A Laplacian Approach to ℓ_1-Norm Minimization
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We propose a novel differentiable reformulation of the linearly-constrained ℓ_1 minimization problem, also known as the basis pursuit problem. The reformulation is inspired by the Laplacian paradigm of network theory and leads to a new family of gradient-based, matrix-free methods for the solution of ℓ_1 minimization problems. We analyze the iteration complexity of a natural solution approach to the reformulation, based on a multiplicative weights update scheme, as well as the iteration complexity of an accelerated gradient scheme. The accelerated method, in particular, yields an improved worst-case bound on the complexity of matrix-free methods of basis pursuit.
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