  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.
