Variable Metric Forward-Backward Algorithm

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1 Variable Metric Forward-Backward Algorithm 1/37 Variable Metric Forward-Backward Algorithm for minimizing the sum of a differentiable function and a convex function E. Chouzenoux in collaboration with A. Repetti and J.-C. Pesquet Laboratoire d Informatique Gaspard Monge - UMR CNRS 8049, Université Paris-Est, France. Séminaire PGMO, 15 mai 2013

2 Variable Metric Forward-Backward Algorithm 2/37 Outline 1. Problem statement Minimization problem Existence of minimizers 2. Theoretical background Variational Analysis Proximity operator Kurdyka- Lojasiewicz Inequality 3. Variable Metric Forward-Backward Algorithm Majorize-Minimize algorithm Proposed algorithm Convergence results 4. Application to image reconstruction Signal-dependent Gaussian noise Results

3 Variable Metric Forward-Backward Algorithm 3/37 Minimization problem Problem with : Find ˆx Argmin{G = F +R}, (1) R: R N (,+ ] proper, lsc, convex, R is continuous on its domain, F: R N R differentiable, F has an L-Lipschitz gradient on domr, i.e. ( (x,y) (domr) 2 ) F(x) F(y) L x y, G is coercive, i.e. lim x + G(x) = +.

4 Variable Metric Forward-Backward Algorithm 4/37 Existence of minimizers Find ˆx Argmin{G = F +R}, R: R N (,+ ] proper, lsc, convex, R is continuous on its domain, F: R N R differentiable, F has an L-Lipschitz gradient on domr, i.e. ( (x,y) (domr) 2 ) F(x) F(y) L x y, G is coercive, i.e. lim x + G(x) = +. domr domf domr = domg.

5 Variable Metric Forward-Backward Algorithm 5/37 Existence of minimizers Find ˆx Argmin{G = F +R}, R: R N (,+ ] proper, lsc, convex, R is continuous on its domain, F: R N R differentiable, F has an L-Lipschitz gradient on domr, i.e. ( (x,y) (domr) 2 ) F(x) F(y) L x y, G is coercive, i.e. lim x + G(x) = +. domg (= domr) is a nonempty convex set.

6 Variable Metric Forward-Backward Algorithm 6/37 Existence of minimizers Find ˆx Argmin{G = F +R}, R: R N (,+ ] proper, lsc, convex, R is continuous on its domain, F: R N R differentiable, F has an L-Lipschitz gradient on domr, i.e. ( (x,y) (domr) 2 ) F(x) F(y) L x y, G is coercive, i.e. lim x + G(x) = +. G proper lsc on R N and continuous on domg.

7 Variable Metric Forward-Backward Algorithm 7/37 Existence of minimizers Find ˆx Argmin{G = F +R}, R: R N (,+ ] proper, lsc, convex, R is continuous on its domain, F: R N R differentiable, F has an L-Lipschitz gradient on domr, i.e. ( (x,y) (domr) 2 ) F(x) F(y) L x y, G is coercive, i.e. lim x + G(x) = +. ( x domg) lev G(x) G is compact and ArgminG.

8 Variable Metric Forward-Backward Algorithm 8/37 Variational Analysis Let ψ: R N (,+ ] Domain of ψ: domψ = {x R N ψ(x) < + }. Proper function: ψ is proper if domψ. δ R ψ Level set of ψ at height δ R: lev δ ψ = {x R N ψ(x) δ}. lev δ ψ domψ R

9 Variable Metric Forward-Backward Algorithm 9/37 Variational Analysis [Rockafellar ] Sub-differential Let ψ: R N (,+ ] be a lsc proper function. Let x domψ. Fréchet sub-differential: { ˆ ψ(x) = t R N liminf y x y x Limiting sub-differential: } 1 ( ) ψ(y) ψ(x) (y x) t 0. x y ψ(x) = {ˆt R N y k x, ψ(y k ) ψ(x), t k ˆ ψ(y k ) ˆt }.

