  In this work we consider the approximability of $\textsf{Max-CSP}(f)$ in the
context of sketching algorithms and completely characterize the approximability
of all Boolean CSPs. Specifically, given $f$, $\gamma$ and $\beta$ we show that
either (1) the $(\gamma,\beta)$-approximation version of $\textsf{Max-CSP}(f)$
has a linear sketching algorithm using $O(\log n)$ space, or (2) for every
$\epsilon > 0$ the $(\gamma-\epsilon,\beta+\epsilon)$-approximation version of
$\textsf{Max-CSP}(f)$ requires $\Omega(\sqrt{n})$ space for any sketching
algorithm. We also prove lower bounds against streaming algorithms for several
CSPs. In particular, we recover the streaming dichotomy of [CGV20] for $k=2$
and show streaming approximation resistance of all CSPs for which $f^{-1}(1)$
supports a distribution with uniform marginals. Our positive results show wider
applicability of bias-based algorithms used previously by [GVV17] and [CGV20]
by giving a systematic way to discover biases. Our negative results combine the
Fourier analytic methods of [KKS15], which we extend to a wider class of CSPs,
with a rich collection of reductions among communication complexity problems
that lie at the heart of the negative results.
