Generative Human Trajectory Recovery via Embedding-Space Conditional Diffusion
Authors: Kaijun Liu
Sijie Ruan
Liang Zhang
Cheng Long
Shuliang Wang
liang yu
Venue: ICML 2025 poster (2025)
Keywords: Trajectory recovery
Diffusion model
Self-supervised learning
Human mobility
Primary area: deep_learning->everything_else
TL;DR: This paper presents a novel embedding-space conditional diffusion based method for recovering human trajectories from incomplete data, outperforming existing approaches by an average of 11% in recall.

Abstract: Recovering human trajectories from incomplete or missing data is crucial for many mobility-based urban applications, e.g., urban planning, transportation, and location-based services. Existing methods mainly rely on recurrent neural networks or attention mechanisms. Though promising, they encounter limitations in capturing complex spatial-temporal dependencies in low-sampling trajectories. Recently, diffusion models show potential in content generation. However, most of proposed methods are used to generate contents in continuous numerical representations, which cannot be directly adapted to the human location trajectory recovery. In this paper, we introduce a conditional diffusion-based trajectory recovery method, namely, DiffMove. It first transforms locations in trajectories into the embedding space, in which the embedding denoising is performed, and then missing locations are recovered by an embedding decoder. DiffMove not only improves accuracy by introducing high-quality generative methods in the trajectory recovery, but also carefully models the transition, periodicity, and temporal patterns in human mobility. Extensive experiments based on two representative real-world mobility datasets are conducted, and the results show significant improvements (an average of 11% in recall) over the best baselines.

Generative Human Trajectory Recovery via
Empedding-Space Conditional Diffusion
Kaijun Liu
Sijie Ruan
Liang Zhang?
Cheng Long?
Shuliang Wang? Liang Yu"
Abstract
Han al., 2025), epidemic prevention (Tang et al., 2023) and urban planning (Yuan et al., 2014). However, due to pri- vacy concerns or device malfunctions, users may not report their locations to the service provider continuously, which makes human trajectories sparsely distributed in space and unevenly observed in time, and thus affects the effectiveness of downstream applications. Accurate recovery of missing trajectory points is also vital for related mrajectory-modeling tasks; for example, on-road trajectory modeling frameworks such as MMTraj+ (Liu et al., 2025) rely on complete se- quence inputs to learn trajectory patterns and destination cues via multi-task learning. Recovered human trajectories, especially those involving Points of Interest (POIs), can greatly benefit downstream applications such as POI recom- mendation, targeted advertising, and ride-sharing services by enabling more personalized and context-aware decisions. For this reason, human trajectory recovery, which infers human trajectories at a fine-grained level, raised more and more attention recently.
Recovering human trajectories from incomplete or missing data is crucial for many mobility-base -based urban applications, e.g., urban planning, trans- portation, and location- based services. Existing methods mainly rely on recurrent neural networks attention mechanisms. Though promising. they encounter limitations in capturing complex spatial-temporal dependencies in low-sampling trajectories. Recently, diffusion models show po- tential in content generation. However, most of proposed methods are used to generate contents in continuous numerical representations, which cannot be directly adapted to the human location trajectory recovery. In this paper, introduce conditional diffusion-based trajectory recovery method, namely, DiftMove. It first transforms locations in trajectories into the embedding space, in which the embedding denoising is performed, and then missing locations are recovered by an embedding decoder. DiffMove not only improves accuracy by introducing high-quality generative methods in the trajectory recovery, but also care- fully models the transition, periodicity. and tem- poral patterns in human mobility. Extensive exper- iments based on two representative real- world mo- bility datasets are conducted, and the results show significant improvements (an average of 1 11% in recall) over the best baselines.
Existing human trajectory recovery work leverages Recur- rent Neural Network (Liu et al., 2016; Wang et al., 2019) attention mechanism (Xia et al., 2021) for capturing the spatial- dependencies and resort sparse time in- terval encoding modules for handling unevenly observed trajectory records. Recent studies (Xia et al., 2021; Sun et al., 2021; Deng et al., 2023) further found that explicitly utilizing historical trajectory can enhance the performance due the strong periodicity nature of the human trajectories. However, these approaches face significant limitations in handling other key characteristics of human mobility. First, they struggle to capture intricate spatial-temporaa dependen- cies - the interplay between spatial relationships (proximity and spatial transitions between locations) and temporal pat- terns (sequential dependencies or periodicity of behaviors in historical trajectory). Second, is difficult for modeling uncertainty caused by sparse incomplete observations, which is common in real-world mobility data. Most de- terministic methods produce single imputation sample, potentially oversimplifying the variability in human behav- ior.
1. Introduction
Mobility data plays prominent role many urban applica- tions, c.g, next location recommendations (Feng et al., 2018:
'Alibaba-NTU Singapore Joint Rescarch Institute, Singapore Cooleeee of Computing and Data Science, Nanyang Technological University, Singapore "School of Computer Science and Tech- nology. Beijing Institute of Technology. China "Alibaba Cloud, Hangzhou, Zhejiang, China. Correspondence Sijie Ruan <sjruan bit..edu.cn...
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Although existing methods achieve appealing performance. these approaches all made the traditional predictive recov-

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Generative Human Trajectory Recovery via Embedding-Space Conditional Diffusion
ery, which has above limitations in complex and uncertain scenarios inherently in human mobility. Consequently, the recovery accuracy and scenarios of downstream applica- tions are limited. For instance, a person may follow some routines from home to office daily but occasionally he/she may follow different routes or change his/her preference. In such scenarios, traditional methods typically provide a biased deterministic imputed trajectory. However, with a generative approach to inference, set of imputed trajec- tory locations can be generated through sampling or various averaging techniques on imputation samples.
which provides a solution impute discrete locations leveraging diffusion models by performing the denoising process in embedding space and decoding the inferred embeddings back to discrete locations. To the authors' knowledge, we are the first to design spatial temporal con- ditional diffusion models for human trajectory recovery task.
We design Spatial Conditional Block, Target Conditional Block and Denoising Network Block to fully fuse the knowledge of the current trajectories and historical trajec- tories during the conditional diffusion process and tackle the above challenges.
To address these fundamental limitations, we leverage dif- fusion models, which have shown superior performance in many tasks against other generative models, e.g., image gen- eration (Ho et al., 2020) and audio synthesis (Kong et al., 2021). Furthermore, conditional diffusion models are cently developed for the time-series imputation (Tashiro et al., 2021) given observed entries as input, which inspires us to design trajectory recovery model in the conditional diffusion manner.
Extensive experiments on two real-world mobility datasets demonstrate that DiftMove significantly outper- forms state-of-the-ee af-the-rrt baselines, achieving average im- provement of 11 % in Recall.
2. Related works
Human Trajectory Recovery: The human trajectory re- covery problem we address focuses on free-space settings, unlike MTrajRec (Ren et al., 2021) and RNTrajRee (Chen et al., 2023b), which focus on vehicles' trajectories con- strained by road networks (road segments). Human trajec- tory recovery can be categorized into two types: the former treats missing locations in trajectories as continuous two dimensional values, i.e., latitude and longitude, to be im- puted (Alwan & Roberts, 1988; Moritz & Bartz-Beielstein, 2017; Wang et al., 2019), while the latter infers locations from a discrete candidate location pool (Liu et al., 2016 Xia et al., 2021). The former is suitable to recover trajecto- ries with high sampling frequency, e.g., vehicle trajectories, where the local context plays more important role for the imputation, while the latter is more feasible for highly sparse trajectories, e.g., human trajectories, where the tran- sition and periodicity dependency modeling are the main focus. In human trajectory recovery, the de-facto approach is to explicitly utilize historical trajectory when imputat- ing the current trajectory. For example, AttnMove (Xia et al., 2021) utilizes a multi-stage attention mechanism to recover missing locations. PeriodicMove (Sun al., 2021) constructs day-level graphs to model complex transition patterns among locations. TRILL (Deng et al., 2023) is a trajectory recovery model utilizing graph convolutional net- works, combining global and local mobility patterns. Chen et al. (2023a) proposes a framework called TERI, to tackle trajectory recovery in a two-stage process, with a different problem setting focusing on addressing the special cases of irregular time interval. Ours is more addressing on the generative manner using diffusion model to solve the regu- lar trajectory recovery problem. Existing human trajectory recovery work imputes missing locations in a deterministic manner, which omits the uncertain nature of trajectories and thus constrains the recovery accuracy and scenarios
However, is non-trivial to apply the conditional diffusion model to solve the trajectory recovery problem due to two issues. Firstly, the imputation targets of conventional dif- fusion models are continuous numerical values, which can be directly obtained via the denoising process, while those in trajectory recovery are discrete ID-represented locations (c.g., POI locations or geographical grid cells)- in this case, the transition and periodicity patterns of human trajectories are required to be fully exploited. Secondly, the model must simultaneously consider those abovementioned limitations in handling key characteristics of human mobility both temporal dependencies and the complex spatial relation- ships between current and historical trajectories during the denoising process.
To tackle these issues, we propose a novel conditional diffusion model for human trajectory recovery, namely, DiffMove. Our model incorporates these specialized com- ponents: 1) a novel embedding-basee (with encoding and decoding) conditional diffusion framework that handles dis- crete locations while preserving spatial relationships, 2) a Spatial Conditional Block equipped with ceffusiion-orennter graph neural network and attention mechanism, which cap- tures the sparse spatial transition patterns and periodicity temporal patterns from the current trajectories and historical trajectories. 3) a Target Conditional Block that effectively utilizes historical information despite sparse sampling. 4) a Denoising Network Block to handle uncertainty. Our contributions are three-fold:
* We propose a trajectory recovery framework D
'The code of this paper will be released in the link http://githhh.com/aajnn////fff
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Generative Human Trajectory Recovery via Embedding--Sacce Conditional Diffusion
of downstream applications. In addition, the relationship between locations to be imputed and historical trajectories are not well-modeled.
with historical trajectories Tu Tu.... Tu-1,, the task is to recover the missing locations, i.e., V null in Tw', thereby reconstructing the complete trajectory for the current day.
