Sparse Approximation Over the Cube
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This paper presents an anlysis of the NP-hard minimization problem min{b - Ax_2: x ∈ [0,1]^n, | supp(x) | ≤σ}, where supp(x) = {i ∈ [n]: x_i ≠ 0} and σ is a positive integer. The object of investigation is a natural relaxation where we replace | supp(x) | ≤σ by ∑_i x_i ≤σ. Our analysis includes a probabilistic view on when the relaxation is exact. We also consider the problem from a deterministic point of view and provide a bound on the distance between the images of optimal solutions of the original problem and its relaxation under A. This leads to an algorithm for generic matrices A ∈ℤ^m × n and achieves a polynomial running time provided that m and A_∞ are fixed.
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