Trajectory Inference

Recent advances in generative planning have made trajectory inpainting a promising approach to offline goal-conditioned reinforcement learning. However, these methods typically specify the planning horizon before generating plan content, even though the appropriate horizon depends on the route itself. A horizon that is too short can force infeasible transitions, whereas one that is too long can introduce redundant motion. We introduce HorizonFlow, a hierarchical planner that treats plan length as an output of generation rather than a prescribed input. Its subgoal route planner guides its action-prefix controller through a sequence of latent subgoals. Both components combine insertion-based generation with flow matching to jointly generate continuous plan content and length, using the partially generated plan to guide token insertion. HorizonFlow reuses the resulting length information to select candidates and steer generation toward shorter plans without a separate learned value model. Across Maze2D, Multi2D, and OGBench navigation and visual manipulation benchmarks, HorizonFlow achieves the highest average performance among the compared methods.

Gromov--Wasserstein optimal transport (GW-OT) extends classical optimal transport by introducing structure-aware transport cost. This is particularly relevant for spatial transcriptomics, where dynamical reconstruction should preserve tissue structure in addition to matching expression patterns. While static formulations have been widely used for such structure-aware alignment, a general dynamic formulation for reconstructing continuous trajectories is still missing. We introduce Travelling Pair Dynamical Alignment and Trajectory Estimation (TP-DATE), a theoretical and computational framework to generalize GW-OT dynamically in a simulation-free manner. We formulate a broad class of static and dynamic Quadratic-form OT (QOT) through path actions and prove the static dynamic equivalence. We further develop travelling-pair flow matching, which allows interacting conditional paths and marginalizes their interactions into a single vector field. On synthetic and real spatial transcriptomics data, TP-DATE better preserves spatial structure and improves continuous 3D dynamics reconstruction.

Objective: Incomplete navigation of anatomy during ureteroscopic kidney stone surgeries can contribute to repeat interventions. While skilled surgeons have lower reintervention rates, there are no objective metrics to quantify scope-navigation performance to evaluate when a trainee becomes skilled. This work aims to recover ureteroscope trajectories from endoscopic video and derive navigation metrics to quantify differences in skill. Methods: We propose RAUL, a reference-assisted reconstruction framework for recovering ureteroscope trajectories from ureteroscope videos only in phantoms. For each phantom, we use a slow, high-quality reference exploration video to generate a reference reconstruction. We localize subsequent exploration videos against this reference. We evaluate localization accuracy against electromagnetically tracked scope pose. We compute navigation metrics from phantom exploration trajectories to compare surgical residents across experience levels. Results: The proposed reference-assisted framework achieves a mean translation root mean square error of $0.5 \pm 0.1$ mm across 9 phantoms. Compared to standard Structure-from-Motion (SfM), the proposed pipeline increases frame-wise localization coverage from $50.5 \pm 14.9\%$ to $86.1 \pm 7.2\%$ of all video frames. The reconstructed trajectories revealed significant differences between high- and low-experience trainees in established navigation metrics. Conclusion: RAUL enables substantially more complete recovery of ureteroscope trajectories from videos compared to standard SfM pipelines, enabling trajectory-based skill assessment without additional tracking equipment. Significance: To the best of our knowledge, this is the first use of video-only recovery of ureteroscope trajectories without external tracking sensors for skill assessment, supporting scalable automated assessment of ureteroscopy navigation skill.

Trajectory inference is a fundamental problem in many scientific domains: given a collection of unpaired snapshots of observations at discrete time points, the goal is to generate smooth trajectories that best resemble and interpolate the data. Existing algorithms exhibit computational challenges: they either rely on preprocessing subroutines to enforce smoothness or on simulation-based training objectives, both of which can be expensive. In order to overcome these limitations, we propose a new algorithm called Acceleration Matching (\texttt{AM}). Our approach consists of lifting the original interpolation problem to phase space and then regressing onto an explicit conditional acceleration field that induces random, smooth trajectories that agree with the prescribed marginals. Importantly, our resulting training algorithm only requires positional data, avoids trajectory simulation during training, and is devoid of expensive preprocessing. We provide ample numerical evidence suggesting that \texttt{AM} is competitive with or superior to existing algorithms on several benchmark problems from the existing literature.