Diffusion Model for Temporal and Spatio-temporal
3.2. Denoising Diffusion Probabilistic Model
Data: Diffusion models have found extensive applications in tasks related to time series and spatio-temporal data gen- eration (Wang et al., 2024), imputation (Tashiro et al., 2021), and forecasting (Wen et al., 2023) due to their competence in modeling high-dimensiona data distributions. From the time series generation aspect, diffusion models can be used for the synthesis of electronic health records (EHR) (Alcaraz & Strodthoff, 2023; He et al., 2023; Yuan et al., 2023). Many of these studies adopt the denoising network architecture initially proposed in DiffWave (Kong et al., 2021), which utilizes bidirectional dilated convolution to capture corre- lations between different time steps. CSDI (Tashiro et al., 2021) leverages diffusion models for probabilistic time se- ries imputation, i.e., generating missing values conditioned on observed data points. Trff (Zhu et al., 2024) repre- sents the first attempt to generate GPS trajectories using an unconditioned diffusion probabilistic model. However, it focuses on generating task of raw GPS data in continuous space instead of discrete sparse locations that human trajec- tories always involve. TrajGDM (Chu al., 2023) employs a diffusion model to capture universal mobility patterns, for trajectory generation, but it focuses on simulating synthetic human mobility instead of recovery task on current trajec- tory. A recent work DifISGG (Wen et al., 2023) studied the apatial-temporal graph forecasting problem and introduced denoising network UGnet, which is capable of capturing spatial-temporal dependencies among various geographical locations. However, DifISTG focuses predicting numeri- cal readings of geographical sensors in different locations across different time, while we focus on recovering discrete locations in human trajectories.
Denoising Diffusion Probabilistic Models (DDPM) (Ho et al., 2020) are deep generative models, which map data from the normal distribution to another distribution via learnable denoising network step by step that we can eas- ily generate data sample following the similar distribution of g(zro) by sampling a random Gaussian noise. DDPM is composed of a forward process and a reverse process.
In the forward process, Gaussian noise is gradually added to the data sample wo ~ q(20) by a Markov chain. A closed form exists to transform the initial data sample 2o to the data sample æt at arbitrary time step by the reparameterization trick: 2; = vārao + v1 ā,e (1)
where a₂ = a102........... og, 02 = 1 - Br. 3r E (0, 1) denotes the noise level and is sampled from a Gaussian noise (O,
The reverse process iteratively denoises pure Gaussian noise ær NO, I) to generate the data sample æ follow- ing the similar distribution of g( The transformation be- tween data of two consecutive steps can be formulated as fol- lows: po (=( = (N (1 t), c0 t) (2)
where 9 is shared among different denoising time steps. The parameters of po p0( are calculated as fol- lows: (z a. (x) ví co t)), oz
3
1 - B, (3)
where Eg is the denoising network, which takes the noise- added data I, and the time step t as inputs and produces the predicted noise. By iteratively sampling according to Eq. (2), the generated data å is finally obtained. During the training stage, the denoising network parameters can be learned by minimizing L(0) = Blle ca (æx, t) I12. where xt can be obtained given æg based on Eq. (1).
3. Preliminaries
3.1. Problem Statement
Definition 1 (Trajectory). The trajectory is a chronological sequence of a user's locations within a single day. Lel Tē : + 15,2...- 1i..... + Ki.K represent the trajectory of user u on the j-th day, where 1 denotes the visited location during the k-th time slot within a specified time interval. the location for the k-th time slot is not observed, tisk is marked as null, ie., 12.k is missing,
4. Methodology
The main idea of diffusion-batte trajectory recovery is to first transform discrete locations in trajectories into the dense embedding space, then generate the recovered lo- cation embeddings via diffusion model, and finally rebuild the missing locations by embedding matching process.
u Definition 2 (Current and Historical Trajectory). For a given targeted day J and user u's ''s trajectory Tw,, we define TH as the user's 's current trajectory, while the historical trajectories comprise u's 's trajectories in the past (J 1) days, denoted as (Ty) T?. -11-
To generate satisfactory location embeddings, the true con- ditional data distribution g(et T "..,... -1) in the embedding space should be estimated well, where ef are embeddings of missing locations. Incorporating those conditions, based on the idea of the diffusion model, we need to learn the conditional transformation between
We follow (Xia et al., 2021) to formulate the human trajec- tory recovery problem as follows:
Problem Definition. Given user u's trajectory Ti! along
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Generative Human Trajectory Recovery via Sibedddig-Space Conditional Diffusion
1. I.Trjjectory Location Encoder (TLE)
2. 2.Conditional Embedding Denoiser (CED)
Diflusion Time Step t
Location Embedding
Spatial Conditional
Location Mask
Raw Current Trajectory Historical Trajectories
Historical Location Embeddings ehist
Block
hist
Temporal Length
Regularization Loss L
e
Random Masking
Denoising Network Block
Target
Conditional Block
Current Trajectory
Observed Location Embeddings e
3.Missing Location
Decoder (MLD)
Embedding
Imputation Target Embeddings
Noisy Target Embeddings e\
Estimated Noise Eo
Imputation
Sample
Noise Estimation Loss Laiffu
Embedding
Masked Locations len
Gaussian Distribution
Noise
Recovered Locations lea
Decoding
Cross Entrogy Loss heg
Legend:9 Recorded location
Missing location
Current observed location
Masked location
Operations:
Addition
Conv Concateenntoon
R
Figure 1. Overview of training stage of Diff
consecutive steps (from to 1): (o(e c Trl, (Tu) T :..,, Tod No
and gives the embeddings of observed locations eg and missing locations et in the current trajectory, and historical trajectories hiist, which is shown in the left part of Figure 1. During the training stage, et" is obtained by adding random Gaussian noise to embeddings of masked locations, while during the inference stage, et is directly sampled from the Gaussian distribution. We now elaborate on its training stage detail as follows.
Mo(e)" | Tod [Tu T2 u 1)) aa(e)", t T.', (Tu. To To I)I)
(4)
Specifically, the parameterization of DDPM in Eq. (3) is also extended to the conditional case below similar to the proof Tashiro et al. (2021):
As shown in Figure 1, the blue and white location icons in trajectories represent observed and missing locations re- spectively. During the training stage, we first randomly mask some observed locations in the raw current trajectory as imputation targets, and thus separate into the current trajectory (which is the actual input during the inference stage) and pseudo missing locations (icons orange). i.e., masked locations 4a- Locations in trajectories / E L are represented by discrete IDs, and we also assign a special ID to the missing location, i.e., "null". After that, we feed the current trajectory, masked locations and historical trajecto- ries into a location embedding layer, where each location 7 would be transformed into a dense representation e' E Rd by an embedding layer: el Embedding, (note: other embedding methods are also welcome). The imputation target embeddings et are added with Gaussian noise by Eq. (1) to form noisy target embeddings e/a.
p T.' (Tu) To T..... Ta [)=
u DDPM ta (, (e)" ta t| Tu,, (Tu) Tn .... Tod ()),
co(e) t T [T. T2 ...,,., I)) = o DDPM (e
(5)
where s4 (2tt,)) 1) and (o 1) in Eq. (3) are denoted as y DDPM (e,, t, Eæ(e,, t)) and o DDPM t) here respectively, and general variable x is replaced by the embedding of missing location e.
As can be observed, it essentially requires our denoising network to incorporate observations in the current tra- jectory and historical trajectories. To well encode those conditions and realize the diffusion-basee trajectory recov- ery, we present DifiMove, the training stage of which is shown in Figure 1. As the existing imputation work (Xia et al., 2021) did, DiffMove is trained in self-supervised manner, which randomly masks some observed locations in the current trajectory and treats them as supervision sig- nals, i.c., missing locations. To facilitate the description DiffMove, we decompose the whole process into three com- ponents: Trajectory Location Encoder (TLE), Conditional Embedding Denoiser (CED) and Missing Location Decoder (MLD).
Finally. historical location embeddings hist observed lo- cation embeddings eg and noisy target embeddings et would be fed into Conditional Embedding Denoiser for the noise estimation. The learned embedding table, i.c.. E, R([[++1xxd would also be used to perform matching to recover missing locations, which would be introduced later in Section 4.3.
4.1. Trajectory Location Encoder (TLE)
4.2. Conditional Embedding Denoiser (CED)
Conditional Embedding Denoiser (CED) takes diffusion time step f, noisy target embeddings eta and conditions
Trajectory Location Encoder (TLE) takes the current tra- jectory and historical trajectories of days as inputs,
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Generative Human Trajectory Recovery via Embedding Conditional Diffusion
B Gu C C Bill!
Secondly, a Diffusion Step T Gated Graph Neural Net- work (TGGNN) is proposed make diffusion-time-step aware spatial pattern learning. Two TGGNN are intro- duced to learn patterns from current trajectory and histor- ical trajectories, separately. We first transform the diffu- sion time step into a dense representation exte by sinu- soidal functions DiflTEmeedr()) (Kong et al., 2021; Tashiro et al., 2021), following by fully connected layer: er DiffEmbedy (t) + b². Then, embeddings of current and historical trajectories would be passed into TGGNN for several times. In the s-th layer of TGGNN as shown in Figure 3, (1) the information propagation from neighbor- hood is performed based on node embeddings of s-th layer [eg..... .,,] and two adjacency matrices, i.e., A and A to obtain incoming/outgoing aggregated node embedding e, er, respectively; (2) an intermediate representations as+1 is created by concatenating those aggregated node em- heddings with embedded diffusion time step er to enhance the representations; (3) gating mechanism (Li et al., 2016) is used to fuse the node embeddings of the s-th layer and (s+1)-th 1- layer:
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Sprid Conditena Block
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Tranlarg
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(A) Overall Archstecture Design for CED
(C) Targel Canditional Black
Figure 2. The Architecture for Conditional Embedding Denoiser
(i.e.. historical location embeddings e hist observed location embeddings eg ) and estimates the noise added to the target embeddings ei- at the time step, which is shown in the top right part of Figure 1. To fully exploit the power of conditions, Spatial Conditional Block is devised to model the transition and periodicity patterns, Target Conditional Block is designed to capture the relationship between the missing locations and historical trajectories, and Denoising Network Block is developed to capture the local context and produce the noise estimation. The design of each block is elaborated as follows.
