Projected Nesterov's Proximal-Gradient Algorithm for Sparse Signal Recovery

PNPG Algorithm Applications References References References Acronyms

Projected Nesterov’s Proximal-Gradient Algorithmfor Sparse Signal Recovery�

Aleksandar Dogandžić

Electrical and Computer Engineering

� joint work with Renliang Gu, Ph.D. student

supported by

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Terminology and Notation

soft-thresholding operator for a D .ai /NiD1 2 RN :�T �.a/

�iD sign.ai /max

�jai j � �; 0�I“�” is the elementwise version of “�”;proximal operator for function r.x/ scaled by �:

prox�r a D arg minx

12kx � ak22 C �r.x/:

"-subgradient (Rockafellar 1970, Sec. 23):

@"r.x/ ,˚g 2 Rp j r.z/ � r.x/C .z � x/Tg � ";8z 2 Rp

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Introduction I

For most natural signals x,

# significant coefficients of .x/ � signal size p

where .x/ W Rp 7! Rp

is sparsifying transform.

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Sparsifying Transforms I

p pixels

lineartransform$DWT

# significant coeffs � p

.x/ D ‰Tx

where ‰ 2 Rp�p0 is a known sparsifying dictionary matrix.4 / 71

Sparsifying Transforms II

p pixels

transform$gradient map

# significant coeffs � p

Œ .x/�p0

iD1 ,sXj2Ni

.xi � xj /2

Ni is the index set of neighbors of xi .5 / 71

Convex-Set Constraint

x 2 Cwhere C is a nonempty closed convex set.Example: the nonnegative signal set

C D RpC

is of significant practical interest and applicable to X-ray CT,SPECT, PET, and MRI.

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Sense the significant components of .x/ using a small number ofmeasurements.Define the noiseless measurement vector �.x/, where

�.�/ W Rp 7! RN (N � p).

Example: Linear model

�.x/ D ˆx

where ˆ 2 RN�p is a known sensing matrix.

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Penalized NLL

objectivefunction

f .x/ D L.x/ C u�k .x/k1 C IC .x/

��

convex differentiablenegative log-likelihood (NLL)convex penalty termu > 0 is a scalar tuning constantC � cl

�domL.x/

�8 / 71

Comment

Our objective function f .x/ isconvex

has a convex set as minimum (unique if L.x/ is stronglyconvex),

not differentiable with respect to the signal xcannot apply usual gradient- or Newton-type algorithms,need proximal-gradient (PG) schemes.

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Develop a fast algorithm withO�k�2

�convergence-rate and

iterate convergence guaranteesfor minimizing f .x/ that

is general (for a diverse set of NLLs),requires minimal tuning, andis matrix-free�, a must for solving large-scale problems.

�involves only matrix-vector multiplications implementable using, e.g.,function handle in Matlab

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Majorizing Function

Define the quadratic approximation of the NLL L.x/:

Qˇ�x j xx� D L.xx/C .x � xx/TrL.xx/C 1

2ˇkx � xxk22

with ˇ chosen so that Qˇ�x j xx� majorizes L.x/ in the

neighborhood of x D xx.

x̄(j)x̄(i)

L(x)Qβ(i) (x|x̄(i))Qβ(j) (x|x̄(j))

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x̄(j)x̄(i)

Q10β(j) (x|x̄(j))

Figure 1: Majorizing function: Impact of ˇ.

No need for strict majorization, sufficient to majorize in theneighborhood of xx where we wish to move next!

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PNPG Method: Iteration i

B.i/ D ˇ.i�1/=ˇ.i/

� .i/ D˚1; i � 11 Cqb C B.i/�� .i�1/�2; i > 1

xx.i/ D PC�x.i�1/ C � .i�1/ � 1

� .i/

�x.i�1/ � x.i�2/

��accel. step

x.i/ D proxˇ .i/ur

�xx.i/ � ˇ.i/rL�xx.i/�� PG step

where ˇ.i/ > 0 is an adaptive step size:satisfies

L�x.i/

� � Qˇ .i/

�x.i/ j xx.i/

�majorization condition,

is as large as possible.more

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allows general domL

B.i/ D ˇ.i�1/=ˇ.i/

� .i/ D˚i; i � 11 Cqb C B.i/�� .i�1/�2; i > 1

xx.i/ D PC

�x.i�1/ C � .i�1/ � 1

� .i/

�x.i�1/ � x.i�2/

��accel. step

x.i/ D proxˇ .i/ur

L�x.i/

� � Qˇ .i/

�x.i/ j xx.i/

is as large as possible.

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B.i/ D ˇ.i�1/=ˇ.i/

� .i/ D˚i; i � 11 Cqb C B.i/�� .i�1/�2; i > 1

xx.i/ D PC�x.i�1/ C � .i�1/ � 1

� .i/

�x.i�1/ � x.i�2/

��accel. step

x.i/ D proxˇ .i/ur

L�x.i/

� � Qˇ .i/

�x.i/ j xx.i/

needs to hold for x.i/, not for all x!

