at any time l is learned using Principal Component Analysis (PCA). To predict P(e (curr, v) curr, d, !), we concatenate the representations of curr, d and traffic status time I, and pass them through a Multi-layered Perceptron (MLP) to predict the transition probability. The entire network is trained end-to-end. We next discuss each the sub-components. By convention, we used bold font for vectors and matrices.
4.1 Constructing Node Attributes
In this section, describe the process of constructing node attributes for our GCN. We wish to learn embeddings where nodes with similar routes to common destinations are close in the embedding space. While latitude and longitude may be used node attributes, they do characterize node positions accurately since movement of vehicles is constrained by the network structure. Rather, we need learn node attributes that reflect road network distances. Towards that end, we use Lipschitz embeddings.

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Definition 4 (Attribute Embedding). Let A = {ai." .ax S V be randomly selected subset of nodes. We call them anchors. The distance d(u, u) between two nodes u, v E V is defined as splue))) where sp(u, v) is the spatial shortest path distance from u to v. We embed all nodes in V in a k-dimensional feature space "z(u) 1=1,... ,kk. where x; - d(u,a,).
2
The dimensionality of the attribute space dictates how well the original shortest path distances are preserved. To gain formal understanding of distance preservation, introduce the definition of distortion.
Definition 5 (Distortion). Given two metric spaces (O, d) and (O', d') and an embedding function f:0- O', f has distortion if Vo,, 02 E O, _d(or. 02) S {d((foo)). f(2)) S d()),0)).
In our case, d(01, 02) is the average way shortest path distance. d'(, along with dimensionality k. remains to be defined. To define them, we use Bourgain's 's Theorem. Bourgain's 's Theorem establishes that low distortion Lipschitz embedding exists for any metric space.
Theorem 1 (Bourgain's 's Theorem (2). Given any finite metric space (O,d) with distance function d(-), there exists an embedding of in (O, d) into R* under any Lp metric, where = O(log" n) and the distortion of the embedding (loo n), where n =O].
To apply Bourgain's 's Theorem on our problem, we need to show that d(u, v) is metric.
Lemma 1. d(u, v) is metric distance function. For proof, refer App.
4.2 Learning Node Representations through GCN
In addition to capturing road network distances in node representations through Lipschitz embeddings, would also like to generalize to unseen nodes in the road network. To illustrate, let assume node V13 in Fig. has not appeared in any training trajectory. If we fine-tune representations for only seen nodes, then the representation of Via would remain unchanged from its Lipschitz embedding. A GCN avoids this scenario by message passing among neighbors. More specifically, information is shared among L-hop neighbors in GCN, where L is the number of layers. Thus, a subset of these neighbors have appeared in training trajectories, then this information is shared in its neighborhood. which infuses information beyond Lipschitz embeddings even for unseen nodes.
To train the GCN, Lipschitz embeddings v(.) corresponding to | nodes are stacked as original input features. Specifically, Vu V. ha (u)). Next, compute:
ho
h = a
Wi Σ
(5)
V(N(())+ +
vEN(M)Uu
Here, Wi stands for layer-specific learnable weight matrix for 4 layer, h² the embedding of node at layer and af-) denotes activation function (ReLU in our implementation). Furthermore, (u)) = {w E (u, v) E } denotes the neighbors of node u in the road network. The vector formed in the final layer L is Zu h
4.3 Traffic representation
The simplest option is to partition D into various time slots (Ex: 8AM-1IAM, 11-AM-2PM, etc.) and learn a model separately for each time slot. This strategy, however, assumes that traffic is homogeneous in each time slot all days. Clearly, this assumption is not true in real life (Ex: weekends vs. week days). Further, this scheme is data inefficient since the traffic-dependent travel patterns across different time-slots might be similar.
In order characterize the real-time traffic conditions for the entire road network at time t, we obtain the top-5% of the most frequently traversed edges E C È in D. The raw traffic representation at time i the E -d-mmensioonal vector of speeds on these edges. Specifically, r, = (Tx(e) e E E). We use only the top-5% edges since the frequency distribution of edges follows a power-law (See App. H Consequently, we do not use noisy speed information on tess-travelled edges.
It is natural for neighboring road segments (edges) to exhibit co-variance in speed. To remove such information redundancy, we learn a low-dimensional representation of r2 through PCA. Specifically, let the training trajectories in D span the time range [fmin, tmaz]. Thus, we have a collection of
We assume that the road network is strongly connected, which is typically true.
