Length-Aware Adversarial Training for Variable-Length Trajectories: Digital Twins for Mall Shopper Paths

TL;DR

The paper addresses generating variable-length trajectories and the instability caused by mixing short and long sequences during mini-batch training. It introduces length-aware sampling (LAS), a simple batching strategy that groups trajectories by length, and integrates LAS with a conditional trajectory GAN augmented by time-alignment losses to improve distributional fidelity of derived trajectory statistics. The authors provide theoretical results, including a Wasserstein-based bound and an IPM interpretation, showing how LAS reduces length-only shortcuts and focuses learning on within-bucket structure. Empirically, LAS yields substantial improvements in derived-variable distributions across a multi-mall shopper dataset and several public sequential datasets (GPS, education, e-commerce, movies), while remaining a drop-in training modification. The work demonstrates the practical impact of LAS for digital twins and counterfactual analysis of shopper paths and other sequence data, enabling more accurate forward simulations and what-if analyses.

Abstract

We study generative modeling of \emph{variable-length trajectories} -- sequences of visited locations/items with associated timestamps -- for downstream simulation and counterfactual analysis. A recurring practical issue is that standard mini-batch training can be unstable when trajectory lengths are highly heterogeneous, which in turn degrades \emph{distribution matching} for trajectory-derived statistics. We propose \textbf{length-aware sampling (LAS)}, a simple batching strategy that groups trajectories by length and samples batches from a single length bucket, reducing within-batch length heterogeneity (and making updates more consistent) without changing the model class. We integrate LAS into a conditional trajectory GAN with auxiliary time-alignment losses and provide (i) a distribution-level guarantee for derived variables under mild boundedness assumptions, and (ii) an IPM/Wasserstein mechanism explaining why LAS improves distribution matching by removing length-only shortcut critics and targeting within-bucket discrepancies. Empirically, LAS consistently improves matching of derived-variable distributions on a multi-mall dataset of shopper trajectories and on diverse public sequence datasets (GPS, education, e-commerce, and movies), outperforming random sampling across dataset-specific metrics.

Figures (19)

• Figure 1: Representative mall distributions. LAS improves agreement with the ground-truth marginals without changing the GAN objective.
• Figure 2: Amazon marginals. LAS improves agreement on diversity and timing-related derived variables.
• Figure 3: Movie marginals. LAS improves both trajectory-length and inter-event timing distributions.
• Figure 4: Education dataset: representative marginals under RS and LAS.
• Figure 5: GPS dataset: representative marginals under RS and LAS.

Theorems & Definitions (21)

• Theorem 1: Distributional closeness for derived variables
• Corollary 2: From $W_1$ control to CDF control (informal)
• Lemma 3: Bucket-only (length-only) critics are a null space
• Lemma 4: Global Wasserstein dominated by length mismatch + within-bucket discrepancy
• Proposition 5: LAS removes length-only shortcut critics
• Lemma 6: Lipschitz control of derived variables
• proof
• Lemma 7: Matched-step control via L1 losses
• proof
• Lemma 8: Length tail controlled by divergence