L�x� — Qˇ .i/

�x j xx.i/

�in general, for an arbitrary x! more

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Algorithm 1: PNPG method

Input: x.�1/, u, n, m, �, �, and threshold �Output: arg min

xf .x/

Initialization: � .0/ 0, x.0/ 0, i 0, � 0, ˇ.1/ by the BB methodrepeat

i i C 1 and � � C 1while true do // backtracking search

evaluate B.i/, � .i/, and xx.i/if xx.i/ … domL then // domain restart

� .i�1/ 1 and continue

solve the PG stepif majorization condition holds then

breakelse

if ˇ.i/ > ˇ.i�1/ then // increase nn nCm

ˇ.i/ �ˇ.i/ and � 0

if i > 1 and f�x.i/

�> f

�x.i�1/

�then // function restart

� .i�1/ 1, i i � 1, and continue

if convergence condition holds thendeclare convergence

if � � n then // adapt step size� 0 and ˇ.iC1/ ˇ.i/=�

elseˇ.iC1/ ˇ.i/

until convergence declared or maximum number of iterations exceeded16 / 71

10�7

10�6

10�5

10�4

0 50 100 150 200 250 300

Sizeˇ.i/

# of iterations

PNPG .n D 4/PNPG .n D1/

��

Figure 2: Illustration of step-size selection for Poisson generalized linearmodel (GLM) with identity link.

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Momentum Illustration

xx.i/ D PC� yx.i/

¥x.i�1/ C �.i�1/�1

�.i/

�x.i�1/ � x.i�2/�

momentum termprevents zigzagging

x(i−2)

x(i−1)

x̂(i)

−∇L(x(i))

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Comments on the extrapolation term � .i/ I

� .i/ D 1 Cqb C B.i/�� .i�1/�2; i � 2

� 2; b 2 �0; 1=4�are momentum tuning constants.

To establish O�k�2

�convergence of PNPG, need

� .i/ � 12Cq14C B.i/�� .i�1/�2; i � 2:

controls the rate of increase of � .i/.

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Comments on the extrapolation term � .i/ II

� .i/x? implies stronger momentum.

Effect of step size

In the “steady state” where ˇ.i�1/ D ˇ.i/, � .i/x? aproximatelylinearly with i , with slope 1= .Changes in the step size affect � .i/:ˇ.i/ < ˇ.i�1/ step size decrease, faster increase of � .i/,ˇ.i/ > ˇ.i�1/ step size increase, decrease or slower increase of � .i/

than in the steady state.

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Proximal Mapping

To compute

prox�r a D arg minx

12kx � ak22 C �r.x/

usefor `1-norm penalty with .x/ D ‰Tx, alternating directionmethod of multipliers (ADMM)

o iterative

for total-variation (TV)-norm penalty with gradient map .x/,an inner iteration with the TV-based denoising method in(Beck and Teboulle 2009b).

o iterative

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Inexact PG Steps

B.i/ D ˇ.i�1/=ˇ.i/

� .i/ D˚i; i � 11 Cqb C B.i/�� .i�1/�2; i > 1

xx.i/ D PC

�x.i�1/ C � .i�1/ � 1

� .i/

�x.i�1/ � x.i�2/�� accel. step

x.i/ Ñ".i/ proxˇ .i/ur

Because of their iterative nature, PG steps are inexact: ".i/

quantifies the precision of the PG step in Iteration i .more

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Remark (Monotonicity)

The projected Nesterov’s proximal-gradient (PNPG) iteration withrestart is non-increasing:

f�x.i/

� � f �x.i�1/�if the inexact PG steps are sufficiently accurate and satisfy

".i/ �pı.i/

ı.i/ , x.i/ � x.i�1/ 2

is the local variation of signal iterates.

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Convergence Criterion

pı.i/ < �

x.i/ 2

where � > 0 is the convergence threshold.more

RestartThe goal of function and domain restarts is to ensure that

the PNPG iteration is monotonic andxx.i/ and x.i/ remain within domf .

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Summary of PNPG Approach

Combineconvex-set projection withNesterov acceleration.

Applyadaptive step size,restart.

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Thanks to step-size adaptation, no need for Lipschitzcontinuity of the gradient of the NLL.domL does not have to be Rp.

Extends the application of the Nesterov’s acceleration� to moregeneral measurement models than those used previously.

�Y. Nesterov, “A method of solving a convex programming problem withconvergence rate O.1=k2/,” Sov. Math. Dokl., vol. 27, 1983, pp. 372–376.

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Theorem (Convergence of the Objective Function)

Assume

NLL L.x/ is convex and differentiable and r.x/ is convex,C � domL: no need for domain restart.

Consider the PNPG iteration without restart.