5

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traffic representations in the form of R = {re t [trii (mau]). We perform PCA to learn the eigenvectors on R. Given any raw traffic vector Tt. it is projected on the top-k eigenvectors to construct the low-dimensional representation zt. Mathematically, =t rt V, where V glE|x The columns of V contain the cigenvectors with the k largest eigenvalues.
4.4 Aggregating Node embedding with Traffic representation
Cellcurr, d, θ) is a function of four input features: condition at time i, and the transition node corresponding to edge e = (curr, v). GCN provides the embeddings Zeurr, Zu, Zd of curr, v, respectively and the traffic representation 2 is constructed as discussed above. All these embeddings are concatenated as:
current node curr, destination d, traffic
z Zourr || |za
(6)
4.5 Model Training
Following the pass through GCN, the concatenated embedding z (Eq. 6 is passed through an MLP to convert the vector into scalar (unnormalised) confidence value i. f(u, curr, d,
MLP(z). The predicted transition value is defined as softmax over all possible transitions from cury. Mathematically,
eap (f (v, curr, d,t))
Q ((curr,v)leurr, d, t, 0)
E Ed'EN exp (f (v', curr, d,t))
Loss(0)
log ([R.ej]R.Uj, R.d, R.I;
(8)
[D]
VRED j=1
Finally, in Eq. 8 above, the model parameters O are optimized through cross-entropy loss over
trajectories in The pseudocode the training procedure can found Alg. 3 App. D
4.6 Inference Phase
Given any query 9 (s, d, t), we follow one of the route search algorithms discussed in Both search mechanisms require only the transition probabilities as edge weights. Prediction a transition probability Pr(e (curr, v) curr, d, () simply involves forward pass through the architecture. Complexity analysis: Time complexity of predicting Pr(e (curr, u) curr, d, f) is O(g Las Lms-d)) Here, Lm is the number of the layers in the MLP, g is the average degree in the road network, dy is output feature vector dimension of GCN, Refer Appendix, E for detailed derivation.
Experiments
In this section, we benchmark NEUROMLR against DEEPST and CssRNN and establish that:
Accuracy: NEUROMLR is more accurate in terms of precision and recall when compared to the
state-of-the-ard algorithms of DEEPST 100 and CssRNN (27].
Reachability: NEUROMLR, with its greedy route search mechanism, is more efficient, and achieves significantly higher reachability than DEEPST and CSSRNN.
Inductive Learning: Due to its inductive learning capability, NEUROMLR learns more effectively and generalizes significantly better to unseen/lesser seen parts of the road network.
Scalability: NEUROMLR generates high quality routes on large road networks. The performance CssRNN, on the contrary, deteriorates heavily with increase in road network size(Fig [c].
Our code-base is available at wtthtts:/// hnttps://github.com com idea- Wett///eeuooiii
5.1 Experimental Setup
The system configuration details are present in App. All experiments are repeated 5 times and we report the average of the metric being measured.
Datasets: We use publicly available real datasets from five different cities. Table ! summarizes
the statistics of the datasets. The first four cities namely Chengdu Porto (15) Harbin [II] and
P We also tried the Attention mechanism to capture the importance of each the explanatory factors. Details can found in Appendix.
Chengdu dataset link

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Statistics
Chengdu CHG)
Porto(PT)
Harbin(HRBB)
Beijing(BJG
CityIndia CTT)
No. of nodes
3,3
5,330
6,398
31, 199
105, 873
No. of edges
9. 255
11,491
16, 292
72, 156
281, 086
No. of trajectories
3, 600, 503
1. 4262, 312
1, 133, 548
1,382, 948
451,
Avg trip length (km)
4.54
5.27
10.92
7.39
3.27
Avg number of edges/trip
22.93
51.07
56,81
36.08
42.68
Table I: Dataset statistics after pre- pre-processing.
