In order to reduce the quadratic cost of matrix-vector multiplications in dense and attention layers, Monarch matrices have been recently introduced, achieving a sub-quadratic complexity. It consists in factorizing a matrix using fixed permutations and learned block diagonal matrices, at the price of a small performance drop. We propose a more general model where some permutations are learned. The optimization algorithm explores the space of permutations using a Straight-Through Estimator (STE) inspired by the support exploration algorithm designed for sparse support recovery. Our experimental results demonstrate performance improvement in the context of sparse matrix factorization and of end-to-end sparse learning.

The *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: *sparse support recovery*. We introduce the *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 *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.