A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
Citations
17,433 citations
Cites background from "A Fast Iterative Shrinkage-Threshol..."
...[173] compares and benchmarks a number of representative algorithms, including gradient projection [73, 102], homotopy methods [52], iterative shrinkage-thresholding [45], proximal gradient [132, 133, 11, 12], augmented Lagrangian methods [175], and interior-point methods [103]....
[...]
8,059 citations
Cites methods from "A Fast Iterative Shrinkage-Threshol..."
...When this method is combined with the iterative soft thresholding technique (for R(θ) = λ||θ||1), plus a continuation method that gradually reduces λ, we get a fast method for the BPDN problem known as the fast iterative shrinkage thesholding algorithm or FISTA (Beck and Teboulle 2009)....
[...]
6,783 citations
4,487 citations
Cites background or methods from "A Fast Iterative Shrinkage-Threshol..."
...Is shown in [2, 25, 27, 28] that if G or F ∗ is uniformly convex (such that G∗, or respectively F , has a Lipschitz continuous gradient), O(1/N2) convergence can be guaranteed....
[...]
...Remark 4 In [2, 25, 27], the O(1/N2) estimate is theoretically better than ours since it is on the dual energy G∗(−K∗yN)+F ∗(yN)−(G∗(−K∗ŷ)+F ∗(ŷ)) (which can easily be shown to bound ‖xN − x̂‖2, see for instance [13])....
[...]
...• FISTA: O(1/N2) fast iterative shrinkage thresholding algorithm on the dual ROF problem (66) [2, 25]....
[...]
...(35) In that case one can show that ∇G∗ is 1/γ -Lipschitz so that the dual problem (4) can be solved in O(1/N2) using any of the accelerated first order methods of [2, 25, 27], in the sense that the objective (in this case, the dual energy) approaches its optimal value at the rate O(1/N2), where N is the number of first order iterations....
[...]
...• NEST: Restarted version of Nesterov’s algorithm [2, 25, 28], on the dual Huber-ROF problem....
[...]
[...]
3,627 citations
Cites background or methods from "A Fast Iterative Shrinkage-Threshol..."
...Important papers on forward-backward splitting include those by Passty [159], Lions and Mercier [129], Fukushima and Mine [88], Gabay [90], Lemaire [120], Eckstein [78], Chen [54], Chen and Rockafellar [55], Tseng [184, 185, 187], Combettes and Wajs [62], and Beck and Teboulle [17, 18]....
[...]
...50 k f(k )− fs ta r Subgradient method Generalized gradient 9 # iterations 1This is taken from the lecture notes of Geoff Gordon and Ryan Tibshirani; “generalized gradient” in the legend means ISTA....
[...]
...This is also called the iterative soft thresholding algorithm, or ISTA....
[...]
...Here are typical runs2 for the LASSO, which compares the standard proximal gradient method (ISTA) to its accelerated version (FISTA): f (xk)− f ?...
[...]
...Hence generalized gradient update step is: x+ = S t(x + tA T (y Ax)) Resulting algorithm called ISTA (Iterative Soft-Thresholding Algorithm)....
[...]
References
9,950 citations
Additional excerpts
...R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp :// w w w .s ia m .o rg /jo ur na ls /o js a. ph p Copyright © by SIAM....
[...]
7,669 citations
4,699 citations
Additional excerpts
...R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp :// w w w .s ia m .o rg /jo ur na ls /o js a. ph p Copyright © by SIAM....
[...]
4,690 citations