Algorithm
Precision (%)
Recall (%)
Reachability (%)
Reachability Distance(km)
HRB
BJG
CTI
HRB
BJG
CTT
HRB
BJG
CTI
HRB
BIG
CTT
NEUROMLR-D
66.1
77.9
77.9
49.6
76.5
73.1
NEUROMER-G
59.6
75.6
14.3
48.6
74.5
70.1
99.1
99.1
96.1
0.02
0.01
0.03
CSSRNN
49.8
59.5
36.9
51.1
68.8
53.2
95.3
91.7
50.2
0.16
0.83
2.03
DEEPST
51.9
60.3
67.4
27.3
33.2
34.9
8.7
6.7
1.96
2.75
1.07
SP
46.4
59.2
62.1
31.3
55.5
53.7
QP
40.7
51.4
47.6
28.6
50.0
44.0
Table 2: Comparison of NEUROMER against the benchmarked algorithms on the four different metrics of average precision, average recall, percentage of trips that reached destination, and average distance from the true destination. The best performance for each dataset is highlighted in bold.
Beijing [2] are taxi datasets. The fifth dataset is a publicly available food delivery dataset [ The authors did not reveal the name of the city, other than the fact that, in terms of food delivery volume, this city ranks among the highest in India. We extract the road network of each city from OpenStreetMap [17]. To align the GPS sequences to the road network we use map- mapp-matching(29 The temporal edge weight (e) is set to the average travel time of all vehicles going over in the past one hour.
Baselines:
We benchmark the performance of NEUROMLR against, (1) DEEPST, (2) CsSRNN, (3) shortest path (SP), and (4) the quickest path (QP). For NEUROMLR, we consider two versions: (NEUROMLR-)
(NEUROMLR-O) correspond- ing to the two route search algorithms described in $ 3 We do not consider personalized route recommendation algorithms (25]. since we do not consider the personalization aspect. The codebase of CssRNN, shared by the authors, is implemented in TensorFlow 1.15. DEEPST and NEUROMLR are implemented in PyTorch 1.6.0.
NEUROMLRRDijjkse
and NEUROMLR-
Train-Validation Validation-Tess setup: train all models till convergence of the validation loss. Before splitting, we sort the trajectories the basis of the start timestamp. Unless specifically mentioned, we use the first 60% of the trajectories for training, next 20% for validation and remaining 20% for inference.
For a fair comparison of NEUROMIR with DEEPST and CssRNN,
Evaluation metries: For evaluation, we pick each route R in the test set and issue the corresponding query (s, d, f), where
R.U,, d = R.UyR|+1, and is the time at which R was initiated. The predicted trajectory R" for q is then generated and compared with R. We use four metrics to evaluate prediction quality.
Precision Eec(Rem 8(e)
Recall EEE(RAVR 5(e)
EeER. 8(e) [1 if d dt
EEER ō(e)
Reachability
Reachability Distance Haversine (d, d²)
otherwise"
In the above equations, 8(e) denotes the length of edge and d' denotes the last node in R*. Note that the reachability metrics relevant for only NEUROMLR- -rredd, DEEPST and CssRNN since the rest of the algorithms guarantee reachability.
Parameters: The default parameters for NEUROMLR are provided in App. G
5.2 Accuracy and Reachability
Table 2 presents the performance of the various algorithms across the three larger datasets Harbin, Beijing and City India The following observations emerge from Table 2
'Due space limitations, the results for the other datasets (Porto, Chengdu) be found in App.
7

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Fl-score (%)
Reachability (%)
78
NzuroMLR
NeuroMi R-D
CSSRNN
DeepST
25
SP
QP
0-2
5:10
10225
20-10
10-25
25100
True trip longth (km)
True trip length (km)
(a) Fi-score
(b) Reachability
Figure Impact of trip length on accuracy and reachability on the Beijing dataset.
Both versions of NEUROMLR consistently outperform the baselines As expected, NEUROMI.R-Dijkki
Precision and Recall:
is marginally better than NEUROMIR MRUROMLLR-Greddy, Among the four considered baselines, CSSRNN achieves the best result. Furthermore, NEUROM.. significantly outperform both SP and QP, which validates past work that people rarely follow shortest/quickest paths; automobile travellers select routes based on several latent factors such as the presence of tolls [16 [9], travel time reliability [3, traffic lights [18), and road aesthetics [21 32
Reachability:
NEUROMLR-Greede outperforms both DEEPST and CSSRNN and achieves more than 96% reachability across all datasets. This superior performance of NEUROMLE R-Greddy establishes the efficacy of Lipschitz embeddings with GCN in modeling transitions that govern vehicle movements. The reachability performance is weakest in DEEPST due to the clustering based approach it adopts. Specifically, several nodes are allotted the same destination representation and hence reachability is compromised.