Trajectory data collected in real-world settings is frequently incomplete due to sensor failure, communication loss, or occlusion. We address the task of \emph{trajectory inpainting}: reconstructing contiguous missing segments from observed context. We propose a Temporal Convolutional Network (TCN) with symmetric dilation that relaxes the standard causality constraint, allowing each time step to draw on both past and future observations, a property that is essential for inpainting, but absent from forecasting-oriented architectures. The model is trained with a composite loss that combines weighted mean squared error, boundary--continuity penalties, and a smoothness regularizer. Trained on a synthetic dataset of $1,000$ (train), $200$ (validation), and $300$ (test) two-dimensional trajectories with randomly placed 20% masked segments, the model achieves good R$^{2}$, MSE and MAE metrics.

Long-range vehicle trajectories provide important spatio-temporal evidence for traffic safety analysis, autonomous driving evaluation, and data-driven traffic management, yet continuously recovering them from fixed highway cameras remains difficult. As vehicles recede into distant road regions, perspective compression and scale decay often fragment or prematurely terminate automatic tracklets, even when their continuation remains identifiable from motion consistency across neighboring frames. We formulate this problem as recovering the far-range continuation of a vehicle trajectory from a reliable near-field tracklet. We introduce LoRFT, to our knowledge the first open benchmark dedicated to long-range vehicle trajectory reconstruction from fixed highway cameras. LoRFT comprises 22 expressway surveillance scenes, 366,109 video frames, 6,601 manually verified trajectories, 2,694,889 bounding boxes, road-geometry annotations, scene-level splits, and evaluation scripts. We further propose Map-RSTNet, a map-aware residual sequence-to-sequence model that reconstructs distant trajectories in a road-geometry-aligned state space and dynamically refreshes local road geometry during decoding. On LoRFT, Map-RSTNet reduces ADE, FDE, and 5-second RMSE by 11.0%, 15.4%, and 10.5%, respectively, relative to the strongest baseline. These results demonstrate that road-geometry-aware reconstruction can extend usable trajectory records from existing fixed-camera infrastructure. LoRFT provides a reproducible testbed for long-range vehicle trajectory reconstruction.

Over the past few years, diffusion-based Schrödinger bridge models have been proposed to approximate optimal transport dynamics between two prescribed boundary distributions, with successful applications to generative modeling. More precisely, these methods aim to estimate a path measure whose initial and terminal marginals match the two boundary distributions, while minimizing the Kullback-Leibler divergence with respect to a reference Markov process. In this work, we consider the generalized Schrödinger bridge problem, in which the reference process is a twisted Brownian motion, that is, a Feynman-Kac transform of a Brownian motion induced by a time-dependent differentiable potential. Building on the Iterative Markovian Fitting (IMF) paradigm, and in particular on its special case Diffusion Schrödinger Bridge Matching (DSBM), which corresponds to the zero potential case, we introduce Twisted Schrödinger Bridge Matching (TSBM), a diffusion-based method designed to handle both continuous- and discrete-time potentials. Unlike previous approaches, TSBM provides a rigorous extension of the IMF scheme to the generalized Schrödinger bridge problem. This derivation leads to a new bridge-matching loss that depends explicitly on the gradient of the potential and recovers the DSBM objective when the potential vanishes, yielding improved performance. We further introduce trajectory-based variance-reduction techniques that substantially stabilize optimization and may be useful beyond the present setting. Finally, we empirically demonstrate the benefits of TSBM for trajectory inference across increasingly high-dimensional settings, including crowd navigation and single-cell data. Code available at https://github.com/maxencenoble/twisted-sb-matching.

