  We study the sparsity and optimality properties of crowd navigation and find
that existing techniques do not satisfy both criteria simultaneously: either
they achieve optimality with a prohibitive number of samples or tractability
assumptions make them fragile to catastrophe. For example, if the human and
robot are modeled independently, then tractability is attained but the planner
is prone to overcautious or overaggressive behavior. For sampling based motion
planning of joint human-robot cost functions, for $n_t$ agents and $T$ step
lookahead, $\mathcal O(2^{2n_t T})$ samples are needed for coverage of the
action space. Advanced approaches statically partition the action space into
free-space and then sample in those convex regions. However, if the human is
\emph{moving} into free-space, then the partition is misleading and sampling is
unsafe: free space will soon be occupied. We diagnose the cause of these
deficiencies---optimization happens over \emph{trajectory} space---and propose
a novel solution: optimize over trajectory \emph{distribution} space by using a
Gaussian process (GP) basis. We exploit the "kernel trick" of GPs, where a
continuum of trajectories are captured with a mean and covariance function. By
using the mean and covariance as proxies for a trajectory family we reason
about collective trajectory behavior without resorting to sampling. The GP
basis is sparse and optimal with respect to collision avoidance and robot and
crowd intention and flexibility. GP sparsity leans heavily on the insight that
joint action space decomposes into free regions; however, the decomposition
contains feasible solutions only if the partition is dynamically generated. We
call our approach \emph{$\mathcal O(2^{n_t})$-sparse interacting Gaussian
processes}.
