Kronecker Embeddings: Byte-Level Structured Token Representations for Parameter-Efficient Language Models

Large language models route every input through a learned embedding table of shape |V| x d_model, consuming hundreds of millions to billions of trainable parameters at frontier scale. We introduce Kronecker Embeddings, a deterministic byte-level character-position factorization that replaces this table with a fixed encoder and a single learned projection, compatible with standard BPE tokenizers, eliminating 91--94% of input-side trainable parameters at frontier scale. We provide five contributions. First, a cross-model probe across six LMs (135M-671B parameters) shows trained input embeddings cluster typographic variants of the probe word far more than morphological relatives; Kronecker escapes this clustering at the embedding layer. Second, a controlled three-seed comparison on nanoGPT GPT-2 124M over 2.5B tokens of FineWeb-Edu shows Kronecker reaching 2.5 +- 0.2% lower validation loss than the BPE-tied baseline (gap 0.083 +- 0.007 nats, ~9% lower perplexity), needing ~1.43x fewer steps to reach BPE's converged loss. Third, a spelling-robustness probe over 110 clean/typo pairs shows Kronecker preserves the top-1 prediction on 55.5% of pairs vs. 47.3% for BPE (+8.2 pp) and lowers KL by 7.6%, winning or tying in 10 of 11 categories; a generation probe shows Kronecker echoes byte-novel strings and typos through generation where BPE forgets them. Fourth, BPE embedding norm drifts during training while Kronecker projection norm stays near 1.0, consistent with a stable representational target. Fifth, an on-the-fly runtime variant reconstructs embeddings from a 4.5 MB byte buffer rather than a 2.15 GB table at vocabulary 131,072, with 0.01--0.24% step-time overhead. Byte-level locality has a tradeoff: byte-similar but semantically distant pairs (compute/commute, nation/notion) cluster together, shifting disambiguation to early attention layers.