To further evaluate robustness, we examine the performance against the distance between the source and destination in the query. As the trip length increases, more possible paths emerge, and the modeling task gets harder. This pattern is visible Fig. 3 Consistent with previous results, MLu continues to be the most accurate algorithm.
Impact of Trip Length:
5.3 Inductive Learning
To showcase the benefits of inductive learning, we compare the two best performing algorithms of NEUROMLR with CSSRNNB
Impact of training data: In Figs. 4a 4b
we examine how effectively each algorithm learns as the volume of training data is varied. As clearly evident, NEUROMLR obtains significantly more accurate performance at low volumes of training data than CSSRNN. This a direct consequence of the inductive ability to share information among nodes and thereby generalize for unseen test data.
we investigate how NEUROMLR and CssRNN perform on test trips that originate or end at umpopular nodes. A node is termed unpopular if its frequency of occurrence in the training trips less than a threshold frequency. We segregate the test trips into four categories based on the popularity of source and destination nodes: P-P, U-P, P-U and U-U. For example, P-U refers to trips starting at a popular node and ending at an unpopular one. As expected, the majority of test trips are of type P-P and thus the performance on P-P (Figures[4d 4g) is similar to the aggregate results mentioned in Table 2 We have not included the performance variation of trips of type U-U for different thresholds, since the percentage of such trips is relatively insignificant [Fig/cc. The performance on P-U trips (Figs.
Performance on unseen data: In Figs. 4c 4i
undergoes a dramatic drop for CSSRNN since the destination is unpopular and there is not much information to direct the model. For U-P (Figs. 4h 4e). the performance lies between P-P and P-U since even after a rocky start, could transit popular node and from there may reach the destination. In all cases NEUROMLR adapts more gracefully and highlights the benefits of inductive learning. Note that at zero threshold frequency, unpopular nodes are equivalent to nodes unseen during training.
'In absolute terms, the precision and recall might seem low. This is because, for the input (source- destination pair), we may have different outputs (routes). For more details, refer to App. P
For similar analysis on other datasets, refer to App.
'Here, we have restricted our model's comparisons to CSSRNN since DEEPST performed poorly on all metries. Some potential reasons can found in App. M
For similar analysis other datasets, refer App.
8

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80
100
Percentage of trips
U-U
Reachablityy (%)
Fl-scare (%)
20
NeuroMLR-G
NearoMiLR.G
CSSRNN
CSSRNN
0-
5 10
100
100
20
40
Percent training data kept
Percent of training data kept
Threshold frequency
(a) Fl-score
(b) Reachability
(c) Percentage of trips
1810-
01)
90:
90
90-
Reachability (%)
Reachablitty (%)
Reachablitty (%)
70-
60:
50:
50
50
40
30
30-
30-
LLuroMLRR.G
NeuroMLR-G
20-
NeuroMLR-G
CSSRNN
10
CSSRNN
10-
CSSRNN
022
10
201
30
40
02
10
02
6
10
20
30
40
Threshald frequency
Threshold frequency
Threshold frequency
(d) Popular Popular
(c) Unpopular Popular
Popular Unpopular
80
80
10
70
70
Fl-soare (%)
F1 F1-score (%)
50-
300
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CSSRNN
CSSRNN
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02 Threshold frequency
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Threshold frequency
(g) Popular to Popular
(h) Unpopular to Popular
(i) Popular Unpopular
Figure 4: All these experiments are performed on the Beijing dataset. (a, b) Performance of
NEUROMLR-G and CssRNN with different percentages of training data. (c) Percentage of test data trips in P-U, U-P and U-U categories. (d-i) Variation of Reachability & F1-score with the threshold frequency for different popularity categories.
5.4 Ablation study
Impact GCN and Lipschitz Embeddings: investigate the individual impact of using Lips- chitz embeddings and employing GCNs in Fags.5ap5bbbr 5blby comparing the performance of NEUROMLR on the four possible combinations. As visible, the combined combination of Lipschitz embeddings and GCN imparts significant improvement in both prediction accuracy and reachability. Employing either one individually also enhances the model's performance, thereby justifying their importance. For similar studies other datasets [pp.
Impact Traffic:
The impact traffic on route prediction can be found in App.

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100
80
75
70
80
Fl-ccore (%)
66
Reachability (%)
70
60:
56
50
50
40
45
10:
35
Lipuchizz
Kindori-CCC
Lipschita Kandors
10:
Lioschitz -CNN
40 45 50 Random CCN
Lioschiz Random
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40 45
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35 Number of epochs
20
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Number of epochs
(a) Fl-score
(b) Reachability
Figure 5: Impact of GCN further fine-tuning Lipschitz embeddings in NEUROMLR- Greedy. All plots in this figure use the Beijing dataset.