Cr, = (A! ([o)......... o W + +b))
e0,. (A, (le)..... e w + bbo)
(6)
Spatial Conditional Block. Spatial Conditional Block takes cgh, e hist and diffusion step t, and gives the spatial con- dition expe, which captures the transition and periodicity patterns from historical trajectories. In addition to e an intermediate result, i.c., the diffusion time step embedding er, is also passed to Denoising Network Block, and a his- torical trajectory embedding hist is also obtained to better capture the relationship between the missing locations and historical trajectories in Target Conditional Block.
as+1 = C₁,s eo,s || er, c = Gates(as41.C) (a.41. c.)
where || is the concatenation, WT wo WT E Rddd are learnable parameters, and b', b bT E Rd are bias vectors, N is the number of unique locations in the trajectory, and Gates denotes several gates (Li et al., 2016). i.c., update gate, reset gate, to fuse node embeddings in consecutive layers. We denote the final node embedding of TGGNN as ef for simplicity. By pooling final node embeddings from the historical branch, the historical trajectory embedding hist is derived. G
Since graph neural network (GNN) has demonstrated its capability to capture the consecutive relationship between different entities (Xu et al., 2019; Wu et al., 2019) and attention mechanism is good at capturing the periodicity information (Liang et al., 2018), we propose to use GNN to learn the transition pattern and attention mechanism to learn the periodicity pattern. Since the degree of noise among different diffusion time steps is different, the importance of spatial conditions may also vary, we further incorporate the diffusion time step into the spatial condition learning. Considering above insights, we give the detailed structure of Spatial Conditional Block in Figure 2B. Firstly, we construct location transition graphs for both historical and current trajectories. For each trajectory, we construct in- coming and an outgoing transition graph, where all unique locations appearing in it serve as graph nodes, embeddings of locations from TLE serve as node embedding, and con- secutive locations together form two adjacency matrices, ie.. A and A A similar to (Xu et al., 2019).
TGGNN
e
e
5+1
sigmoid
sigmoid
01x
er
a1st
Element-wise multipliaation
+ Addition
Concatenation
Figure 3. Design of TGGNN.
Thirdly, we employ CrossAttention (Xia et al., 2021) to capture the periodicity among the current trajectory and historical trajectories. For each user u, the head, calculate
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Generative Human Trajectory Recovery via Embedding-SSacce Conditional Diffusion
the cross attention between the i-th time slot of the current trajectory embedding (e and the k-th time slot of the j-th historical trajectory embedding () The final spatial
putation, which transforms noises into meaningful location embeddings based the estimated noise from CED, and Embedding Decoding, which decodes the estimated target embeddings to locations to recover the trajectory.
condition expa is generated by a linear projection of the concatenation of H number of heads as shown in Eq. (7).
Embedding Imputation. Embedding Imputation is to ob- tain robust estimated target embeddings, we perform target embeddings generation for M times, and the means of tar- get embeddings ej are used for the location decoding. For each time of generation, random noise cla is sampled from N I), then we perform the reverse process of diffusion from step I to 1 gradually according to Eg. (4) and Eq. (5) to obtain one estimated target embedding c
head,, = CroossAttteniion((" I. ai
e e = (reLU((([eee) ... h))d + +
Target Conditional Block, Target Conditional Block takes target ID mask, temporal length and historical trajectory embedding chilst, and gives eind, which captures the corre- lations between the missing locations and historical trajec- tories to help the inference. The target ID mask represen- tations are in one-hot form: non-target locations are repre- sented with all zeros, while target locations are represented with ones. This method creates a placeholder embedding that signifies the absence of data at specific positions. This embedding is then concatenated with ehist and will be fused with the output of the Temporal Length Embedding layer. We incorporate temporal length embedding k- = (ki.)) } as auxiliary information. We adopt 128-dimensional tem- poral embedding, consistent with prior research (Vaswani et al., 2017; Zuo et al., 2020):
Embedding Decoding, After the mean imputed target em- beddings e are obtained, we calculate the inner product between ej and location embeddings in embedding table E, R which is from E, in TLE after excluding the em- hedding of "null" item. For each imputed target embedding ata,t its similarities to different locations 2 E R14 are
calculated as follows: 2,aa ====6 6 Tta.'
Subsequently, we apply a softmax function to obtain the location likelihood vector y' for each imputation target: y' = softmax (z)"²). During the inference stage, the loca- tion with the highest probability would be used to recover the trajectory.
Kembedding (kz) (sin(kt/>0/01).. sin(ki/773/00))
4.4. Model Training
cos cos(kt/7 /onts///
cos(kk// (8)
Since trajectory recovery results are discrete, which can- not be easily obtained by the denoising network, multiple losses are introduced as shown in Figure 1 when we train DiffMove.
Here, r 10000. This temporal length embedding en- riches the model with sequential information, enhancing its ability to recover trajectories. Finally, the concatenated representation forms eindd
The first loss is diffusion loss Laitlu which calculates noise estimation accuracy. We sample noise and obtain the noisy target embeddings ct at the diffusion time step t by Eq. (1). Then, DiffMove estimates the added noise condi- tioned on observed locations in the current trajectory and the historical trajectories. The expectation of the mean squared error between the actual noise and the estimated noise is served Laiffuu which is defined as follows:
Denoising Network Block. This block is the function of DiffMove to model Eg in Eq. (5). It receives the inputs from TLE together with er, epa and eind The concate- nation of <g and ctth from TLE as well as ca are passed through a ID convolution layer and a ReLU,, and the results of both would be added to er to form the the total input of a temporal transformer (Tashiro et al., 2021) (with multi- head self- attentientnn to learn the temporal sequence features. Then it will be passed through 1D convolution layer and added with the ID convoluted result of eind, Following a gated activation unit (Ramachandran et al., 2017), part of outputs is directed to the next residual layer as input, whereas the remainder is incorporated into the final output via a skip connection. The Convl x blocks in the net- work facilitate the mapping of data to suitable dimensions. Ultimately, the output é is the culmination of data passed through skip-connections from each residual layer.
Taimuu (0) = ((e)) (eneeei"")eN((.)))))) (0..))
co(o," t & (T,, T...TT---1 T2-1
(9)
The second loss characterizes the location recovery accuracy, which is the cross entropy loss. Given the one-hot repre- sentations of masked locations in the raw current trajectory Y = yy,,,,,, y Kin and the predicted likelihood of the imputed locations Y = (y y } (Kea is the number of masked locations in the raw current trajectory), LCB is defined as
y...,
4.3. Missing Location Decoder (MLD)
Kra Σ
Missing Location Decoder (MLD) leverages CED to re- cover locations. It consists of two steps: Embedding Im-
LCE (Y, (,,YYYY)
YY=
of log((()
(10)
1=1
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Generative Human Trajectory Recovery via Embedding-Space Conditional Diffusion
The third loss is an L.2 loss for regularization, which is suggested in (Gong al., 2022). It regularizes the learning of the location embeddings of the raw current trajectory, i.e.,
validation (20%) and test (20%) sets. We utilize the widely adopted metries Recall @ K and Mean Average Precision (MAP) (Wang et al., 2019). Recall K measures whether the ground truth is present in the top K predictions, averaged over all test cases. MAP evaluates the overall ranking qual- ity by considering the entire prediction list. Larger values for both metrics indicate better performance. Additionally, we use Distance K, which computes the smallest geograph- ical distance between the centers of locations in the top-K ranked list and the ground truth, averaged across test cases. Lower Distance @ K signifies better performance. We report experimental results for Recall @ K and Distance @ K at K = 1, and 10. This allows a comprehensive assessment of our model's ability to rank ground truth locations.
Kea
K
Kea
LL2(e)", e = Kd
100
1 112
+
Σ
1100
1)?
(11)
Consequenlly, DifiMove is trained end to end by jointly optimizing the above three types of losses: LEgE = Luita
X1 ICE + >2LLL, where X1 and >g are multi-task learning weights.
5. Experiments
5.1. Datasets
5.4. Experiment Results
This dataset (Yang et al., 2014) was ob- tained from the Foursquare API, covering the period from April 2012 February 2013. Each record in the dataset in- cludes user ID, timestamp, GPS location, and POI ID. We standardize the timestamps to a one-week format while preserving the original trajectory order.
Foursquare::
As shown in Table 1, firstly, rule-based methods fail to achieve high accuracy, exhibiting the worst performance for all evaluation metrics on both datasets. Secondly, RNN- based methods perform better than rule-based methods as they can model simple sequential patterns among locations. Bidirectional RNNs perform better than unidirectional ones, indicating the importance of spatial-temporal constraints for human mobility recovery. State-of-the-ar deep learning methods, including AttnMove, PeriodicMove and TRILL achieve satisfactory performance by capturing sequential patterns and simple periodicity human mobility. However, DiffMove outperforms all the baselines for all evaluation metrics on both datasets. Specifically, for Recall, DiftMove outperforms the best baseline, TRILL, by 9.57% Geolife dataset and by 11.56% on Foursquare dataset. For Distance, DiffMove outperforms the best baseline, TRILL, by 19.66% on Geolife dataset and by 7.93% on Foursquare dataset. For MAP, DiffMove outperforms the best baseline, TRILL, by 7.05% on Geolife dataset and by 9.56% on Foursquare dataset. These significant improvements indicate that our proposed DiffMove can better learn spatial temporal pat- terns of both current and historical trajectories and recover the details of human mobility. We also change the number of generated samples M (in Section 4.3) from 4 to 1, which simulates the normal single prediction method. We observe the reduced performances in Table 1 (-1.85% Recall on Geolife and -1.7% Recall @ on Foursquare) due to lack- ing probabilistic generation and sampling, which highlights the significance of the probabilistic generation instead of deterministic single imputed embedding.
Geolife:: This publicly available dataset is sourced from the Microsoft Research Asia Geolife project (Zheng et al., 2010), involving 182 users and spanning from April 2007 to August 2012 globally. Each trajectory is represented by a sequence of time-stamped points, providing longitude and altitude information (Zheng et al., 2010).
5.2. Baselines
We evaluate the proposed approach against baseline meth- ods, including both traditional approaches grounded in our understanding of human mobility and advanced deep leam- ing models capable of capturing intricate mobility patterns. We evaluate the proposed approach against below baselines: Rule-based methods: 1): Top, 2) Markov (Gambs et al., 2012), 3) PMF (Mnih& & Salakhutdinov, 2007). Deep learn- ing based methods: 4) LSTM (Liu et al., 2016), 5) BiL- STM (Zhao et al., 2018), 6) DeepMove (Feng al., 2018), 7) AttnMove (Xia et al., 2021), 8) PeriodicMove (Sun et al., 2021), 9) TRILL (Deng et al., 2023). Selections of base- lines are to ensure fair comparisons in the same setting of free-space human trajectory recovery. More details about baselines will be provided in the Appendices.