10 Variable Metric Forward-Backward Algorithm 9/37 Variational Analysis [Rockafellar ] Sub-differential Let ψ: R N (,+ ] be a lsc proper function. Let x domψ. Fréchet sub-differential: { ˆ ψ(x) = t R N liminf y x y x Limiting sub-differential: } 1 ( ) ψ(y) ψ(x) (y x) t 0. x y ψ(x) = {ˆt R N y k x, ψ(y k ) ψ(x), t k ˆ ψ(y k ) ˆt }. ψ is convex ψ corresponds to the usual sub-differential for convex functions x R N is a critical point of ψ iff 0 ψ(x). Convex particular case: x minimizer of ψ 0 ψ(x). General case: x minimizer of ψ 0 ψ(x).

11 Variable Metric Forward-Backward Algorithm 10/37 Proximity operator Proximity operator Let ψ: R N (,+ ] proper, lsc, convex. Let x R N. prox ψ (x) = argmin y R N ψ(y)+ 1 2 y x 2. Characterization of the proximity operator: p = prox ψ (x) x p ψ(p).

12 Variable Metric Forward-Backward Algorithm 11/37 Proximity operator relative to a metric Let U R N N be a symmetric positive definite matrix. Let x R N. Weighted norm: x U = ( x Ux ) 1/2. Loewner partial ordering on R N N ( (U 1,U 2 ) (R N N ) 2) U 1 U 2 x U 1 x x U 2 x.

13 Variable Metric Forward-Backward Algorithm 11/37 Proximity operator relative to a metric Let U R N N be a symmetric positive definite matrix. Let x R N. Weighted norm: x U = ( x Ux ) 1/2. Loewner partial ordering on R N N ( (U 1,U 2 ) (R N N ) 2) U 1 U 2 x U 1 x x U 2 x. Proximity operator relative to the metric induced by U Let ψ: R N (,+ ] proper, lsc, convex. prox U,ψ (x) = argmin y R N ψ(y)+ 1 2 y x U 2. prox IN,ψ = prox ψ.

14 Variable Metric Forward-Backward Algorithm 12/37 Proximity operator relative to a metric Characterization of the proximity operator: Let ψ: R N (,+ ] be a proper, lsc, convex function. ( x R N ) p = prox U,ψ (x) U(x p) ψ(p). Property ( x R N ) ψ(x) = N ψ (n) (x (n) ) n=1 U = Diag(u (1),...,u (N) ) with (u (n) ) 1 n N (0,+ ) N ( x R N ) prox U,ψ (x) = ( ) prox ψ (n) /u (n)(x(n) ) 1 n N Other properties can be found in [Becker and Fadili ].

15 Variable Metric Forward-Backward Algorithm 13/37 Kurdyka- Lojasiewicz inequality Kurdyka- Lojasiewicz Function G satisfies the Kurdyka- Lojasiewicz inequality i.e., for every ξ R, and, for every bounded subset E of R N, there exist three constants κ > 0, ζ > 0 and θ [0,1) such that ( t(x) G(x) ) t(x) κ G(x) ξ θ, for every x E such that G(x) ξ ζ (with the convention 0 0 = 0). Note that other forms of the KL inequality can be found in the literature [Bolte et al ][Bolte et al ]. Satisfied for a wide class of functions : real analytic functions semi-algebraic functions...

16 Variable Metric Forward-Backward Algorithm 14/37 Forward-Backward algorithm FB Algorithm x 0 R N For k = 0,1,... ȳ k = x k γ k F(x k ), y k = prox γk R(ȳ k ), x k+1 = x k +λ k (y k x k ), Convergence is established if: F convex with L-Lipschitzian gradient, R convex lsc proper, and 0 < inf l N γ l sup l N γ l < 2L 1, ( k N) 0 < inf l N λ l λ k 1. [Combettes and Pesquet ] F and R non convex, F with a Lipschitzian gradient, λ k 1. [Attouch, Bolte and Svaiter ]

17 Variable Metric Forward-Backward Algorithm 15/37 Variable Metric Forward-Backward algorithm VMFB Algorithm x 0 R N For k = 0,1,... ȳ k = x k γ k A 1 k F(x k ), y k = prox γ 1 k A k,r (ȳ k), x k+1 = x k +λ k (y k x k ), Convergence is established ([Combettes and Vũ ]) if F convex with L-Lipschitzian gradient, R convex lsc proper, and (η k ) k N l + 1 (N), such that ( k N) (1+η k)a k+1 A k, (ν,ν) (0,+ ) 2 such that ( k N) νi N A k νi N, ( ε 0, min{1, ] 2 } such that ( k N) L(1+ν) { ε γ k 2 Lν ε ν ε λ k 1.