Spatial and Single-cell transcriptomics are transformative in deciphering cellular dynamics. As the fundamental paradigm for reconstructing cell developmental paths, trajectory inference (TI) is critical. However, existing methods require extensive manual intervention and proficiency in heterogeneous tools, posing a significant barrier to efficient TI analysis. To bridge this gap, we propose SpaCellAgent, an autonomous large language model (LLM) multi-agent framework that automates end-to-end spatiotemporal analysis and narrative generation. SpaCellAgent utilizes a multi-agent architecture for strategic workflow planning, a dynamic tool-orchestration engine for adaptive algorithm selection, and a self-evolution module that iteratively refines performance through feedback. We evaluate SpaCellAgent on six heterogeneous datasets encompassing complex temporal developmental trajectories, diverse sequencing platforms, and spatially-resolved tissue architectures. SpaCellAgent consistently demonstrates over 40% improvement in analytical efficiency while maintaining expert-aligned performance. By converting natural language specifications into optimized analytical workflows and fully automating the pipeline, SpaCellAgent democratizes advanced spatiotemporal modeling and establishes a scalable, agent-driven paradigm for computational biology. The code and materials are available at https://github.com/LittleXH-shw/SpaCellAgent.

While humans readily repurpose a book, a stone, or a shoe to drive a nail, robots trained on specific tools fail to transfer the same function to novel ones -- a gap we formalize as functional generalization. Functionally equivalent tools share visually recognizable functional intent, such as where contact can occur and how a contact region should move to the target. However, this perceptual similarity does not directly carry over to action space, where each tool demands a different motor pattern to realize the function. To bridge this gap, we explore intermediate representations including affordance images, human video prompts, functional videos and object masks, and 2D keypoint trajectories, finding that keypoint trajectories best balance functional expressiveness and action groundability. Building on this, we present FuncBridge, a two-stage framework that decouples functional reasoning from action execution: learning to predict generalizable keypoint trajectories from action-free data, then grounding them into robot actions with limited demonstrations. Across a benchmark spanning ten tools and three functions, including hitting, sweeping, and hooking, FuncBridge consistently outperforms state-of-the-art methods on unseen tools in both simulation and the real world.

Reconstructing population dynamics is a central problem in the physical and data sciences. Often, the dynamics are modeled as a Wasserstein gradient flow (WGF): a curve of distributions driven by an energy functional. Though there are multiple mathematical characterizations of a WGF, the dominant algorithmic approach relies on the Jordan--Kinderlehrer--Otto (JKO) scheme. JKO-based methods are inflexible to time discretisation and require solving costly optimal transport problems. We take a residual approach, enforcing the continuity equations via a non-negative loss function whose minimum is the WGF. Combined with a data-fitting divergence, this gives a single global objective. This perspective unifies several existing methods and leads to a new particle-based method, stitching, that is simulation-free and robust to large gaps between observations. We demonstrate that the stitching method achieves state-of-the-art performance across trajectory inference benchmarks. For code see github.com/BasisResearch/wasserstein-residuals.

DNA methylation (DNAm) serves as one of the most robust molecular biomarkers of biological aging. While conventional epigenetic clocks accurately predict chronological age from high-dimensional CpG profiles, they treat aging as a static regression task, meaning they can only output a single score rather than simulating how an entire profile continuously changes over time. To reconstruct these continuous dynamics, we frame lifelong human epigenetic aging as a trajectory inference problem across discrete age snapshots derived from widely available cross-sectional data. We introduce a two-stage computational pipeline: first, an age-regularized Variational Autoencoder (VAE) maps high-dimensional CpG profiles onto a chronologically ordered latent manifold while preserving a generative decoder bridge back to the original methylation space. Second, we model the continuous movement across this latent space via Regularized Unbalanced Optimal Transport (RUOT) that unifies deterministic drift, random diffusion, and non-conservative mass changes. By resolving this RUOT formulation using the DeepRUOT framework, our model fluidly accommodates population-level density shifts like survivorship bias and cellular attrition without requiring rigid biological priors. Evaluated on a large-scale, 80-year pan-tissue dataset, our model demonstrates robust distribution interpolation and uncovers a prominent late-life surge in the learned growth field that mathematically captures the variance expansion driven by stochastic epigenetic drift. Finally, by decoding continuous latent paths back to individual CpG sites, we reconstruct and empirically verify distinct biological aging archetypes, offering a rigorous, generative paradigm for simulating human molecular aging.