5.3. Experimental Settings
5.5. Ablation Analysis
Following (Deng et al., 2023), we mask randomly 10 time slots per day for both the Geolife and Foursquare dataset. The trajectories are split chronologically into training (60%),
We conduct ablations by systematically removing individual components. The results of Foursquare dataset are presented in Table 2. The recall, distance and MAP performance of the first ablation with unconditional diffusion (No observed location, no spatial, and target condition involved) drops sig- nificantly to almost nonfunctional status. This emphasizes
dataset/ http://wwwwwaiiioooooocoeeeeetts hwtp::/wce....too.oooonooooooooo
7

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Generative Human Trajectory Recovery via Embedding-Spacce Conditional Diffusion
Table 1. Overall performance comparison in terms of Recall@ K, Distance K, and MAP.
Dataset
Methods
Recall @
Distance @
MAP
Recall@
Recal
Recall @ 10
Distance@1
D5stance@5
Distance 10
Gcolife
Top
0.1148
0.2451
0.3166
7863
6259
5176
0.1812
Markov
0.1417
0.3263
0,3974
6909
4974
4259
0.2304
PMF
0.1941
0.3436
0,4059
6506
4389
3555
0.2752
L.TTM
0.2086
0.3917
0.4720
6318
3928
3068
0.2965
BiL.STM
0.2285
0.4538
0.5773
6209
3620
2255
0.3298
DeepMove
0,3045
0,5380
0.6371
5370
2052
1358
0.4131
AttnMove
0.3920
0.6696
0.7213
5342
2007
975
0,5046
PeriodicMove
0.4199
0.6893
0,7681
4209
1443
863
0,5385
TRILL
0.4721
0.7563
0.8364
3484
1112
603
0.5985
DitfMove single gen-sample
0,4988
0.7701
0.8350
2905
973
601
0.6180
DiffMove
0.5173
0.7987
0.8578
2799
0.6407
% Improvement
9.57%
5.61 %
2.56%
19.66 %
36.33%
26.37 %
7.05%
Foursquare
Top
0,0865
0.1673
0.2268
8427
4919
3483
0.1347
Markov
0.1090
0,2010
0.2575
8345
4402
3125
0.1792
PMF
0.1215
0.2468
0.2887
8116
3971
3229
0.2358
LSTM
0.1393
0.2540
0.3143
7913
3804
2801
0.2519
BiLSTM
0.2323
0.3968
0,4703
6206
2745
1849
0,3154
DeepMove
0,2612
0,4631
0.5337
5189
2648
1649
0.3789
AttnMove
0.2975
0,5172
0.5746
4942
2396
1482
0.4078
PeriodicMove
0,3125
0,5534
0.6264
4704
1758
1197
0.4245
TRILL
0.3227
0.5636
0.6372
4639
1650
1074
0,4341
DitfMove w/ single gen-sample
0.3430
0.4614
0.5009
5206
1964
1339
0.4035
DifIMove
0.3600
0.6090
0.6876
4271
1548
989
0.4756
% Imppovvement
11.56%
8.06%
7.91%
7.93%
6.18%
7.91%
9.56%
the inadequacy of relying solely the default diffusion probabilistic model for the trajectory recovery task in la- tent space and underscores the importance of integrating multiple spatial and temporal related specific conditional modules for effective learning and training. The removal of the Temporal Transformer or Spatial Conditional Block sig- nificantly impacts performance, emphasizing their critical roles in reinforeing spatial and temporal constraints for miss- ing locations, resulting in substantial improvement when leveraging historical information. The removal of the Target Conditional Block leads to decreased model performance, highlighting the role of the target condition in guiding the model to reconstruct specific embeddings in the locations through the diffusion process. Additionally, the Missing Location Decoder also identified crucial component. It can not be compared in the table since its removal ren- ders the model nonfunctional, as this module plays vital role in converting the reconstructed embeddings of missing locations into decoded discrete ID space.
5.6. Robustness Study
As shown in Table our proposed model, DiffMove, consis- tently outperforms the baseline models, AttnMove, Periodic- Move and TRILL across various missing ratios. The second best results are underlined and the improvements are listed in the brackets. Notably, as the percentage of missing loca- tions in historical trajectories increases from 20% to 80%, DiffMove exhibits superior performance, achieving higher Recall@ 10, lower Dist @ 10, and improved MAP scores compared to the baselines. This suggests that DiffMove is more robust in scenarios with higher missing percent- ages of historical trajectories and sparser locations. The significant reduction in Dist@ 10 for DiffMove indicates its effectiveness in accurately recovering missing locations. Re- markably, the Distance metric performance of our DiffMove with 80% missing ratio even outperforms TRILL with 40% missing rate and surpasses both PeriodicMove and Attn- move, even when they have lower missing rates 20%. This serves as one aspect of scalability and further reinforces the efficacy and good potential of DiffMove in handling larger datasets since it shows better performance even when the model is utilizing a smaller portion of the same existing data (larger missing ratio than those of baselines), which provides insights into its applicability across various scala- bility of missing ratio scenarios. These results underscore the robust, scalable and superior performance of DiffMove in more challenging task of trajectory recovery, making it a
Table Impact af components Foursquare dataset, where denoted the performance decline.
Ablation
R)cail())
Dix.(3))
MAP))KK)
Unconditional
(-88.4 (8.44)
7913 (-85.77)
0.0944 (-80.15%)
Spusial Conditional Block
0.3382 (-6.06%)
4591 (.7.49%)
0.4496 (-5.47%)
Target Conditional Block
0.3493 (-2.97%)
4177 (-2.48%)
0.4632 (-2.61%)
Temporal Transformer
0.3023 (-16.03%)
4783 (-11.99%)
0.4166 (-12.41%)
8

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Generative Human Trajectory Recovery via Embedding-SSpan Conditional Diffusion
Recall Metrics &r
Distance Metrics Br
Recall Metrics vs Embed_Size
Distance Metrics vs Embed Size
6000
Dishrrnsk
4000
Didlncoog
0.6
0.5
Reco 10
Reco 5
w0y
Beca 1
values
2000
Disiancegg 10
Disianceg's 5
nce@1
Values
0.2
0.4
Reca
Roca
Values [m)
4000
2000
Mitarcegi 10
0.4
Rackig0D
1000
M66
0.4
0.2
0.4
0.6
0.0
256
128
64
32
16
0
256
126
64
32
16
Pr
Pr
Embed_Size
Embed_Se Size
Figure 4. Recall vs ST
Figure 5. Distance vs Br
Figure 6. Recall vs Embed_ Size Figure 7. Distance vs Embed_Size
promising model for real-world applications.
ing a novel model, DiffMove. Leveraging an embedding- space conditional diffusion framework, excels trajectory recovery by constructing and utilizing conditional informa- tion with trajectory spatial patterns, inter-trajectory depen- dencies, temporal and target location patterns. The model is innovatively designed and integrated with multiple condi- tional feature extraction modules, tackling the complexity of spatial temporal dependencies in a principled manner.
Table 3. Performance w.r.t Missing Ratios on Geolife
Methods
Metrics
Missing Rate
20%
40%
60%
80%
Attn
@ 10
0.7117
0.6985
0.6785
0.6160
Dist@
987
1037
1174
1371
MAP
0.4815
0.4657
0.4226
0.4112
PeriodicMove
Recall 10
0.7451
0.7392
0.7186
0.6857
Our extensive experiments demonstrate that DiffMove out- performs all state-of-the of-the-art baselines, showcasing its effec- tiveness in recovering missing locations. While DiftMove
@ist@ 10
884
954
1059
1176
MAP
0.5175
0.4750
0,4413
0.4076
TRILL
Recall 10
0.8216
0.8038
0.7627
0.7436
Dist@ 10
682
720
915
1089
demonstrates strong performance under random masking,
MAP
0.5760
0.5534
0,5111
0.5044
wolld uncertainty may involve structured missingness (e.g,, persistent gaps due to device failures). Future work could explore models conditioned on specific missing pat- terns.
DiffMove
@ 10
0.8344
0.8163
0.7931
0.7863
Dist@ 10
507
617
681
695
MAP
0.6107
0,5730
0,5495
0.5099
5.7. Parameter Study
A Acknowledgements
We also conduct some experiments to provide insights into the performance of our model (DiffMove) across different values of Br and embedding size.
This work was supported by the National Key R&D Program of China (No. 2023YFC2308003 and the National Natural Science Foundation of China (No. 62306033), This research is also supported in part by Alibaba Group through Alibaba- NTU Singapore Joint Research Institute (JRI), Nanyang Technological University, Singapore, and by the Ministry of Education, Singapore, under its Academic Research Fund (Tier 2 Award NOE--2EP20221-00 -0001 and Tier 1 Award (RG20/24))- Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of the Ministry of Education, Singapore.
Beta_end inde_nnda 3T: Figure 4 and illustrate the interplay be- tween Recall @ K, MAP, and Distance @K K across different values of By. We vary the 31 to change the noise schedule, the Recall@1 1 and @istance@1 1 performance are more impor- tant and seem to have increasing trends but drop when 3T is too large although there are some fluctuations for @ and Recall@ @ 10. We try to choose the optimal value at 0.6 af- ter consideration of all tradeoffs. The relationship between 37 and spatial Distance@) accuracy reveals specific 3rr values that result in optimal spatial alignment, indicating the importance of Br in shaping spatial aspects of trajectory recovery performance.
Impact Statement
Embedding Size: In addition to 37. Figure 6 and illustrate the variation in all metrics across different embedding sizes. As expected, the initial increase of the embedding size con- tributes to the increase of @ecall@e 1 and Distance@) 1 since more information is recorded by embedded vectors. How- ever, too large embedding could also bring some uncertain information and lead to saturation of prediction accuracy. As a result, we choose the optimal value at 128.
This paper presents work whose goal is to advance the field of Machine Learning. There are potential societal conse- quenees of our work, none which we feel must be specifi- cally highlighted here.