18 Variable Metric Forward-Backward Algorithm 16/37 Our contribution [Chouzenoux et al ] Convergence of the VMFB algorithm for F non convex? Kurdyka- Lojasiewicz Inequality. Choice of variable metric (A k ) k N? Majorize-Minimize principle. Calculation of the proximity operator? Inexact VMFB algorithm.

19 Variable Metric Forward-Backward Algorithm 17/37 Majorize-Minimize assumption MM Assumption For every k N, there exists a symmetric positive definite matrix A k R N N such that for every x R N Q(x,x k ) = F(x k )+(x x k ) F(x k )+ 1 2 (x x k) A k (x x k ), is a majorant function of F at x k on domr, i.e., F(x k ) = Q(x k,x k ) and ( x domr) F(x) Q(x,x k ). There exists (ν,ν) (0,+ ) 2 such that ( k N) νi N A k νi N. F is differentiable with an L-Lipschitzian gradient on domr A k LI N satisfies the above assumption [Bertsekas ]

20 Variable Metric Forward-Backward Algorithm 18/37 Majorize-Minimize algorithm [Jacobson and Fessler ] MM Algorithm x k+1 ArgminQ(x,x k ) x Q(.,x k ) F VMFB Algorithm with R 0 λk 1 γk 1 x k x k+1

21 Variable Metric Forward-Backward Algorithm 18/37 Majorize-Minimize algorithm [Jacobson and Fessler ] MM Algorithm x k+1 ArgminQ(x,x k )+R(x) x Q(.,x k ) F VMFB Algorithm with λk 1 γk 1 x k x k+1

22 Variable Metric Forward-Backward Algorithm 19/37 Proposed algorithm VMFB Algorithm x 0 domr For k = 0,1,... ȳ k = x k γ k A 1 k F(x k), y k = prox γ 1 k A k,r (ȳ k), x k+1 = (1 λ k )x k +λ k y k, where (η,η) (0,+ ) 2 such that ( k N) η γ k λ k 2 η. λ (0,+ ) such that ( k N) λ λ k 1.

23 Variable Metric Forward-Backward Algorithm 19/37 Proposed algorithm Inexact VMFB Algorithm x 0 domr,τ (0,+ ) For k = 0,1,... Find y k R N and r(y k ) R(y k ) such that R(y k )+(y k x k ) F(x k )+γ 1 k y k x k 2 A k R(x k ), F(x k )+r(y k ) τ y k x k Ak, x k+1 = (1 λ k )x k +λ k y k, where (η,η) (0,+ ) 2 such that ( k N) η γ k λ k 2 η. λ (0,+ ) such that ( k N) λ λ k 1.

24 Variable Metric Forward-Backward Algorithm 20/37 Inexact proximal step { yk = prox γ 1 k A k,r (x k γ k A 1 k F(x k)) Convexity of R ( r(y k ) R(y k )) { r(y k ) = F(x k )+γ 1 k A k(x k y k ) (y k x k ) r(y k ) R(y k ) R(x k ). R(y k )+(y k x k ) F(x k )+γ 1 k y k x k 2 A k R(x k ), F(x k )+r(y k ) = γ 1 k A k(y k x k ) γ 1 k ν yk x k Ak η 1 ν y k x k Ak τ = η 1 ν

25 Variable Metric Forward-Backward Algorithm 21/37 Assumptions R proper lsc convex and continuous on domr, F differentiable, F L-Lipschitz on dom R, G is coercive. G satisfies the Kurdyka- Lojasiewicz inequality. (A k ) k N satisfies the majorization conditions. (λ k ) k N and (γ k ) k N bounded.

26 Variable Metric Forward-Backward Algorithm 21/37 Assumptions R proper lsc convex and continuous on domr, F differentiable, F L-Lipschitz on dom R, G is coercive. G satisfies the Kurdyka- Lojasiewicz inequality. (A k ) k N satisfies the majorization conditions. (λ k ) k N and (γ k ) k N bounded. There exists α (0,1] such that ( k N) G(x k+1 ) (1 α)g(x k )+αg(y k ).

