Rethinking Positional Encoding
This provides a more general theoretical framework for analyzing positional encoding, which is incremental but clarifies a known bottleneck in neural representation learning.
The paper tackles the problem of understanding why positional encodings work in coordinate-based MLPs, showing that non-Fourier embeddings can be effective and that performance depends on a trade-off between stable rank and distance preservation, with Fourier features as a special case.
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.