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6. Conclusion
In conclusion, this research addresses the problem of trajec- tory recovery from sparse human mobility data by introduc-
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Generative Human Trajectory Recovery via Embedding-Space Conditional Diffusion
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Generative Human Trajectory Recovery via Embedding- Conditional Diffusion
A. Appendix / Supplemental material
A.1. More Details on Transition Graph Construction of Spatial Conditional Block
The prior investigations (Xu ct al., 2019; Wu et al., 2019; Sun et al., 2021) have demonstrated that a gated graph neural network (GGNN) is adept capturing intricate transition patterns among nodes. This characteristic renders the gated GNN well-suited for addressing our specific problem. In the graph neural network layer, handle each trajectory independently unveil the complex transition patterns concealed within each trajectory. To elaborate, we initiate the process by establishing directed graph for each trajectory. Subsequently, the gated GNN is employed each of these directed graphs refine the location embeddings, thereby capturing the transition patterns into the model. Besides this, we manage to conduct data fusion diffusion step embedding into the gated GNN to make the transition pattern learning adaptive to the diffusion time step.
Trajectory Graph Construction: The initial step of the graph neural network layer involves constructing transition graph representation for each historical and current trajectory in the context of trajectory recovery. Similar to session recommendation, given location IDs' trajectory 4 + l2 - (K. we consider each location 4 as a node and (4-1. 4) as an edge, representing the user's movement from 4 4 in the trajectory conceptualized directed graph. The graph structure is learned by facilitating communication among distinet nodes, Specifically, let A! Ao denote the weighted transitions of incoming and outgoing edges in the trajectory graph, respectively. To address the possibility of repeated occurrences of locations in a trajectory, calculated as the edge's 's occurrence divided by the outdegree of the start node of that edge. Consider transitions in a trajectory [4, 12, 4,, 12, 14), the corresponding graph, the incoming matrix A and the outgoing matrix are shown in Figure
Consequently, each trajectory can be
assign each edge a normalized weight,
Outgoing Matrix
Incoming Matrix
1
3
1
3
0
0
0
0
0
1/2
1/2
1/2
1/2
0
0
0
0
0
0
0
2
3
Figure 8. A example of trajectory transition graph and the incoming and outgoing matrix A
A.2. Details on Diffusion Step Embedding
Ostep is the 128-dimension diffusion step embedding obtained from a special embedding layer DiffEmbedr,()) by sinusoidal functions following previous works (Kong et al., 2021; Tashiro et al., 2021):
DiffEmbedr-() (1i
sin(1063-4/631) cos(1100-/66))) 5 cos( (1063-4663))
(12)
and it further processed through a fully connected layer to obtain er.
A.3. Details Cross Attention in Spatial Conditional Block Further details of Eq. (7) are elucidated in Eq. (13).
12

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Generative Human Trajectory Recovery via Embedding-Spaae Conditional Diffusion
(h) exp( (el E((a))
0i4 Eg exp(o(()(E o)))))
(()) (2 2 = (W w 2 W et
(13)
( L
(W (1) E
(2) ...
2u (
u
ea = ReLU + t
where W wh( wh(h) E Rd' xd are transformation matrices, and (.) denotes the inner product function. Next, we Q compute the representation of time slot for each head by aggregating information from all locations in other time slots based on the coefficients (h) heads.
i,k. The symbol denotes the concatenation operator, and H represents the total number of
A.4. Imputation (Sampling) Algorithm with DiffMove
Algorithm Imputation (Sampling) with DiffMove
I: Input: Location Embedding sample eo, No. of generated samples M, trained denoising function eø
2: Output: Imputed missing value a
3: Construct observation condition of e, as e
4: for m = to M do
5: e7 [O, I) where the dimension of eit" corresponds to the missing indices of eg
6: for
to do 7: Sample etth using Eq. (4) and Eq. (5)
8: end for
9: Record ata
10: end for
11: Calculate mean value o by mean 6%)
A.5. Study of Number of Generated Samples M
We change the number generated samples M from 4 to (meaning only predict one single embedding and use it directly), which simulates the normal deterministic way as an ablation study to some extent. We observe the Table results of reduced performances of the original DiffMove (-1.85% Recall @ on Geolife and -1.7% Recall@1 on Foursquare) due to lack of considering effects of probabilistic generation and sampling, which highlights the significance of the probabilistic generation instead of deterministic single imputed embedding.
A.6. Implementation Details
DiffMove is trained using batch gradient descent with the Adam optimizer (Kingma & Ba, 2014), implemented in Python and PyTorch (Paszke al., 2019), Linux server equipped with an NVIDIA RTX A5000. We set random seed as 2021. Multi-task learning weights A and A2 are set after experimental study. We employed a learning rate of 0.001 with a weight decay of Ic-6. We set the location embedding size as 128, the steps (loops) of TGGNN as 2, the number of heads for cross attention as 4, diffusion step embedding dimension and temporal length embedding dimension are 128.
Dataset
Methods
Recall@
Recall @
Distance@
Distance @
MAP
DiffMove single gen-sample
0.4988
0.7701
0.8350
2905
973
601
0.6180
Geolife
DiffMove
0,5173
0.7987
0.8578
2799
708
444
0.6407
DiffMove w/ single gen-sample
0.3430
0.4614
0.5009
5206
1964
1339
0.4035
Foursquare
DiffMove
0.3600
0.6090
0.6876
4271
1548
0.4756
Table 4, Performance comparison between DiffMove variants
13

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Generative Human Trajectory Recovery via Embedding-Space Conditional Diffusion
We the number of residual layers as 4, residual channels 128, and attention heads for the temporal transformer as 8. We set the number of the diffusion step T 50, the minimum noise level f1 0.0001, and the maximum noise level BT = 0.6. We tuned hyperparameters for each dataset to achieve optimal results. Following recent studies (Song et al., 2021; Nichol & Dhariwal, 2021), quadratic schedule was adopted for decay of a to enhance sample quality:
8. +
A.7. Details of Baselines
We evaluate the proposed approach against several baseline methods, including both traditional approaches grounded in understanding of human mobility and advanced deep learning models capable of capturing intricate mobility patterns:
Top: A straightforward counting- bousting-based method that selects the most frequently visited location in the training the recovery for each user.
Markov (Gambs al., 2012): A commonly used method treating visited locations as states and constructing a transition matrix capture first-order transition probabilities.
PMF (Mnih& & Salakhutdinov, 2007): An advanced model rooted in conventional collaborative filtering, based on the user location matrix.
LSTM (Liu et al., 2016): A deep learning model that captures sequential patterns through recurrent neural networks, using the predicted next time slot as recovery.
BiLSTM (Zhao et al., 2018): An extension of LSTM with bidirectional recurrent neural networks, incorporating spatial-temporal constraints from all observed locations.
DeepMove (Feng al., 2018): A model that jointly considers user preferences and sequential dependencies for predicting the next location used for recovery.
AttnMove (Xia al., 2021): A method leveraging various attention mechanisms to capture the regularity and patterns in user's mobility.
PeriodicMove (Sun et al., 2021):A recent model that considers factors such as transition patterns among locations and periodicity in human mobility.
TRILL (Deng et al., 2023): The latest state-of the-te- model which is capturing global mobility patterns leveraging graph convolutional networks for mobility patterns.
Table 5. Basic statistics mobility datasets.
#Historical Trajs
#Current Trajs
#Distinctive Locations
Total #IDs Processed in Training (approx.)
Dataset
City
Foursquare
Tokyo
11,430
2,286 3,912
1,411
404.9K
Geolife
Beijing
15,648
1,124
563.3K
A.8. Pre-processing
For our location representation, We collect the cities' street map data of Tokyo and Beijing from an online map source and partition the region into distinct blocks. Each of these blocks is considered as an individual location, with an average area size of approximately 0.25 km2 (500m 500m for both datasets). Other pre-processings are the same as (Deng et al., 2023). As per (Chen et al., 2019), 30-minute time interval is employed for both datasets. Further details and statistics presented in Table 5. Our framework is not inherently tied to the 30-minute interval and can easily adapt to other interval lengths, such as 10 minutes or even finer resolutions, provided the data supports such granularity. We can always treat the time interval as a parameter to tune on the data side. This flexibility makes our method broadly applicable, as it caters to human mobility patterns by modeling discrete location transitions effectively. Human mobility typically involves meaningful transitions at specific timeframes (e.g., work, shopping, dining), which align naturally with discrete intervals.
14

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Generative Human Trajectory Recovery via Embedding-Space Conditional Diffusion
A.9. Scalability Study
Based on the experimental results shown in Figures 9 and 10, conducted a scalability study comparing the performance of the DiffMove and best baseline model TRILL across different scaling ratios ranging from 20% 100% to vary the scale of the whole training dataset. Regarding recall performance, depicted in Figure 9, DiffMove consistently outperforms TRILL across all scaling ratios. In terms of distance performance, illustrated in Figure 10, both models display an increasing trend with higher scaling ratios. Notably, DiftMove maintains a lower distance value (more accurate) compared to TRILL across all scaling ratios, indicating superior trajectory recovery accuracy. During the experiments, we also found that the average training time per epoch ranges from 4.2s to 22.4s (scaling from 20% to 100% of full training data) which still comparable with the best baseline. The training time can satisfy the common requirement in company services since this model is only for offline applications of trajectory recovery. These findings suggest that Dift exhibits better scalability and accuracy to variations in different data scales, making it promising solution for trajectory recovery tasks across diverse datasets and scaling scenarios.
