FoNE: Precise Single-Token Number Embeddings via Fourier Features
This addresses a specific bottleneck in LLMs for numerical tasks, offering significant efficiency and accuracy improvements, though it is incremental as it builds on existing embedding techniques.
The paper tackles the problem of inefficient number representation in Large Language Models (LLMs) by proposing FoNE, a method that encodes numbers as single tokens using Fourier features, resulting in 64× less data needed for 99% accuracy on 6-digit addition and 100% accuracy on over 100,000 test examples for basic arithmetic tasks.
Large Language Models (LLMs) typically represent numbers using multiple tokens, which requires the model to aggregate these tokens to interpret numerical values. This fragmentation makes both training and inference less efficient and adversely affects the model's performance on number-related tasks. Inspired by the observation that pre-trained LLMs internally learn Fourier-like features for number tokens, we propose Fourier Number Embedding (FoNE), a novel method that directly maps numbers into the embedding space with their Fourier features. FoNE encodes each number as a single token with only two embedding dimensions per digit, effectively capturing numerical values without fragmentation. This compact representation accelerates both training and inference. Compared to traditional subword and digit-wise embeddings, FoNE not only reduces computational overhead but also achieves higher accuracy across various numerical tasks including addition, subtraction and multiplication. On 6-digit decimal addition, FoNE requires 64$\times$ less data to achieve 99% accuracy than subword and digit-wise embeddings while using 3$\times$ and 6$\times$ fewer tokens per number, respectively. Furthermore, FoNE is the only method that yields 100% accuracy on over 100,000 test examples for addition, subtraction, and multiplication. The codes and visualization are available at https://fouriernumber.github.io/.