  We show that static data structure lower bounds in the group (linear) model
imply semi-explicit lower bounds on matrix rigidity. In particular, we prove
that an explicit lower bound of $t \geq \omega(\log^2 n)$ on the cell-probe
complexity of linear data structures in the group model, even against
arbitrarily small linear space $(s= (1+\varepsilon)n)$, would already imply a
semi-explicit ($\bf P^{NP}\rm$) construction of rigid matrices with
significantly better parameters than the current state of art (Alon, Panigrahy
and Yekhanin, 2009). Our results further assert that polynomial ($t\geq
n^{\delta}$) data structure lower bounds against near-optimal space, would
imply super-linear circuit lower bounds for log-depth linear circuits (a
four-decade open question). In the succinct space regime $(s=n+o(n))$, we show
that any improvement on current cell-probe lower bounds in the linear model
would also imply new rigidity bounds. Our results rely on a new connection
between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak,
2006), and on a new reduction from worst-case to average-case rigidity, which
is of independent interest.
