Iterated Integrals as Lexical Fingerprints: Path Signature Embeddings for Lexical Similarity

While sentence embeddings are primarily designed for semantic generalization, their latent geometry implicitly encodes surface-level lexical properties, a relationship that remains poorly understood. Shallow pooling methods discard word order, while deep Transformer models obscure these lexical alignments behind computational complexity and uninterpretable spaces. We propose constructing sentence embeddings via the path signature of Word2Vec trajectories, leveraging the iterated integrals of rough path theory to create representations that explicitly capture sequential and lexical structure in a geometrically tractable manner. Through extensive ablations spanning Word2Vec dimensions, signature depths, and PCA compression, we correlate embedding distances against five lexical string metrics: Levenshtein, Jaro-Winkler, word Jaccard, phoneme Levenshtein, and character TF-IDF. We reveal that while average pooling almost entirely fails to capture character-level edit distances, path signatures strongly align with them, with depth 2 signatures optimally balancing character and word-level sensitivities. Furthermore, we demonstrate that the critical lexical variance is concentrated in a highly compact PCA subspace, and unlike pooling, signatures structurally preserve and quantify word order. These findings establish path signatures not as semantic generalizers, but as rigorous mathematical lenses that explicitly encode discrete lexical structures and word order, offering an interpretable alternative to opaque deep learning for lexical tracking.