Rethinking Positional Encoding

TL;DR

This paper broadens the understanding of positional encoding by showing that non-Fourier embeddings can be effective, and introduces a general theory based on shifted basis functions that explains their performance.

Contribution

It demonstrates that alternative non-Fourier embeddings can be used for positional encoding and develops a general theoretical framework based on shifted basis functions.

Findings

Fourier features are a special case of a broader class of embeddings.

Performance depends on the trade-off between stable rank and distance preservation.

Theoretical predictions are empirically validated.

Abstract

It is well noted that coordinate based MLPs benefit -- in terms of preserving high-frequency information -- through the encoding of coordinate positions as an array of Fourier features. Hitherto, the rationale for the effectiveness of these positional encodings has been solely studied through a Fourier lens. In this paper, we strive to broaden this understanding by showing that alternative non-Fourier embedding functions can indeed be used for positional encoding. Moreover, we show that their performance is entirely determined by a trade-off between the stable rank of the embedded matrix and the distance preservation between embedded coordinates. We further establish that the now ubiquitous Fourier feature mapping of position is a special case that fulfills these conditions. Consequently, we present a more general theory to analyze positional encoding in terms of shifted basis functions. To this end, we develop the necessary theoretical formulae and empirically verify that our theoretical claims hold in practice. Codes available at https://github.com/osiriszjq/Rethinking-positional-encoding.