27 Variable Metric Forward-Backward Algorithm 21/37 Assumptions R proper lsc convex and continuous on domr, F differentiable, F L-Lipschitz on dom R, G is coercive. G satisfies the Kurdyka- Lojasiewicz inequality. (A k ) k N satisfies the majorization conditions. (λ k ) k N and (γ k ) k N bounded. There exists α (0,1] such that ( k N) G(x k+1 ) (1 α)g(x k )+αg(y k ). Satisfied if ( k N) x k+1 is such that G(x k+1 ) G(y k ) and α = 1.

28 Variable Metric Forward-Backward Algorithm 21/37 Assumptions R proper lsc convex and continuous on domr, F differentiable, F L-Lipschitz on dom R, G is coercive. G satisfies the Kurdyka- Lojasiewicz inequality. (A k ) k N satisfies the majorization conditions. (λ k ) k N and (γ k ) k N bounded. There exists α (0,1] such that ( k N) G(x k+1 ) (1 α)g(x k )+αg(y k ). Satisfied if ( k N) x k+1 is such that G(x k+1 ) G(y k ) and α = 1. Satisfied if (λ k ) k N is such that, ( k N) λ k = α = 1.

29 Variable Metric Forward-Backward Algorithm 21/37 Assumptions R proper lsc convex and continuous on domr, F differentiable, F L-Lipschitz on dom R, G is coercive. G satisfies the Kurdyka- Lojasiewicz inequality. (A k ) k N satisfies the majorization conditions. (λ k ) k N and (γ k ) k N bounded. There exists α (0,1] such that ( k N) G(x k+1 ) (1 α)g(x k )+αg(y k ). Satisfied if ( k N) x k+1 is such that G(x k+1 ) G(y k ) and α = 1. Satisfied if (λ k ) k N is such that ( k N) λ k = α = 1. Satisfied if ( k N) G convex on [x k,y k ].

30 Variable Metric Forward-Backward Algorithm 21/37 Assumptions R proper lsc convex and continuous on domr, F differentiable, F L-Lipschitz on dom R, G is coercive. G satisfies the Kurdyka- Lojasiewicz inequality. (A k ) k N satisfies the majorization conditions. (λ k ) k N and (γ k ) k N bounded. There exists α (0,1] such that ( k N) G(x k+1 ) (1 α)g(x k )+αg(y k ). Satisfied if ( k N) x k+1 is such that G(x k+1 ) G(y k ) and α = 1. Satisfied if (λ k ) k N is such that ( k N) λ k = α = 1. Satisfied if ( k N) G convex on [x k,y k ]. Satisfied iff ( k N) there exists α k [α,1] such that G(x k+1 ) (1 α k )G(x k )+α k G(y k ).

31 Variable Metric Forward-Backward Algorithm 22/37 Descent Properties Property 1 There exists µ 1 (0,+ ), such that ( k N) G(x k+1 ) G(x k ) µ 1 2 x k+1 x k 2 G(x k ) µ 1 2 y k x k 2. Property 2 There exists µ 2 (0,+ ), such that ( k N) G(y k ) G(x k ) µ 2 2 y k x k 2.

32 Variable Metric Forward-Backward Algorithm 23/37 Convergence results Convergence theorem (x k ) k N and (y k ) k N, generated by the inexact VMFB algorithm (or the VMFB algorithm), both converge to a critical point ˆx of G. + k=0 x k+1 x k < + and + y k+1 y k < +. k=0 (G(x k )) k N and (G(y k )) k N both converge to G(ˆx). Local convergence to a global minimizer Let (x k ) k N and (y k ) k N, generated by the inexact VMFB algorithm (or the VMFB algorithm). There exists υ > 0 such that if G(x 0 ) inf x R N G(x) + υ, then (x k ) k N and (y k ) k N both converge to a solution to Problem (1).

Signal-dependent noise: w(h x) = ( w (m) ([H x] (m) ) ) 1 m M, with ( m {1,.

33 Variable Metric Forward-Backward Algorithm 24/37 Image reconstruction under signal-dependent noise H + x [0,+ ) N Hx w(hx) z [0,+ ) M Observation matrix: H [0,+ ) M N. Signal-dependent noise: w(h x) = ( w (m) ([H x] (m) ) ) 1 m M, with ( m {1,...,M}) a (m) [0,+ ), b (m) (0,+ ), w (m) ([H x] (m) ) realization of W (m) N ( 0,a (m) [H x] (m) +b (m)). OBJECTIVE: Produce an estimate ˆx [0,+ ) N of the target image x from the observed data z.

