Method paper
Gram Projections for Conversation Trajectories: A Polynomial Equivalent of the DCT
20 July 2026 · 17 pages · PDF
Abstract
Fixed orthogonal transforms are a standard way to turn a variable-length sequence of embeddings into a fixed-size vector: keep the first few transform coefficients. The DCT is the default choice. We introduce the Gram (discrete orthogonal polynomial) projection as its polynomial equivalent for short, aperiodic, drifting sequences such as conversation trajectories, and measure whether the choice matters. The Gram coefficients have stable semantics: degree 0 is exactly the mean of the sequence, truncation never changes retained coefficients, degree 1 is the least-squares drift, and the drift coefficient reports the same end-to-end drift at every sequence length. On three conversation-satisfaction corpora the bases tie almost everywhere; the exception favours the polynomial basis on populations dominated by very short per-role sequences, by a measured 2.3 points. Within the class, choose the basis you can read; the accuracy is free.
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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.