  In this paper, we show that if the optimization function is
restricted-strongly-convex (RSC) and restricted-smooth (RSM) -- a rich subclass
of weakly submodular functions -- then a streaming algorithm with constant
factor approximation guarantee is possible. More generally, our results are
applicable to any monotone weakly submodular function with submodularity ratio
bounded from above. This (positive) result which provides a sufficient
condition for having a constant factor streaming guarantee for weakly
submodular functions may be of special interest given the recent negative
result (Elenberg et al., 2017) for the general class of weakly submodular
functions. We apply our streaming algorithms for creating compact synopsis of
large complex datasets, by selecting $m$ representative elements, by optimizing
a suitable RSC and RSM objective function. Above results hold even with
additional constraints such as learning non-negative weights, for
interpretability, for each selected element indicative of its importance. We
empirically evaluate our algorithms on two real datasets: MNIST- a handwritten
digits dataset and Letters- a UCI dataset containing the alphabet written in
different fonts and styles. We observe that our algorithms are orders of
magnitude faster than the state-of-the-art streaming algorithm for weakly
submodular functions and with our main algorithm still providing equally good
solutions in practice.