Recall vs Scaling Ratio
Distance vs Scaling Ratio
0.4
4000
0.3
Distance (m)
Recall
3000
0.2
2000
0.1
TRILL
1000
TRILL DiffMove
DiffMove
0.0
0
20
40
60
80
100
20
40
60
80
100
Scaling Ratio (%)
Scaling
Ratio (%)
Figure 9. Recall vs Sample%
Figure 10. Distance vs Sample
15


Reviews:
[{"review_id": "eUaUlVtEvq", "reviewer": "Reviewer_bS4d", "summary": "This paper introduces DiffMove, a new framework for recovering human trajectory data based on conditional diffusion model design. DiffMove effectively handles complex spatial-temporal patterns in low-sampling data. It works by transforming trajectory locations into an embedding space, denoising the embeddings, and then decoding them to recover missing points. DiffMove improves recovery accuracy by modeling mobility patterns like transitions and periodic behaviors. Experiments on two real-world datasets show that DiffMove outperforms previous methods by more than 10%. Overall, this research idea is interesting, and the recovering problem has good impact to the field.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "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": "Thanks for your valuable comments and stating our evaluations on multiple datasets are good, claims are clear and convincing. Our responses to other parts are as below:\n\nW(i):\n\nRecovering sparse human trajectories, particularly those involving Points of Interest (POIs), significantly enhances various mobility-related applications. By accurately reconstructing incomplete trajectory data, we can improve POI recommendations, leading to more personalized and relevant suggestions for users. Additionally, comprehensive trajectory data supports better urban planning and traffic management by providing insights into movement patterns and congestion areas. Location-based services, such as targeted advertising and ride-sharing, also benefit from complete trajectory information, resulting in improved user engagement and operational efficiency. Overall, the ability to recover and utilize complete trajectory data is crucial for advancing various applications that rely on understanding human mobility. We will clarify this in the revised manuscript to make readers easier to understand.\n\n\nW(ii):\n\nThe term \"ID-based representations\" refers to how locations are discretized into identifiable points (e.g., POI locations or geographical grid cells) rather than being represented as continuous GPS coordinates. As mentioned in Section A.8 Data Preprocessing, we used publicly available online map services to define the geographical partitioning of the study areas (Tokyo and Beijing) into 500m x 500m blocks. This partitioning provides a grid-based ID representation of locations. This is commonly used in mobility data analysis where raw GPS trajectories are mapped to meaningful locations, making it easier to handle data sparsity and improve interpretability. The baselines—AttnMove proposed this, and PeriodicMove, TRILL focus on the same data preprocessing, which aligns with our problem setting. We will clarify this in the revised manuscript.\n\nW(iii):\n\nYes, recall is a relevant metric for trajectory recovery, especially when evaluating how well the model retrieves missing locations. Given that real-world applications prioritize capturing as many true missing locations as possible, recall helps assess the effectiveness of the model in recovering essential trajectory points. The baselines—AttnMove, PeriodicMove, and TRILL focus on the same trajectory recovery task, which aligns with our problem setting and they also proposed the same evaluation metrics earlier, so we continue to do the fair comparison with them on this. Other works in trajectory imputation and POI recommendations also utilize recall as a key performance indicator, aligning with our evaluation approach.\n\nW(iv):\n\nOk, noted on the minor change. We will modify \"%Improv. \" to \"%Improvement\" in the table of the revised manuscript.\n\nSupplementary Material:\n\nWe think your answer no is by mistake or misunderstanding maybe. Just for information only, we have supplementary materials which include implementation details and code. We include additional details on data processing, model design, and other experiments such as efficiency and scalability study in the supplementary material and appendices.\n\nOverall, thanks for your positive and insightful feedback.", "claims_and_evidence": "The claims are clear and convincing", "methods_and_evaluation_criteria": "yes", "theoretical_claims": ".", "experimental_designs_or_analyses": "yes, evaluations on multiple datasets are good", "relation_to_broader_scientific_literature": "Urban computing, spatio-temporal data science, pandemic control", "essential_references_not_discussed": "references are good", "other_strengths_and_weaknesses": "The proposed model is novel and could be applied to other scenarios. The evaluations are solid, with lots of comparisons on multiple datasets, detailed ablation studies, and solid comparisons. The writing is clear and easy to follow, with lots of figures that make it much easier to follow.  \n\n\n\nThere are also some concerns. \n\n(i) The authors could discuss potential impact of the recovered data. Otherwise, it is hard for broader readers to understand. For example, who can use the recovered results for what applications. More discussion will be appreciated. \n\n(ii) In line 66, trajectories are ID-based representations, this is not easy to understand. Because GPS trajectories could also been continuous, e.g., 36.88284.  \n\n(iii) is Recall a regular metric for this kind of recovery problem?\n\n(iv) The writing could be revised. For example, in Line 343, \"%Improv. \" could be revised to \"%Improvement\" since there are enough space in the table.", "other_comments_or_suggestions": "no", "questions_for_authors": "no", "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": "", "extra_scores": {}}, {"review_id": "SDP8t0l0Qe", "reviewer": "Reviewer_9gdy", "summary": "This paper introduces DiffMove, a conditional diffusion-based model for recovering missing locations in sparse human mobility data. By converting discrete trajectory locations into a continuous embedding space, DiffMove effectively denoises and reconstructs missing locations through an embedding decoder. The model captures spatial and temporal dependencies using modules including the Spatial Conditional Block, which leverages graph neural networks and attention mechanisms, and the Target Conditional Block, which extracts knowledge from historical trajectories. Experiments on Geolife and Foursquare datasets show that DiffMove outperforms leading methods, achieving an average 11% improvement in recall rate. Although highly robust, the model could benefit from additional visualizations and efficiency analyses.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Thank you for your detailed feedback. \n\nClaims&Evidence:\n\nWe clarify how existing experiments explicitly demonstrate DiffMove’s superiority in complex, irregular, and uncertain scenarios:\n1. Probabilistic Generation vs. Deterministic: Table 1 and 3 shows that sampling multiple trajectories (DiffMove) improves Recall@1 by 1.7–1.85% over single-sample generation. This directly validates our claim that probabilistic diffusion captures uncertainty in human mobility, whereas deterministic baselines cannot.\n2. Robustness to Extreme Sparse and Irregularity: Table 6 (Appendix A.10) tests DiffMove under different missing ratios. For example, Distance metric performance of our DiffMove with 80% missing ratio even outperforms TRILL with 40% missing rate and surpasses both PeriodicMove and Attnmove, even when they have lower missing rates 20%. \n3. Due to the page limit, including other unnecessary case study (more on cherry picking certain special cases), while small valuable, would extend the scope and length of the manuscript beyond the intended focus. DiffMove demonstrates significant improvements across different missing ratios (Table6). These experiments simulate sparse, irregular, and uncertain scenarios (e.g., through random masking), and our method consistently outperforms baselines in recall and distance metrics. This already provides empirical evidence of the effectiveness of our approach. We will clarify these in the revised manuscript.\n\nRela To Literature:\n\nDiffMove’s technical design and problem focus are distinct from the cited works, as detailed below:\n\nUnlike CSDI which handles continuous numerical sensor time series (e.g., temperature), DiffMove tackles the fundamentally different challenge of recovering discrete location IDs and is designed as an entirely new framework of embedding-based diffusion and decoding to categorical space (Sec. 4.3) by a single end to end training. It is rather than raw continuous numerical space (unlike CSDI’s simple and single regression-style output).\n\nAnother difference is Trajectory-Specific Architecture: Our model introduces spatial-temporal conditioning (new TGGNNs + cross-attention) features and decoding features for human mobility patterns (Sec. 4.2), which CSDI lacks. \n\nMoreover, unlike RNTrajRec and Diff-RNTraj—which rely on vehicles and road network constraints (next road has to be adjacent to the current road) —our approach is designed for scenarios where such road external priors are unavailable, which is fundamentally different problem setting and technical design. RNTrajRec use a transformer to infer missing points deterministically, leveraging road network constraints (e.g., road segments) to guide the predictions. Diff-RNTraj combines diffusion modeling with road network constraints, processes continuous GPS and road graphs and focuses on synthetic generation, not recovery of real-world human sparse trajectories. In contrast, our innovative Spatial Conditional Block and Target Conditional Block (Sec 4.2) are differently engineered to capture the human trajectories’ complex interplay between spatial and temporal dependencies. \n\nOther Suggestions:\n\nThank you for your valuable feedback. \n1. We add a schematic diagram in below link for TGGNN module would enhance intuitive understanding in the revised manuscript. **https://anonymous.4open.science/r/A01D/README.md** \n\n2. Our method is trained using a masked recovery framework, where we mask certain locations in the trajectory and train the model to recover them based on the observed data, with ground truth available for validation. Once the model is fully trained, we simply apply the same recovery mechanism to infer the real missing locations. DiffMove’s inference process for missing locations is explicitly detailed in Section 4.3 and Appendix A.4. During inference:\n\nStep 1: The model uses observed locations in the current trajectory and historical data to condition the diffusion process. Step 2: Noisy embeddings for missing slots are iteratively denoised via the reverse diffusion process (Eq. 4–5), guided by Spatial/Target Conditional Blocks. Step 3: Decode denoised embeddings.\n\nQuestions:\n\nDiffMove leverages contextual information from adjacent time slots and the overall spatial-temporal patterns learned during training to infer the most plausible value for that slot. 1. Transition Patterns: The Sec. 4.2.1 models transitions between locations, inferring k from neighboring observations. 2. Periodicity: Cross-attention identifies recurring patterns (e.g., daily routines) across historical days, even if k-th slots are missing. 3.Global Mobility Knowledge: The embedding table E encodes universal location semantics (e.g., \"office\" vs. \"home\"), enabling recovery via similarity matching. 4.Empirically it is validated by our robustness study (Table 6). It shows DiffMove with 80% missing data outperforms all baselines. This demonstrates robustness to extreme sparsity, including scenarios where critical slots are missing.", "claims_and_evidence": "Not all.\nThe authors claim that the diffusion model can enhance performance in complex, irregular, and uncertain scenarios, providing\nseveral examples to support this assertion. However, more evidence is needed to substantiate these claims, such as performing case\nstudies, or conducting some specific experiments in these scenarios. The current experiments are insufficient to clearly demonstrate\nthe limitations of existing methods and the superiority of the proposed model, which is crucial for justifying the motivation behind\nthis research.