5.5 Inference Time
Table presenss the number of queries solved by the three best performing algorithms per second. Since the performance of DEEPST was found to be much inferior to these three algorithms (Table 2 we skip its efficiency analysis. As evident, NEUROMLE R-Greedy is the fastest. Consistent with its computational complexity, NEUROMLR-Dijkstn is slower. CssRNN slower than NEUROMLR- Greedy since it uses more complex auto-regressive model where each transition is also conditioned on the path taken so far. Owing to significantly faster inference and accuracy that almost identical to NEUROMLE MOLLR-Diijkee Dijjkstra, recommendation is to use NEUROMLR M--reedyy as the default algorithm.
Algorithm
Chengdu
Porto
Harbin
Beijing
CityIndia
NEUROMLR-D
11.38
10.11
7.62
1.08
0.28
NEUROMLR-G
5612
4539
3286
2084
1522
CSSRNN
4425
1684
1355
467
Table 3: Inference speed (trips processed per second).
Conclusion
For a route recommendation algorithm to be deployable the real world, it must ensure that the rec- ommended route reaches the destination. In addition, it must show good generalization performance on queries over unseen/rarely seen data. Existing techniques for predicting the most likely route lack the above mentioned abilities. In this paper, we propose NEUROMLR which overcomes these limitations through novel combination of Lipschitz embedding and Graph Convolutional networks. This strategy ensures inductive learning and enhances reachability. Specifically, even those nodes that are not seen adequately in training data, get good representations due to Lipschitz embedding capturing network position and GCN ensuring information propagation from neighboring nodes. Further, the model decouples probability transition modelling and route search, allowing us to use popular route search algorithms guarantee reachability. All-in-all, NEUROMLR is more reliable, scales to larger cities, robust to unseen data and more effective in learning from low volume of data.
Linitations: In the future, we would like to work capturing the personalization aspect of the problem and learn to transfer knowledge from one city to another.
Potential for Negative Societal Impact: Our proposed work facilitates robust and reliable computa- tion of the mostly likely route road networks. To the best of our understanding, we do not see any potential of negative socictal impact from this work.
10

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Checklist
For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] See Section O
(C) Did you discuss any potential negative societal impacts of your work? [Yes]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
[Yes]
2. If you are including theoretical results...
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(a) Did you state the full set of assumptions of all theoretical results? [Yes]
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13


Reviews:
[{"review_id": "wcbfJKP4H0", "reviewer": "Reviewer_ABKf", "summary": "This paper proposes a method for recommending travel routes in a city. As shown in Figure 2, in order to consider the meaningful representation of road network, this paper uses a combination of Graph Convolutional Network with Lipschitz embeddings. Multi-layered Perceptron (MLP) is then applied to predict the next visit.", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "- Strengths\n  - This paper is well-structured and easy to follow.\n  - The proposed architecture seems reasonable for this task. Through the ablation study (Sec. 5.4), it has also been shown that each component of the proposed method (Lipschitz embeddings and GCN) is working effectively.\n  - The minimum necessary comparisons with two conventional methods have been made using multiple data sets.\n- Weaknesses\n  - The novelty of the proposed method should be described in more detail. Is the key point of this work is to devise a pre-processing (PCA and Lipschitz embedding)? Does the majority of the proposed architecture relies on existing methods?\n  - I am concerning about whether the comparison experiment was done under fair conditions. (1) The authors set a restriction of 36 hours for the parameter training time, but has this restriction affected the experimental results? (e.g., would the same conclusion be reached if the time were 72 hours?). (2) For both DEEPST and CSSRNN, the authors use the default parameters prescribed by these papers. However, unless the parameters of the existing methods are learned in a way that is tailored to the current data, it may not be a fair comparison between the proposed method and the conventional method. (3) Why did the authors use only one of three datasets for the experiments on \"Accuracy\" and \"Reachability\"? I am also curious about the large difference in performance between each dataset (each town).", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Limitation of the proposed method is mentioned in the conclusion. The negative impacts of this study were not mentioned in the paper, nor were there any points that bothered me when I read the paper.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3 hours"}, "extra_text": {}}, {"review_id": "inUaXHwxsmz", "reviewer": "Reviewer_F1v9", "summary": "The paper introduces a route recommendation model based on graph neural networks with Lipschitz embedding initialization. The proposed model is compared with several baseline models and the evaluation shows that it outperforms baselines in terms of precision, recall, and reachability. Ablation study is done to demonstrate the importance of both Lipschitz embedding and GCN components. The model which uses inductive learning shows that it significantly outperforms baselines when the source/destination node is unpopular. ", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The model is easy to understand and explained clearly in the paper. There are ablation studies to help understand the impact of each component in the model. The results look promising compared with the baseline in the paper.