We present a new representation learning framework, Intensity Profile Projection, for continuous-time dynamic network data. Given triples $(i,j,t)$, each representing a time-stamped ($t$) interaction between two entities ($i,j$), our procedure returns a continuous-time trajectory for each node, representing its behaviour over time. The framework consists of three stages: estimating pairwise intensity functions, e.g. via kernel smoothing; learning a projection which minimises a notion of intensity reconstruction error; and constructing evolving node representations via the learned projection. The trajectories satisfy two properties, known as structural and temporal coherence, which we see as fundamental for reliable inference. Moreoever, we develop estimation theory providing tight control on the error of any estimated trajectory, indicating that the representations could even be used in quite noise-sensitive follow-on analyses. The theory also elucidates the role of smoothing as a bias-variance trade-off, and shows how we can reduce the level of smoothing as the signal-to-noise ratio increases on account of the algorithm `borrowing strength' across the network.

翻译：我们提出了一种新的表示学习框架——强度轮廓投影（Intensity Profile Projection），用于处理连续时间动态网络数据。给定三元组 $(i,j,t)$，每个三元组表示两个实体 $(i,j)$ 之间带有时间戳 ($t$) 的交互，该框架为每个节点返回一条连续时间轨迹，以刻画其随时间变化的行为。该框架由三个阶段组成：首先通过核平滑等方法估计成对强度函数；其次学习一个投影，最小化某种强度重构误差；最后通过所学的投影构建动态的节点表示。这些轨迹满足两个性质——结构与时间连贯性，我们认为它们是可靠推理的基础。此外，我们建立了估计理论，对任意估计轨迹的误差提供严格控制，表明这些表示甚至可用于对噪声敏感的后续分析中。该理论也阐明了平滑作为偏差-方差权衡的作用，并展示了如何随着信噪比的提高而降低平滑程度，这是因为算法能够跨网络"借力"提升性能。