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "There are no proofs or theoretical claims in the paper.", "experimental_designs_or_analyses": "Yes, the existing experimental design is sound and valid.", "relation_to_broader_scientific_literature": "This paper builds upon previous work on human trajectory recovery and the study of diffusion models. It presents the first work to\ndesign spatial-temporal conditional diffusion models for the human trajectory recovery task, achieving significant improvements and\ncomplementing the existing literature. However, this paper shares some similarities with existing works in technical designs such as\nCSDI[1], RNTrajRec[2], and Diff-RNTraj[3].\n\n[1] Tashiro, Yusuke, et al. \"Csdi: Conditional score-based diffusion models for probabilistic time series imputation.\" Advances in\nneural information processing systems 34 (2021): 24804-24816.\n\n[2] Chen, Yuqi, et al. \"Rntrajrec: Road network enhanced trajectory recovery with spatial-temporal transformer.\" 2023 IEEE 39th\nInternational Conference on Data Engineering (ICDE). IEEE, 2023.\n\n[3] Wei, Tonglong, et al. \"Diff-rntraj: A structure-aware diffusion model for road network-constrained trajectory generation.\" IEEE\nTransactions on Knowledge and Data Engineering (2024).", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "Strengths :\n\n1. This paper introduces the first spatial-temporal conditional diffusion model specifically designed for human trajectory recovery,\nshowing significant improvements.\n\n2. It designs and integrates multiple conditional feature extraction modules to tackle the complexity of spatial-temporal dependencies.\n\n3. Extensive experiments on two real-world datasets demonstrate the model's effectiveness and improvements over existing methods.\n\nWeakness:\n\n1.See the Claims and Evidence and Relation to Broader Scientific Literature for details.", "other_comments_or_suggestions": "1. It's better to draw a schematic diagram of the TGGNN module in the article for intuitive understanding.\n\n2. It's better to include how to infer the real missing locations.", "questions_for_authors": "What would be the effect of the model if the k-th position is missing in both the current trajectories and the historical trajectories?", "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": "", "extra_scores": {}}, {"review_id": "qDu2gZNDFm", "reviewer": "Reviewer_Skim", "summary": "The paper proposes the model DiffMove which is a conditional diffusion-based method for human trajectory recovery that leverages embedding denoising.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Thank you for your detailed feedback. \n\nW2:\n\nThey are shown as the first two equations in eq 6. Actually this is to treat trajectory embeddings as sessions and $e_0^{ob}$ and $e^{hist}$ are integrated using session based graph methods (Xu et al., 2019) and explained in Appendix A.1. We will revise section 4.2 and add a schematic diagram in below link for the TGGNN (yes, a new method proposed) module would enhance intuitive understanding. **https://anonymous.4open.science/r/A01D/README.md** \n\nW1&W3:\n\nOur work focuses on free-space trajectory recovery where data can be from either case of discrete normal location grid IDs and sparse irregular check ins and data points are not constrained by roads, a special setting that differs from the listed papers.\n\nTAU is a different problem (next POI) from our work which focuses directly on trajectory recovery. TrajBERT and TrajWeaver either operate under different assumptions (e.g., leveraging road network constraints, normal grid IDs only or continuous GPS) or target slightly different objectives,e.g., next POI or trajectory generation (creating new trajectories from scratch). TrajWeaver works directly with continuous GPS coordinates, and the datasets used are smooth taxi trajectories (points laid on road networks) collected in Xi’an and Cheng Du. These differences highlight a key distinction in problem settings. Our baselines’ problems proposed by AttnMove, PeriodicMove, and TRILL, have been widely used in the literature for this specific problem setting, ensuring fair comparisons. \n\nGiven the most listed models are not providing their source code yet, regarding new recent baselines, we discussed TERI (Chen et al., VLDB 2024) cited in our related work. We have been working hard on setting up a common problem setting, manage to settle the necessary data pipeline to fit the TERI and get this model run, where our model still outperforms SOTA baselines as shown below in this problem setting.\n\n|Dataset|Methods|Recall|Dataset|Methods|Recall|\n|-------|-------|------|-------|-------|------|\n|Foursquare|AttnMove|0.2975|Geolife|AttnMove|0.3920|\n||PeriodicMove|0.3125||PeriodicMove|0.4199|\n||TRILL|0.3227||TRILL|0.4721|\n||TERI|0.3355||TERI|0.4922|\n||DiffMove|0.3600||DiffMove|0.5173|\n\nW4:\n\nWhile we agree that visualizations can provide intuitive insights, the denoising process in our method occurs at the embedding level rather than in the direct data space (e.g., raw trajectory points or maps). Visualizing these high-dimensional embeddings would offer very limited interpretability. Instead, the effectiveness is shown quantitatively through our results as shown in Table1,2,6,Fig3-9.\n\nTheoretical Claims:\n\nEq5 can be surely derived based on eq3 (refer to page4 of cited CSDI) due to space limit, we will add to revised manuscript.\n\nQ2 &Q4:\n\nIt refers to the limitations of deterministic trajectory recovery methods in capturing the full complexity of spatial-temporal dependencies in low-sampling scenarios. Next location recom do not address the challenges of reconstructing entire trajectories with many missing points. Our choice to use a diffusion model is driven by iteratively refining noisy inputs, naturally generate a distribution of plausible trajectories rather than a single deterministic outcome. Other generative models such as VAE or GAN usually suffer from issues like mode collapse, unstable training dynamics, and difficulty in accurately modeling complex spatial-temporal dependencies; Our diffusion-based approach leverages a probabilistic framework that not only captures the inherent uncertainty in the data but also better models the interplay between spatial relationships and temporal patterns.  **We include an intro schematic here with the link in W2**.\n\nQ3:\n\nTime slot 2 would be treated as missing. It is important to emphasize that our framework is **not inherently tied to the 30-minute/1 hour interval** and can easily adapt to other interval lengths, **such as 10 minutes or even finer resolutions** like a parameter for preprocessing. Regarding multiple check-ins within the same time slot, we aggregate these such as the most frequent check-in (adopted in prior works). Repeated check-ins across consecutive slots would be considered observed for each slot. The distance factors are considered in the evaluation metrics. This setting is proposed by prior baselines such as AttnMove. Spatial relationships are instead modeled by our Spatial Conditional Block. \n\nQ5:\n\nThe two losses serve complementary purposes, a good balance between predictive accuracy and model generalization. Experimentally, these losses are initially printed during training and there seems to be not big differences in scale. Since we also haven’t treated this weight choices as a more advanced optimization problem (which is not our main focus) and we follow this also inspired by Gong et al. (2022) referenced in line 302, which demonstrated the utility of similar balanced weighting in diffusion models applied in NLP.", "claims_and_evidence": "I am confused about the research questions or challenges raised in this paper. I have listed them in detail in the question section.", "methods_and_evaluation_criteria": "Two datasets are useful, but the baselines are not new to this work.", "theoretical_claims": "It seems that only Equation 5 needs to be derived in the main text, and the rest is a description of the method. However, I am confused how Equation 5 is derived from Equation 3, and the paper does not seem to give the derivation process.", "experimental_designs_or_analyses": "Yes, for example, baselines, datasets, parameter setting experimental, etc.. I list the detail in the weakness part.", "relation_to_broader_scientific_literature": "This paper mainly applies the diffusion model to the field of human trajectory recovery, and it is unclear how it is related to the broader scientific literature.", "essential_references_not_discussed": "This paper lacks many discussions or experimental comparisons of related works. I list them in the below weakness section in detail.", "other_strengths_and_weaknesses": "Strength:\n1. DiffMove effectively captures both spatial transitions and periodicity patterns in human mobility, improving trajectory recovery accuracy.  \n\n2. The embedding-space conditional diffusion framework enables better handling of missing locations by incorporating uncertainty and historical trajectory dependencies.  \n\n3. Experiments demonstrate significant performance improvements over baselines.\n\nWeakness:\n1. “Second, existing methods lack systematic mechanisms for handling irregular data sampling from incomplete check-ins. Most of their deterministic approaches could not effectively capture the inherent uncertainty in human mobility.”\nI believe some studies have already started addressing this issue, such as:  \n[1] Zhuang, Zhuang, et al. \"TAU: trajectory data augmentation with uncertainty for next POI recommendation.\" Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 38. No. 20. 2024.\n\nHowever, although related works are still relatively few, this can indeed be considered a technical innovation. At least, the authors should thoroughly discuss relevant research and clearly explain the differences between their approach and existing studies.  \n\n2. Section 4.3 is very difficult to understand. \nThe authors should first introduce Figure 2A to ensure that readers at least understand the data flow. Additionally, I do not clearly understand how e_0^{ob}  and e^{hist}  in Figure 2B are combined with the defined graph structure, nor do I see related equations. The authors primarily describe the methodology using textual explanations rather than mathematical formulations, which makes it easy for readers to become confused. Furthermore, TGGNN appears to be a new method proposed in this paper, yet it is not separately presented in detail but rather mixed in with other sections, making it difficult to understand.  \n\n3. The selected baselines are not new. \nFor example, in the field of human trajectory recovery, at least the following works should be considered:  \n[1] Si, Junjun, et al. \"TrajBERT: BERT-based trajectory recovery with spatial-temporal refinement for implicit sparse trajectories.\" IEEE Transactions on Mobile Computing 23.5 (2023): 4849-4860.  \n[2] Long, Wangchen, et al. \"Learning semantic behavior for human mobility trajectory recovery.\" IEEE Transactions on Intelligent Transportation Systems 25.8 (2024): 8849-8864.  \n[3] Wang, Jinming, et al. \"TrajWeaver: Trajectory Recovery with State Propagation Diffusion Model.\" arXiv preprint arXiv:2409.02124 (2024).  \n\nAdditionally, DeepMove is a classic model for the next POI prediction task. You could also include some state-of-the-art POI prediction methods, such as:  \n[1] Zhuang, Zhuang, et al. \"TAU: trajectory data augmentation with uncertainty for next POI recommendation.\" Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 38. No. 20. 2024.  \n[2] Wu, Junhang, et al. \"Where have you gone: Category-aware multigraph embedding for missing point-of-interest identification.\" Neural Processing Letters 55.3 (2023): 3025-3044.  \n\nMoreover, some trajectory generation methods could also be considered as SOTA baselines:  \n[1] Zhu, Yuanshao, et al. \"Difftraj: Generating GPS trajectory with diffusion probabilistic model.\" Advances in Neural Information Processing Systems 36 (2023): 65168-65188.  \n[2] Wang, Jiawei, et al. \"Large language models as urban residents: An LLM agent framework for personal mobility generation.\" arXiv preprint arXiv:2402.14744 (2024).  \n\nIn summary, there are various baseline options available, and the current selection of baselines in the paper is not sufficiently up-to-date.  \n\n4. The interpretability of the experimental results is insufficient.  \nThe authors could enhance interpretability by adding some visual analyses of the experimental results.", "other_comments_or_suggestions": "1. A strange word \"de-facto\" in line 77. I think human trajectory recovery methods are unrelated to any laws.