  The article concerns low-rank approximation of matrices generated by sampling
a smooth function of two $m$-dimensional variables. We refute an argument made
in the literature that, for a specific class of analytic functions, such
matrices admit accurate entrywise approximation of rank that is independent of
$m$. We provide a theoretical explanation of the numerical results presented in
support of this argument, describing three narrower classes of functions for
which $n \times n$ function-generated matrices can be approximated within an
entrywise error of order $\varepsilon$ with rank $\mathcal{O}(\log(n)
\varepsilon^{-2} \mathrm{polylog}(\varepsilon^{-1}))$ that is independent of
the dimension $m$: (i) functions of the inner product of the two variables,
(ii) functions of the squared Euclidean distance between the variables, and
(iii) shift-invariant positive-definite kernels. We extend our argument to
low-rank tensor-train approximation of tensors generated with functions of the
multi-linear product of their $m$-dimensional variables. We discuss our results
in the context of low-rank approximation of attention in transformer neural
networks.