34 Variable Metric Forward-Backward Algorithm 25/37 Optimization problem Solve Problem (1): Find ˆx Argmin{G = F +R} where F: data fidelity term neg-log-likelihood of the data. R: penalty function serving to incorporate a priori information.

35 Variable Metric Forward-Backward Algorithm 25/37 Optimization problem Solve Problem (1): Find ˆx Argmin{G = F +R} where Data fidelity term (neg-log-likelihood of the data) { F 1(x)+F 2(x) if x [0,+ ) N F(x) =, where + otherwise F 1(x) = 1 M ([Hx] (m) z (m) ) 2 Convex function. 2 m=1 a (m) [Hx] (m) +b (m) F 2(x) = 1 M ) log (a (m) [Hx] (m) +b (m) Concave function. 2 Penalization term m=1 ( x R N ) R(x) = R 1(x)+R 2(x), where { R 1 ι [xmin,x N(x) = 0 if x [x min,x max] N max] + otherwise R 2 Sparsity prior in analysis frame or Total Variation..

36 Variable Metric Forward-Backward Algorithm 26/37 Majorization strategy for F 1 F 1 Convex and additive separable function. ( x [0,+ ) N ) F 1(x) = M m=1 ρ(m) 1 ([Hx] (m) ), where ( m {1,...,M}) ( u [0,+ )) ρ (m) 1 (u) = 1 (u z (m) ) 2 2 a (m) u +b (m).

37 Variable Metric Forward-Backward Algorithm 26/37 Majorization strategy for F 1 F 1 Convex and additive separable function. ( x [0,+ ) N ) F 1(x) = M m=1 ρ(m) 1 ([Hx] (m) ), where ( m {1,...,M}) ( u [0,+ )) ρ (m) 1 (u) = 1 (u z (m) ) 2 2 a (m) u +b (m). Then ( k N) a majorant function of F 1 on [0,+ ) N at x k is given by { Q 1(,x k ) = F 1(x k )+( x k ) F 1(x k )+( x k ) A k ( x k ) A k = Diag(P ω(hx k ))+εi N for ε 0. ( ) with ω: (v (m) ) 1 m M [0,+ ) M ω (m) (v (m) ) ( m {1,...,M}) ω (m) (u) = { ρ (m) 1 m M R M, where 1 (0) if u = 0, 2 ρ(m) 1 (0) ρ (m) 1 (u)+u ρ (m) 1 (u) u 2 if u > 0. ( m {1,...,M}) ( n {1,...,N}) P (m,n) = H (m,n) N p=1 H(m,p). Proof based on the strict concavity of ρ (m) 1 and Jensen s inequality ([Erdogan and Fessler ]).

38 Variable Metric Forward-Backward Algorithm 27/37 Implementation Construction of the majorant { F 1(x)+F 2(x) if x [0,+ ) N F(x) =, where + otherwise F 1 Convex function. Majorized at x k by Q 1(,x k ). F 2 Concave function. Majorized at x k by Q 2(,x k ) = F 2(x k )+( x k ) F 2(x k ).

39 Variable Metric Forward-Backward Algorithm 27/37 Implementation Construction of the majorant { F 1(x)+F 2(x) if x [0,+ ) N F(x) =, where + otherwise F 1 Convex function. Majorized at x k by Q 1(,x k ). F 2 Concave function. Majorized at x k by Q 2(,x k ) = F 2(x k )+( x k ) F 2(x k ). Backward step { y k = argmin R(x)+ 1 x R N 2 x ȳ k 2 γ 1 k { R(γ 1/2 A k } with ȳ k = x k γ k A 1 k F(x k ) x)+ 1 } x γ 1/2 k A 1/2 k ȳ k 2 2 y k = γ 1/2 k A 1/2 k argmin k A 1/2 k x R N Dual Forward-Backward Algorithm [Combettes et al ]

projections from 128 acquisitions lines and 128 angles. ( m {1,.