This paper presents SD-RouteFusion, a deployable end-to-end ego-trajectory prediction method that fuses a front-facing camera, vehicle kinematics, and a navigation route derived from a Standard Definition (SD) map. Unlike approaches that rely on High Definition (HD) map geometry, SD-RouteFusion aligns the learning objective with scalable and production-ready SD-map route inputs, enabling route-aware prediction without requiring HD-map infrastructure. First, we demonstrate that SD-map route prior provides a powerful long-horizon semantic prior. Through a comprehensive study on a large-scale real-world dataset comprising 480k driving scenarios across 10 European countries and the U.S., we quantify the value of SD-route conditioning: incorporating SD-map routes yields a 10.5% ADE improvement over an image-and-kinematics baseline, while our full fusion strategy achieves a 16.9% ADE reduction given a prediction horizon of 8 seconds. The fusion strategy consists of a dual-hypothesis design paired with a gated classifier, to ensure robustness under route corruption and visual uncertainty. Finally, to support broader evaluation, we release an SD-route generation toolkit that enables SD-route-conditioned ego-trajectory prediction on all datasets containing ego pose and future trajectories. Together, SD-RouteFusion establishes a practical path toward robust, route-aware ego-trajectory prediction at scale.

In stealth-constrained swarm robotics, visual communication provides a critical alternative to active radio transmissions, which might be jammed. This research investigates motion-based communication for non-active information exchange, utilizing modular, dynamically feasible planar trajectories as visual cues. On the receiver drone end, a pose estimator tracks the transmitting drone's pose, feeding it into our custom 3DTrajDecoder. The decoder is designed to classify and segment the spatiotemporal sequence while simultaneously regressing its size and normal vector. To robustly train the decoder on both communicative and non-communicative trajectories, we developed a configurable online procedural generation pipeline. We validate our system through real-world testing and simulation to define its operating domain, supported by an extensive ablation study detailing our architectural choices and system limitations.

Continuous 6-DoF pose estimation is essential for autonomous UAV operations. Yet, existing visual odometry and SLAM methods accumulate drift and yield only relative, up-to-scale trajectories. Single-frame geo-localization, in turn, discards temporal continuity and remains too slow for real-time use. We present OrthoTrack, a training-free system that estimates continuous 6-DoF UAV trajectories using only publicly available orthophotos and surface models as a map prior. OrthoTrack matches keyframes against the orthophoto and lifts correspondences to metric 3D via the surface model. It then propagates these map-anchored correspondences to intermediate frames with optical flow, producing absolute, metrically scaled poses at every frame without GPS or post-hoc alignment. We also introduce the MovingDrone Dataset, a large-scale benchmark pairing photorealistic UAV sequences with dense 6-DoF ground truth and co-registered multi-modal geodata including multi-temporal orthophotos. On MovingDrone and real-world benchmarks, OrthoTrack runs in real time on a single GPU. It outperforms all baselines by a large margin, even those receiving oracle scale and alignment. By relying on publicly available geodata, OrthoTrack enables deployment to new regions without site-specific adaptation.

Predicting how a person's first-person view will evolve (what action will follow, what plan completes a task, whether an in-progress shot will score) is fundamentally under-specified: the same context admits many plausible futures, and a model trained to minimize prediction error is forced to hedge or average across them, getting it wrong either way. Two findings shape our approach. First, the future camera trajectory, the path the head carves through space, lets the model commit to one of those futures: it carries the operator's intent in a form fine enough to determine how an action will unfold, substantially outperforming language as a conditioning signal. Second, this same intent makes the trajectory itself partially predictable from the context at hand, enough that trajectory need not be observed at test time to recover most of the gain. We instantiate these findings as TrajPilot, a model that predicts candidate future trajectories from egocentric context and uses them to pilot action prediction in an action-aligned embedding space where language shapes the structure but is never used as a conditioning input. TrajPilot beats VLM and structured-planner baselines on procedural planning across Ego-Exo4D atomic, Ego-Exo4D Keystep, Ego4D GoalStep, and EgoPER, with the trajectory advantage widening with horizon (exactly where prior planners collapse) and holding under RGB-only camera-pose estimation. With the goal masked at inference, the same model performs goal-free anticipation, beating VLM baselines on Ego-Exo4D atomic and extending to EPIC-Kitchens-100 and basketball shot-outcome prediction.

