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

Token embeddings are the basic representational units that connect discrete tokens with continuous computation in language models. Although modern language models learn embeddings from random initialization through gradient-based training, the dynamical mechanism by which meaningful embedding structures emerge remains unclear. In this work, we identify that the evolving embedding structures are closely related to token-conditioned label and contextual distributions, which we formalize as probability signatures. We observe a progressive learning process, which we term Context Staircase: embeddings learn the low-order statistic signatures of the data before the high-order ones. More specifically, we observe that early in training they align with the simplest, context-free signature linking a token to its label, and as training proceeds, they progressively reflect signatures involving more and more context tokens. We then analyze the gradient flow of embeddings under small initialization to explain this phenomenon, deriving embedding evolution equations for feed-forward and self-attention architectures. We further extend these observations to real language-model training. Finally, we show that these embedding structures play an important role in both task learning and the incorporation of semantic structure into the embedding space. Overall, our results provide a dynamic explanation of how data statistics and architecture jointly shape token embeddings in language models, and reveal an implicit bias in the space of data statistics: training proceeds from simpler, low-order statistical relations toward increasingly complex, context-dependent ones.

Path signatures provide a rich representation of sequential data, with strong theoretical guarantees and good performance in a variety of machine-learning tasks. While signatures have progressed from fixed feature extractors to trainable components of machine-learning models, existing libraries often lack the required scalability for large-scale, gradient-based learning. To address this gap, this paper introduces pathsig, a PyTorch-native library that computes path signatures directly in the word basis. By using CUDA kernels to update signature coefficients in parallel over prefix-closed word sets, pathsig achieves high GPU throughput and near-minimal peak memory. Compared with other libraries, pathsig achieves 10-30x speedups for computation of truncated signatures and up to 4-10x speedups in training that require backpropagation through the signature. Beyond regular truncation, pathsig supports projections of the (infinite-dimensional) signature onto user-specified sets of words and anisotropic truncation motivated by inhomogeneous path regularity, enabling more compact representations that can reduce dimensionality, redundancy, and computational cost.

Studying the shape of text embedding spaces enhances our understanding of model behavior and how geometry affects downstream task performance. While a range of topological and geometric metrics have been explored, they are typically analyzed in isolation, and the connections between them remain largely unknown. In this paper, we present Unified Topological Signatures (UTS) for characterizing the structure of embedding spaces. We evaluate our method across a wide range of text embedding models and retrieval datasets, finding that it outperforms individual metrics in downstream performance prediction. Through several experiments, we demonstrate that the signatures characterize embedding spaces more accurately, improve the prediction of model-specific properties and biases, and reveal novel insights into representational similarity that challenge the global claims of the Platonic Representation Hypothesis.