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While deep neural networks (DNNs) have achieved state-of-the-art results in many fields, they are typically over-parameterized. Parameter redundancy, in turn, leads to inefficiency. Sparse signal recovery (SSR) techniques, on the other hand, find compact solutions to over-complete linear problems. Therefore, a logical step is to draw the connection between SSR and DNNs. In this paper, we explore the application of iterative reweighting methods popular in SSR to learning efficient DNNs. By efficient, we mean sparse networks that require less computation and storage than the original, dense network. We propose a reweighting framework to learn sparse connections within a given architecture without biasing the optimization process, by utilizing the affine scaling transformation strategy. The resulting algorithm, referred to as Sparsity-promoting Stochastic Gradient Descent (SSGD), has simple gradient-based updates which can be easily implemented in existing deep learning libraries. We demonstrate the sparsification ability of SSGD on image classification tasks and show that it outperforms existing methods on the MNIST and CIFAR-10 datasets.