1. What have the authors contributed in "Sparse reconstruction by separable approximation" ?
The authors present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex, sparsity-inducing function.. The authors propose iterative methods in which each step is an optimization subproblem involving a separable quadratic term ( diagonal Hessian ) plus the original sparsity-inducing term.. Their approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem.. In addition to solving the standard l2 − l1 case, their approach handles other problems, e. g., lp regularizers with p 6= 1, or group-separable ( GS ) regularizers.. Experiments with CS problems show that their approach provides state-of-the-art speed for the standard l2 − l1 problem, and is also efficient on problems with GS regularizers.
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2. What is the key to the efficiency of SpaRSA and IST algorithms?
If c has the separable form (3), the subproblem (6) is also separable and can be written asxk+1i ∈ arg min z(z − uki ) 22 +ταk ci(z), i = 1, 2, . . . , n. (7)Separability is key to the efficiency of SpaRSA and IST algorithms.
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3. What is the simplest variant of the SpaRSA scheme?
In the simplest variant of the SpaRSA scheme, the acceptance criterion is trivial: accept whatever z solves the subproblem (5) as the new iterate xk+1, even if it yields an increase in the objective function φ.
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4. What is the simplest way to solve a SpaRSA subproblem?
The existence of a value of αk sufficiently large to ensure a decrease in the objective at each iteration can be inferred from the connection between (6) and the following trust-region subproblem:min z∇f(xk)T (z− xk) + τc(z) subject to ‖z− xk‖2 ≤ ∆k.
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