\n\nHowever, I feel like there should be more exploration on the model component as GCN is only one possible way of implementing graph neural networks. What if different GNN architectures are used? Have you tried them? Are there any GNN based model for route recommendations? Maybe it's more fair to compare with them? It seems to me that GNN is only used to learn node embeddings and then an MLP is applied to generate the transition probability. Have you tried GNN models that can learn edge embeddings as well and include edge embeddings in the MLP input? Have you tried different ways to learn these embeddings and compare them? It seems that the proposed model mostly stack existing models together so the real novelty is not very significant. It would be better if the paper could explain or explore more on the alternatives, for example, maybe using GNN to also execute graph algorithms (https://openreview.net/forum?id=SkgKO0EtvS) instead of using MLP to generate transition probabilities. \n\nThere should be more explanation on why shortest path or quickest path does not perform well. I would assume in the dataset, people use existing apps to navigate which would potentially give a route based on a combination of route distance and time (which may consider live traffic). So probably crudely using shortest/quickest path without the live traffic is why is not performing well. \n\nFor Reachability Distance in the evaluation metrics, it seems to me network distance is better than Haversine distance? If two locations are not physically reachable in the road network, it really doesn't matter how close they are in terms of Haversine distance.\n\nIn terms of inductively learning, I think a better way to advance it is to handle dynamic network, such as road closures or new road being introduces. This would be more appealing in a practical sense. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "I don't see any potential negative societal impact of their work.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "RIrd-lRJIX", "reviewer": "Reviewer_egXq", "summary": "This paper presents a new approach for route recommendation. The key idea of the approach is to model the mobility pattern as a Markov process and learn the local transition probability from historical data through a GCN with Lipschitz embedding. The proposed approach overcomes two limitations in the existing works - (1) limited reachability and (2) limited generalizability on unseen/lesser-seen nodes. The paper evaluates their approach on public datasets with a few baselines. The improvement is significant and there are also a good amount of ablation studies and case studies to help the readers better understand the effectiveness of the proposed solution. ", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths:\n\nS1 The idea of this paper has its novelty in route recommendation. The way it models the route recommendation problem can inspire future works.   \n\nS2 The paper is well written. It is easy to follow and contains sufficient details. \n\nS3 The experiments are sufficient and well-designed. \n\n\nWeaknesses:\n\nW1 The precision and recall seem low but this may not be an issue of the proposed approach but the metric itself. There might be several reasonable routes going from A to B, and not all the drivers will follow the same route. By this nature, there should be an upper bound of the accuracy. For example, if there are 2 reasonable routes from A to B, and 80% of the drivers take route-1 (most likely route) and 20% of the drivers take route-2 (assume the traffic conditions and all other factors are the same). In this case, you may only be able to achieve 80% precision if the routing queries are all anonymous. Given the existence of this upper bound, the accuracy numbers reported in the paper might be good enough.\n\n\nAdditional Comments:\n\nC1 I wonder if there is a way to estimate the upper bound (see W1) of the accuracy from the data? Showing this estimation can be helpful to understand the nature of the most likely route prediction problem. \n\nC2 At line 254, training all the models for up to 36 hours does not seem to be a fair comparison. It would be better to fix the training epochs rather than training time. \n\nC3 How does the proposed approach handle changes in the road network? For example, there could be temporary road closure caused by construction or crashes. \n\nC4 At line 104, should it be Prob. 1 rather than Prob. 2?\n\n\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "No potential negative societal impact. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a method for route recommendation using graph neural networks and Lipschitz embedding. The idea of the proposed method for route recommendation is interesting. The proposed architecture is reasonable. The paper is well written, and experiments are well-designed. As commented by the reviewers, the novelty needs to be clearly explained in the revised paper. The experimental results showed in the author response would strengthen the paper.
Decision: Accept (Poster)