Abstract
The LASSO problem has been explored extensively for CT image reconstruction, the most useful algorithm to solve the LASSO problem is the FISTA. In this paper, we prove that FISTA has a better linear convergence rate than ISTA. Besides, we observe that the convergence rate of FISTA is closely related to the acceleration parameters used in the algorithm. Based on this finding, an acceleration parameter setting strategy is proposed. Moreover, we adopt the function restart scheme on FISTA to reconstruct CT images. A series of numerical experiments is carried out to show the superiority of FISTA over ISTA on signal processing and CT image reconstruction. The numerical experiments consistently demonstrate our theoretical results.
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References
Beck A, Teboulle M (2009) A fast iterative shrinkage–thresholding algorithm for linear inverse problems. SIAM J Image Sci 2(1):183–202
Chang J, Zhang LJ (2019) Case Mix Index weighted multi-objective optimization of inpatient bed allocation in general hospital. J Comb Optim 37(1):1–19
Chen SS, Donoho DL, Saunders MA (2001) Atomic decomposition by basis pursuit. SIAM Rev 43(1):129–159
Devore R, Jawerth B (1992) Image compression through wavelet transform coding. IEEE Trans Inf Theory 38(2):719–746
Donoho DL (1995) De-noising by soft-thresholding. IEEE Trans Inf Theory 41(3):613–627
Donoho DL (2006) Compressed sensing. IEEE Trans Inf Theory 52(4):1289–1306
Efron B, Hastie T, Johnstone IM, Tibshirani R (2004) Least angle regression. Ann Stat 32(2):407–499
Gai L, Ji JD (2019) An integrated method to solve the healthcare facility layout problem under area constraints. J Comb Optim 37(1):95–113
Hale E, Yin W, Zhang Y (2008) A fixed-point continuation method for \(l_1\)-minimization: methodology and convergence. SIAM J Optim 19(3):1107–1130
Johnstone PR, Moulin P (2015) A Lyapunov analysis of FISTA with local linear convergence for sparse optimization. arXiv preprint arXiv:1502.02281
Nesterov Y (1983) A method for unconstrained convex minimization problem with the rate of convergence. Dokl AN SSSR 269:545–547
O’Donoghue B, Candès E (2015) Adaptive restart for accelerated gradient schemes. Found Comput Math 15(3):715–732
Tao SZ, Boley D, Zhang SZ (2016) Local linear convergence of ISTA and FISTA on the LASSO problem. SIAM J Optim 26(1):313–336
Tseng P (2008) On accelerated proximal gradient methods for convex-concave optimization. Manuscript. University of Washington. Seattle
Wen B, Chen XJ, Pong TK (2017) Linear convergence of proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth minimization problems. SIAM J Optim 27(1):124–145
Xu ZB, Chang XY, Xu FM, Zhang H (2012) \(L_{1/2}\) regularization: a thresholding representation theory and a fast solver. IEEE Trans Neural Netw Learn Syst 23(7):1013–1027
Yang AY, Zhou Z, Balasubramanian AG, Sastry S, Ma Y (2013) Fast \(L_{1}\)-minimization algorithms for robust face recognition. IEEE Trans Image Process 22(8):3234–3246
Zhong LW, Bai YQ (2019) Three-sided stable matching problem with two of them as cooperative partners. J Comb Optim 37(1):286–292
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This research was supported by the National Natural Science Foundation of China (Grant No. 11901382).
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Li, Q., Zhang, W. An improved linear convergence of FISTA for the LASSO problem with application to CT image reconstruction. J Comb Optim 42, 831–847 (2021). https://doi.org/10.1007/s10878-019-00453-7
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DOI: https://doi.org/10.1007/s10878-019-00453-7