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This paper presents a novel online method for sparse signal reconstruction. In particular, the notion of sub-dimensional projections is introduced, which allows a significant complexity reduction in the Adaptive Projection-based Algorithm using Weighted ℓ1 balls (APWL1). This is achieved without sacrificing performance. The proposed method is evaluated in both stationary and time-varying environments and its performance is compared with state-of-the-art online and batch LASSO-based methods.