Understanding how individuals navigate mental health challenges over time is critical yet methodologically challenging. Traditional approaches analyze community-level snapshots, failing to capture dynamic individual recovery trajectories. We introduce a novel framework applying Topological Data Analysis (TDA) specifically persistent homology to model users' longitudinal posting histories as trajectories in semantic embedding space. Our approach reveals topological signatures of trajectory patterns: loops indicate cycling back to similar states (stagnation), while flares suggest exploring new coping strategies (growth). We propose Semantic Recovery Velocity (SRV), a novel metric quantifying the rate users move away from initial distress-focused posts in embedding space. Analyzing 15,847 r/depression trajectories and validating against multiple proxies, we demonstrate topological features predict self-reported improvement with 78.3% accuracy, outperforming sentiment baselines. This work contributes: (1) a TDA methodology for HCI mental health research, (2) interpretable topological signatures, and (3) design implications for adaptive mental health platforms with ethical guardrails.

翻译：理解个体如何随时间应对心理健康挑战至关重要，但在方法论上面临困难。传统方法分析社区层面的快照，无法捕捉动态的个体恢复轨迹。我们引入一种新颖框架，应用拓扑数据分析（TDA），特别是持续同调，将用户的纵向发帖历史建模为语义嵌入空间中的轨迹。我们的方法揭示了轨迹模式的拓扑特征：循环表示状态回退至相似状态（停滞），而辐射状分支则暗示对新应对策略的探索（成长）。我们提出了语义恢复速度（SRV），这是一种新颖的度量标准，用于量化用户在嵌入空间中远离初始以痛苦为中心的帖子的速率。通过分析15,847条r/depression轨迹并针对多个代理指标进行验证，我们证明拓扑特征能以78.3%的准确率预测自我报告的改善情况，优于情感基线方法。本工作的贡献在于：（1）为人机交互心理健康研究提供了一种TDA方法论；（2）可解释的拓扑特征；（3）对具有伦理护栏的自适应心理健康平台的设计启示。