\n\n2. Line 319 and all captions of figure except figure 1 lack period.\n\n3. All equations lack punctuation.\n\n4. Eq 1, 2, 3 should preferably be in a separate line", "questions_for_authors": "1. See weakness.  \n\n2. In the Abstract:  \n\"Though promising, they encounter limitations in capturing complex spatial-temporal dependencies in low-sampling trajectories.\"\n\nWhat does this sentence mean? Does it refer to the following issue:  \n\n\"First, they struggle to capture intricate spatial-temporal dependencies – the interplay between spatial relationships (proximity and spatial transitions between locations) and temporal patterns (sequential dependencies or periodicity of behaviors in historical trajectories).\" \n\nHowever, I don't think this should be considered a problem, as many downstream tasks in trajectory data, such as next location recommendation, have already addressed this issue, for example:  \n[1] Yang, Song, Jiamou Liu, and Kaiqi Zhao. \"GETNext: trajectory flow map enhanced transformer for next POI recommendation.\" Proceedings of the 45th International ACM SIGIR Conference on research and development in information retrieval. 2022.  \n[2] Rao, Xuan, et al. \"Graph-flashback network for next location recommendation.\" Proceedings of the 28th ACM SIGKDD conference on knowledge discovery and data mining. 2022.\n\nDo their methods still fail to solve the aforementioned problem, or is there another unresolved issue? Or is it that many existing methods for Human Trajectory Recovery have overlooked this aspect, even though it has been emphasized in related downstream tasks? Based on the subsequent sections, I also suspect that this issue might arise from the use of diffusion models. If that is the case, is using a diffusion model really necessary? Would other generative models also encounter this problem? I strongly suggest that the authors add a figure in the introduction section to clarify the research problem, as I currently find it somewhat confusing.  \n\n3. I have some confusion regarding the definition of a missing location. I couldn't find the total number of time slots. Let me give an example: suppose there are 24 time slots in a day, and the input consists of tuples in the form of (location ID, time slot), such as {(0, 1), (1, 1), (2, 1), (3, 3)}.  \nDoes the paper consider the location for time slot 2 to be missing? Additionally, if there are multiple check-ins at different locations within the same time slot, are they all retained? This design seems somewhat unusual, as a user could stay at location 2 for more than an hour. Should distance factors also be considered?  \n\n4. Would it be possible to experiment with other generative models to evaluate the necessity of using a diffusion model?  \n\n5. I noticed in the appendix that all \\lambda values are set to 1. Does this mean that all loss components contribute equally? However, intuitively, I believe their contributions should not be the same.", "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": "", "extra_scores": {}}, {"review_id": "lGf7n82wAr", "reviewer": "Reviewer_fD1d", "summary": "This paper proposes a conditional diffusion-based method for human trajectory recovery from incomplete or missing data. The authors aim to address the limitations of existing methods in capturing complex spatial-temporal dependencies and handling irregular sampling in human mobility data. DiffMove first transforms trajectory locations into the embedding space, performs denoising in this space, and then recovers missing locations through an embedding decoder. Experiments on two real-world mobility datasets, Foursquare2 and Geolife, demonstrate that DiffMove outperforms state-of-the-art baselines.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "Thanks for your detailed feedback.\n\nClaim&Evidence, Rela To Literature:\n\nWe introduce critical innovations that distinguish it from existing works: \n\na)Handling Discrete Locations via Embedding-Space Diffusion. Existing diffusion models for trajectories (e.g., DiffTraj) focus on generating continuous GPS coordinates or simulating synthetic mobility. In contrast, human trajectory recovery involves sparse discrete locations (e.g., check ins), which cannot be directly modeled by continuous-value diffusion (sparse GPS numerical points are not enough for such training). \n\nAs mentioned in the Sec 4.1 line 190, one-hot embedding is just one of the methods that can be applied (it is welcome to change by user and not our main research focus). We transform discrete locations into embeddings and performing denoising in this latent space,preserving spatial relationships and decode embeddings back to discrete IDs using an explicit matching process (Sec 4.3) through a stable end to end training (This will be very difficult if people know diffusion well, this model is not only catering for diffusion MSE loss).\n\nb)Conditional Diffusion with Spatial-Temporal Guidance: DiffMove explicitly incorporates historical trajectories and spatial-temporal dependencies through 3 novel modules: 1)Spatial Conditional Block combines our new TGGNN (for transition patterns) and cross-attention (for periodicity) to model complex mobility dynamics. 2)Target Conditional Block fuses temporal length and historical trajectory embeddings to guide imputation targets and 3) Denoising Network Block.\n\nc)To our knowledge, no existing study has designed diffusion models that integrate the diffusion step t embedding with the learning of Graph-based spatial transitions, as detailed in Sec 4.2 TGGNN (also add a figure here in the link **https://anonymous.4open.science/r/A01D/README.md**).\n\nMethod & Eval Criteria:\n\nFoursquare and Geolife datasets with variations (Table 6) serve as standard benchmarks in trajectory recovery research. DiffMove’s design inherently supports adaptability to diverse mobility data:\n\n1. Handling Diverse Mobility Patterns: Foursquare (urban check-ins) and Geolife (normal trajectories) already represent fundamentally different scenarios (sparse POIs vs. normal GPS). \n\n2. Generalization Ability to Varied Missing Ratios:In Table 6, Distance metric performance of DiffMove with 80% missing ratio even outperforms TRILL with 40% missing rate. This demonstrates resilience to extreme sparsity, a key challenge across datasets.\n\n3. Flexible Preprocessing: As noted in Appendix A.8, DiffMove partitions regions into arbitrary geo region sizes (e.g., 0.25 km²) and adapts to variable time intervals. This flexibility ensures applicability to datasets with varied spatial/temporal granularity.\n\t\nExp Design Or Analyses:\n\nSuch comparisons would be inappropriate for our problem setting:\n\nFundamental Task Mismatch: Continuous trajectory diffusion models (e.g., DiffTraj) generate GPS coordinates or simulate synthetic mobility. In contrast, human trajectory recovery deals with discrete locations (e.g., sparse IDs), requiring modeling of transitions between categorical IDs and periodicity. It cannot be directly modeled by continuous-value diffusion (sparse GPS numerical points are not enough for such training). \n\nOur baselines (AttnMove,PeriodicMove,TRILL) are widely recognized and remain the standard benchmarks for this problem setting. Regarding more recent baseline, we discussed TERI (Chen et al., VLDB 2024) cited in our related work. We have been working hard on setting up a common problem setting,where our model still outperforms SOTA baseline as shown below in this problem setting.\n|Dataset|Methods|Recall|Dataset|Methods|Recall|\n|-------|-------|------|-------|-------|------|\n|Foursquare|PeriodicMove|0.3125|Geolife|PeriodicMove|0.4199|\n||TRILL|0.3227||TRILL|0.4721|\n||TERI|0.3355||TERI|0.4922|\n||DiffMove|0.3600||DiffMove|0.5173|\n\nEssential References:\n\nThere are task mismatches, they are primarily designed for trajectory prediction in an image (more specifically is a computer vision trajectory problem), aiming to generate future movement trajectories in continuous space (x,y in the image, using dense CV dataset of ETH and UCY). In contrast, our work specifically addresses trajectory recovery for human mobility in locations (a real-world geographical problem, not the trajectory in image), where trajectories are represented as discrete location IDs requiring historical periodicity and spatial transition modeling. \n\nQ1:\n\nOur method accounts for variations in trajectory length through a standardized data preprocessing pipeline (Appendix A.8). Specifically, we discretize each day into a fixed number N of time slots, where the time interval is a configurable parameter. Trajectories with shorter length than N are padded, ensuring that every trajectory is represented uniformly. This approach enables our model to handle variable-length trajectories effectively.", "claims_and_evidence": "The authors mention: \"In such scenarios, traditional methods typically provide a biased deterministic imputed trajectory. However, with a generative approach to inference, a set of imputed trajectory locations can be generated through sampling or various averaging techniques on imputation samples.\" However, without explicit guidance, diffusion models still follow the data distribution, making it difficult to generate irregular results.\n\nThe novelty of this work is limited. There is a lot of diffusion-based research in trajectory prediction and human motion generation, which all follow a similar fundamental approach. This work merely adds one-hot encoding to handle discrete data conversion. However, in essence, it does not introduce significant innovation regarding the use of diffusion models for trajectory recovery.", "methods_and_evaluation_criteria": "The experiments are mainly conducted on two specific datasets, Foursquare2 and Geolife. It is not clear how well the model will perform on other types of mobility datasets with different characteristics, such as different sampling frequencies, data distributions, or geographical regions. This limits the generalization ability of the model and needs further exploration.", "theoretical_claims": "The method is technically sound", "experimental_designs_or_analyses": "The diffusion-based methods used for continuous trajectory recovery can be slightly modified to adapt to the discrete setting in this paper. Using this as a baseline would better highlight the contributions of this work.\n\nThe baselines are outdated and lack representativeness.", "relation_to_broader_scientific_literature": "This work does not show a significant difference from previous studies.", "essential_references_not_discussed": "It is necessary to discuss the differences compared to previous diffusion-based trajectory prediction methods.\n\nBae, Inhwan, Young-Jae Park, and Hae-Gon Jeon. \"Singulartrajectory: Universal trajectory predictor using diffusion model.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024.\n\nMao, Weibo, et al. \"Leapfrog diffusion model for stochastic trajectory prediction.\" Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2023.", "other_strengths_and_weaknesses": "NA", "other_comments_or_suggestions": "Add the references for the baselines to the table.", "questions_for_authors": "- How to account for changes in trajectory length?", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: Initially, the reviewers raised several questions, but the rebuttal effectively addressed the major concerns, leading three reviewers to improve their ratings.

Following the rebuttal, all four reviewers recommend either a weak accept or accept for this submission. The remaining concerns, pointed out by the reviewers, are: (1) the setup about the uncertain scenarios—specifically, whether random masking can adequately simulate uncertainty. (2) the need to tune down the paper’s claim regarding handling irregular (out-of-distribution) trajectories.

While these issues should be addressed, they do not appear to grounded for rejecting the submission, especially as the uncertain scenario is only one of several tested settings. Therefore, the AC agrees with the reviewers' consensus but encourages the authors to address these points in the final revision. These include explicitly discussing the limitations of using random masking to simulate uncertainty, fixing the overstatement regarding the method's ability to handle irregular trajectories, or adding further experiments to substantiate the current claims.

This meta-review has been discussed with and approved by the SAC.
Decision: Accept (poster)