Language instructions offer a flexible way to specify task semantics but can become ambiguous in cluttered scenes containing multiple visually similar objects and candidate targets. We study Spatially Prompted Visual Trajectory Prediction (SP-VTP), an open-loop setting in which points or bounding boxes in the first egocentric frame specify an object--target pair and condition the prediction of future 3D end-effector (EE) trajectory chunks. To support this setting, we introduce EgoSPT, a controlled dataset comprising 2,841 egocentric fork pick-and-place episodes with first-frame object and target annotations and reconstructed 3D EE trajectories. SP-VTP evaluates whether a static task specification remains informative as the viewpoint and scene configuration evolve throughout execution. We also introduce SPOT, a spatially prompted object--target trajectory predictor that integrates first-frame visual and coordinate prompts with current visual observations and motion history to generate future trajectories. Rather than relying on an explicit object tracker, SPOT maintains a fixed representation of the first-frame task specification. Across three training seeds, spatial prompting consistently reduces translation and endpoint errors relative to a no-prompt baseline, while controlled prompt interventions reveal distinct roles for object and target cues at different stages of execution. Together, EgoSPT and SPOT provide a dataset and baseline for spatially conditioned egocentric trajectory prediction in open-loop pick-and-place settings.

Accurately forecasting human trajectories from an egocentric perspective plays a central role in applications such as humanoid robotics, wearable sensing systems, and assistive navigation. However, progress in this direction remains limited due to the scarcity of egocentric trajectory datasets collected in real-world environments. Addressing this need, we introduce EgoTraj, an egocentric multimodal open dataset recorded using Meta Quest Pro (MQPro). EgoTraj contains 75 sequences of human navigation collected from multiple MQPro wearers in real-world urban environments. Each recording provides synchronized RGB video along with ground-truth data, including continuous time-synchronized 6-degree-of-freedom head poses, per-frame 3D eye gaze vectors, scene annotations. To the best of our knowledge, EgoTraj differs from typical egocentric trajectory datasets by capturing long-horizon, self-directed navigation across diverse urban routes with broad participant diversity. To demonstrate the potential of the dataset, we benchmark several state-of-the-art methods for egocentric trajectory prediction and conduct ablation studies to analyze the contributions of gaze, scene, and motion cues. The results highlight the utility of EgoTraj for AR-based perception, navigation, and assistive systems. The EgoTraj dataset, code, and EgoViz Dashboard are publicly available at https://github.com/yehiahmad/EgoTraj.

Single-cell trajectory inference from destructive time-course snapshots is fundamentally ill-posed: neither cross-time cell correspondences nor continuous trajectories are observed, so the snapshot distributions alone do not uniquely determine the underlying dynamics. Existing optimal transport and flow-based methods typically couple cells by Euclidean proximity at observed clock times, which can misalign trajectories when development is asynchronous and cells sampled at the same experimental time occupy different latent pseudotime stages. We propose PACE, a trajectory inference framework that recovers geometry-consistent continuous transport dynamics from destructive time-course snapshots through three coupled components. First, PACE constructs a state- and time-dependent anisotropic Riemannian metric that assigns low transport cost along locally supported tangent directions while penalizing normal velocity components. Second, it alternates between refining cross-time couplings under the induced path-action cost and fitting endpoint-preserving neural bridges between adjacent snapshots. Third, it distills the learned bridge dynamics into a global continuous-time velocity field over cellular states. Across seven controlled and biological datasets covering nine held-out reconstruction experiments, PACE achieves the strongest overall reconstruction performance, reducing MMD, Wasserstein-1 distance, and Wasserstein-2 distance by 23.7% on average relative to the strongest competing baseline. PACE also improves RNA-velocity alignment by 15.4% on an embryoid body differentiation benchmark, without requiring explicit cell pairing, lineage tracing, or RNA-velocity supervision during training. Code is available at https://github.com/AI4Science-WestlakeU/PACE.

