  The {\it straight-through estimator} (STE) is commonly used to optimize
quantized neural networks, yet its contexts of effective performance are still
unclear despite empirical successes.To make a step forward in this
comprehension, we apply STE to a well-understood problem: {\it sparse support
recovery}. We introduce the {\it Support Exploration Algorithm} (SEA), a novel
algorithm promoting sparsity, and we analyze its performance in support
recovery (a.k.a. model selection) problems. SEA explores more supports than the
state-of-the-art, leading to superior performance in experiments, especially
when the columns of $A$ are strongly coherent.The theoretical analysis
considers recovery guarantees when the linear measurements matrix $A$ satisfies
the {\it Restricted Isometry Property} (RIP).The sufficient conditions of
recovery are comparable but more stringent than those of the state-of-the-art
in sparse support recovery. Their significance lies mainly in their
applicability to an instance of the STE.
