  We study the relationship between the eluder dimension for a function class
and a generalized notion of rank, defined for any monotone "activation" $\sigma
: \mathbb{R} \to \mathbb{R}$, which corresponds to the minimal dimension
required to represent the class as a generalized linear model. When $\sigma$
has derivatives bounded away from $0$, it is known that $\sigma$-rank gives
rise to an upper bound on eluder dimension for any function class; we show
however that eluder dimension can be exponentially smaller than $\sigma$-rank.
We also show that the condition on the derivative is necessary; namely, when
$\sigma$ is the $\mathrm{relu}$ activation, we show that eluder dimension can
be exponentially larger than $\sigma$-rank.
