  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 mainly 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. In addition, we argue
that employing a more complex positional encoding -- that scales exponentially
with the number of modes -- requires only a linear (rather than deep)
coordinate function to achieve comparable performance. Counter-intuitively, we
demonstrate that trading positional embedding complexity for network deepness
is orders of magnitude faster than current state-of-the-art; despite the
additional embedding complexity. To this end, we develop the necessary
theoretical formulae and empirically verify that our theoretical claims hold in
practice.