Large-scale GPS trajectory data offer rich observations of human mobility, yet assigning trip purposes to detected stops remains challenging due to the absence of individual-level ground truth, spatial uncertainty from GPS noise and incomplete points of interest (POIs) coverage, and fundamental behavioral differences across trip purposes. We propose a weakly supervised framework integrating neighborhood-level POI semantic zones with distance-weighted spatial likelihoods, differentiated inference strategies for mandatory and non-mandatory activities, and a multi-phase Pareto optimization that jointly minimizes distributional divergence from household travel survey statistics and maximizes inference reliability without requiring annotated labels. Evaluated on over 81 million staypoints in Los Angeles, the framework reduces activity type frequency Jensen-Shannon distance (JSD) by 23%, start time JSD by 48%, and duration JSD by 12% respectively relative to a comparable baseline. The proposed approach provides a scalable and uncertainty-aware path from raw GPS trajectories to semantically annotated mobility data for travel demand modeling and transportation policy analysis.

Trajectory Inference (TI) seeks to recover latent dynamical processes from snapshot data, where only independent samples from time-indexed marginals are observed. In applications such as single-cell genomics, destructive measurements make path-space laws non-identifiable from finitely many marginals, leaving held-out marginal prediction as the dominant but limited evaluation protocol. We introduce a general framework for estimating the Kullback-Leibler divergence (KL) divergence between probability measures on function space, yielding a tractable, data-driven estimator that is scalable to realistic snapshot datasets. We validate the accuracy of our estimator on a benchmark suite, where the estimated functional KL closely matches the analytic KL. Applying this framework to synthetic and real scRNA-seq datasets, we show that current evaluation metrics often give inconsistent assessments, whereas path-space KL enables a coherent comparison of trajectory inference methods and exposes discrepancies in inferred dynamics, especially in regions with sparse or missing data. These results support functional KL as a principled criterion for evaluating trajectory inference under partial observability.

Current single-cell foundation model benchmarks universally extract final layer embeddings, assuming these represent optimal feature spaces. We systematically evaluate layer-wise representations from scFoundation (100M parameters) and Tahoe-X1 (1.3B parameters) across trajectory inference and perturbation response prediction. Our analysis reveals that optimal layers are task-dependent (trajectory peaks at 60% depth, 31% above final layers) and context-dependent (perturbation optima shift 0-96% across T cell activation states). Notably, first-layer embeddings outperform all deeper layers in quiescent cells, challenging assumptions about hierarchical feature abstraction. These findings demonstrate that "where" to extract features matters as much as "what" the model learns, necessitating systematic layer evaluation tailored to biological task and cellular context rather than defaulting to final-layer embeddings.

Understanding the continuous evolution of populations from discrete temporal snapshots is a critical research challenge, particularly in fields like developmental biology and systems medicine where longitudinal tracking of individual entities is often impossible. Such trajectory inference is vital for unraveling the mechanisms of dynamic processes. While Schrödinger Bridge (SB) offer a potent framework, their traditional application to pairwise time points can be insufficient for systems defined by multiple intermediate snapshots. This paper introduces Multi-Marginal Schrödinger Bridge Matching (MSBM), a novel algorithm specifically designed for the multi-marginal SB problem. MSBM extends iterative Markovian fitting (IMF) to effectively handle multiple marginal constraints. This technique ensures robust enforcement of all intermediate marginals while preserving the continuity of the learned global dynamics across the entire trajectory. Empirical validations on synthetic data and real-world single-cell RNA sequencing datasets demonstrate the competitive or superior performance of MSBM in capturing complex trajectories and respecting intermediate distributions, all with notable computational efficiency.

Learning the dynamics of a process given sampled observations at several time points is an important but difficult task in many scientific applications. When no ground-truth trajectories are available, but one has only snapshots of data taken at discrete time steps, the problem of modelling the dynamics, and thus inferring the underlying trajectories, can be solved by multi-marginal generalisations of flow matching algorithms. This paper proposes a novel flow matching method that overcomes the limitations of existing multi-marginal trajectory inference algorithms. Our proposed method, ALI-CFM, uses a GAN-inspired adversarial loss to fit neurally parametrised interpolant curves between source and target points such that the marginal distributions at intermediate time points are close to the observed distributions. The resulting interpolants are smooth trajectories that, as we show, are unique under mild assumptions. These interpolants are subsequently marginalised by a flow matching algorithm, yielding a trained vector field for the underlying dynamics. We showcase the versatility and scalability of our method by outperforming the existing baselines on spatial transcriptomics and cell tracking datasets, while performing on par with them on single-cell trajectory prediction. Code: https://github.com/mmacosha/adversarially-learned-interpolants.