40 Variable Metric Forward-Backward Algorithm 28/37 Reconstruction with sparsity prior H: Radon matrix modeling M = parallel projections from 128 acquisitions lines and 128 angles. ( m {1,...,M}) a (m) = 0.01 and b (m) = 0.1 Original image Zubal Degraded sinogram

41 Variable Metric Forward-Backward Algorithm 29/37 Results: Restored images FBP: SNR=7 db VMFB: SNR=18.9 db

42 Variable Metric Forward-Backward Algorithm 30/37 Results VMFB Algorithm with λ k 1 and γ k 1 (solid line) λ k 1 and γ k 1.9 (dashed line) FB Algorithm with λ k 1 and γ k 1 (solid line) λ k 1 and γ k 1.9 (dashed line) FISTA G(xk) G(ˆx) xk ˆx Time (s) Time (s)

43 Variable Metric Forward-Backward Algorithm 31/37 Deblurring with Total Variation H: Blur operator corresponding to a truncated Gaussian kernel of standard deviation 1 and size 7 7. ( m {1,...,M}), a (m) = 0.5 and b (m) = 1 Original image Jetplane Degraded image: SNR=21.95 db

44 Variable Metric Forward-Backward Algorithm 32/37 Results: Restored images Degraded image: SNR=21.95 db Restored image: SNR=27.09 db

45 Variable Metric Forward-Backward Algorithm 33/37 Results VMFB Algorithm with λ k 1 and γ k 1 (solid line) λ k 1 and γ k 1.9 (dashed line) FB Algorithm with λ k 1 and γ k 1 (solid line) λ k 1 and γ k 1.9 (dashed line) FISTA G(xk) G(ˆx) xk ˆx Time (s) Time (s)

46 Variable Metric Forward-Backward Algorithm 34/37 Conclusion Convergence of the VMFB algorithm for the sum of a non convex differentiable function F and a non smooth convex function R. Choice of variable metric (A k ) k N based on MM principle. Inexact VMFB algorithm for the calculation of the proximity operator. The variable metric strategy leads to a significant acceleration in terms of decay of both the objective function and the error on the iterates in each experiment.

47 Variable Metric Forward-Backward Algorithm 35/37 Bibliography D. P. Bertsekas. Nonlinear Programming. 2nd. edn. Athena Scientific, Belmont, MA, R. T. Rockafellar and R. J. B. Wets Variational Analysis. 1st edn. Grundlehren der Mathematischen Wissenschaften, vol. 317, Springer, Berlin, H. Attouch, J. Bolte and B. F. Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program., vol. 137, pp , Feb J. Bolte, A. Daniilidis, A. Lewis and M. Shiota. Clarke subgradients of stratifiable functions. SIAM J. Optim., vol. 18, no. 2, pp , J. Bolte, A. Daniilidis, O. Ley and L. Mazet. Characterizations of Lojasiewicz inequalities and applications. Trans. Amer. Math. Soc., vol.362, no. 6, pp , S. Becker and J. Fadili. A quasi-newton proximal splitting method. Tech. Rep., Available on E. Chouzenoux, J.-C. Pesquet and A. Repetti. Variable Metric Forward-Backward algorithm for minimizing the sum of a differentiable function and a convex function. Tech. Rep., Available on HTML/2013/01/3749.html

48 Variable Metric Forward-Backward Algorithm 36/37 Bibliography P. L. Combettes, D. Dũng and B. C. Vũ. Proximity for sums of composite functions. J. Math. Anal. Appl., vol. 380, no. 2, pp , Aug P. L. Combettes and J.-C. Pesquet. Proximal thresholding algorithm for minimization over orthonormal bases. SIAM J. Optim., vol. 18, no. 4, pp , Nov P. L. Combettes and B. C. Vũ. Variable metric forward-backward splitting with applications to monotone inclusions in duality. to appear in Optimization, H. Erdogan and J. A. Fessler. Monotonic algorithms for transmission tomography. IEEE Trans. Med. Imag., vol. 18, no. 9, pp , Nov M. W. Jacobson and J. A. Fessler. An expanded theoretical treatment of iteration-dependent Majorize-Minimize algorithms. IEEE Trans. Image Process., vol. 16, no. 10, pp , Oct J. J. Moreau. Proximité et Dualité dans un espace hilbertien. Bull. Soc. Math. France, vol. 93, pp , 1965.

49 Variable Metric Forward-Backward Algorithm 37/37 Thank you!

A memory gradient algorithm for l 2 -l 0 regularization with applications to image restoration

A memory gradient algorithm for l 2 -l 0 regularization with applications to image restoration E. Chouzenoux, A. Jezierska, J.-C. Pesquet and H. Talbot Université Paris-Est Lab. d Informatique Gaspard