Research paper
Analysis of Conversation Trajectory Representations: Orthogonal Projections versus Scalar Geometry for Satisfaction Prediction
20 July 2026 · 30 pages · PDF
Abstract
A conversation can be represented as a trajectory through high-dimensional semantic space — one point per turn — and a common way to use it is to summarize that trajectory as a fixed-size vector for a simple learner; which summary to use is the question this paper studies. We compare hand-crafted conversational geometry — scalar features of the trajectory, most prominently in TRACE (Gooding and Grefenstette, 2025) — against projecting the trajectory onto a low-order orthogonal-polynomial basis and keeping the coefficients, evaluating both on conversation-level user satisfaction. First, which representation wins is regime-dependent: the projection beats TRACE's geometric features by 5-12 points on all three corpora, but on short, first-person-rated chats the trajectory-shape channels collapse and hand-crafted scalars become a genuine complement — a difference of corpus and label construct, not of conversation length. Second, the projection's structure matters: read as one interleaved sequence, the projection blends the two speakers into shared low-degree coefficients. We therefore split the trajectory by speaker, giving the learner each speaker's content on its own path, and recommend that representation: training-free, and never distinguishably worse than any measured alternative in either regime.
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Reproduction. The paper ships as a self-contained kit on GitHub — a portable pipeline spec plus the generator scripts, regenerating every artifact from public data with one command. Details in the reproducibility statement and appendices.