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Theorem (Convergence of the Objective Function)

f�x.k/

� � f �x?� � 2 x.0/ � x? 22C E.k/

ˇ.1/ CPkiD1

pˇ.i/

�2where

E.k/ ,kXiD1

�� .i/".i/

�2 error term, accounts for inexact PG steps

x? , arg minxf .x/

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Comments

f�x.k/

� � f �x?� � 2 x.0/ � x? 22C E.k/

ˇ.1/ CPkiD1

pˇ.i/

�2Step sizes ˇ.i/

x?, convergence-rate upper bound?y.

better initialization, convergence-rate upper bound #.smaller prox-step approx. error, convergence-rate bound #.

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CorollaryUnder the condition of the Theorem,

f�x.k/

� � f �x?� � 2

x.0/ � x? 22C E.k/

2k2ˇminŸO�k�2

�if E.C1/ < C1

provided that

ˇmin ,C1minkD1

ˇ.k/ > 0:

The assumption that the step-size sequence is lower-bounded by astrictly positive quantity is weaker than Lipschitz continuity ofrL.x/ because it is guaranteed to have ˇmin > �=L if rL.x/ has aLipschitz constant L.

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Theorem (Convergence of Iterates)

Assume that1 NLL L.x/ is convex and differentiable and r.x/ is convex,2 C � domL, hence no need for domain restart,

3 cumulative error term E.k/ converges: E.C1/ < C1,4 momentum tuning constants satisfy > 2 and b 2 �0; 1= 2�,5 the step-size sequence

�ˇ.i/

�C1iD1 is bounded within the range�

ˇmin; ˇmax�; .ˇmin > 0/.

R The sequence of PNPG iterates x.i/ without restart convergesweakly to a minimizer of f .x/. a minimizer of f .x/.

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Theorem (Convergence of Iterates)strict inequality

Assume that1 NLL L.x/ is convex and differentiable and r.x/ is convex,2 C � domL, hence no need for domain restart,

3 cumulative error term E.k/ converges: E.C1/ < C1,

4 momentum tuning const. satisfy > 2 and b 2 �0; 1= 2�,5 the step-size sequence

�ˇ.i/

�C1iD1 is bounded within the range�

ˇmin; ˇmax�; .ˇmin > 0/.

R The sequence of PNPG iterates x.i/ without restart convergesweakly to a minimizer of f .x/. a minimizer of f .x/.

narrower than�0; 1=4

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Idea of Proof.Recall the inequality

� .i/ D 1 Cqb C B.i/�� .i�1/�2 � 1

2Cq14C B.i/�� .i�1/�2

used to establish convergence of the objective function.Assumption 4:

> 2; b 2 �0; 1= 2�creates a sufficient “gap” in this inequality that allows us to

show faster convergence of the objective function than theprevious theorem andestablish the convergence of iterates.

R Inspired by (Chambolle and Dossal 2015).

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Introduction

Signal reconstruction from Poisson-distributed measurements withaffine model for the mean-signal intensity is important for

tomographic (Ollinger and Fessler 1997),astronomic, optical, microscopic (Bertero et al. 2009),hyperspectral (Willett et al. 2014)

imaging.

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PET: Coincidence detection due to positron decay and annihilation(Prince and Links 2015).

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Measurement Model

N independent measurements y D .yn/NnD1 follow the Poissondistribution with means

Œˆx C b�nwhere

ˆ 2 RN�pC ; b

are the known sensing matrix and intercept term�.

�the intercept b models background radiation and scattering, obtained, e.g.,by calibration before the measurements y have been collected

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Existing Work

The sparse Poisson-intensity reconstruction algorithm (SPIRAL)§

approximates the logarithm functionin the underlying NLL by adding asmall positive term to it and thendescends a regularized NLLobjective function with proximalsteps that employ Barzilai-Borwein(BB) step size in each iteration,followed by backtracking.

§Z. T. Harmany et al., “This is SPIRAL-TAP: Sparse Poisson intensityreconstruction algorithms—theory and practice,” IEEE Trans. Image Process.,vol. 21, no. 3, pp. 1084–1096, Mar. 2012.

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PET Image Reconstruction

128 � 128 concentration map x.Collect the photons from 90 equallyspaced directions over 180ı, with 128radial samples at each direction,Background radiation, scattering effect,and accidental coincidence combined to-gether lead to a known intercept term b.The elements of the intercept termare set to a constant equal to 10% of

the sample mean of ˆx: b D 1Tˆx

The model, choices of parameters in the PET system setup, andconcentration map have been adopted from Image ReconstructionToolbox (IRT) (Fessler 2016).

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Numerical Example

Main metric for assessing the performance of the comparedalgorithms is relative square error (RSE)

RSE D kyx � xtruek22

kxtruek22where xtrue and yx are the true and reconstructed signal,respectively.All iterative methods use the convergence threshold

� D 10�6

and have the maximum number of iterations limited to 104.Regularization constant u has the form

u D 10a:We vary a in the range Œ�6; 3� with a grid size of 0.5 andsearch for the reconstructions with the best RSE performance.

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Compared Methods

Filtered backprojection (FBP) (Ollinger and Fessler 1997) andPG methods that aim at minimizing f .x/ with nonnegative x:

C D RpC:

All iterative methods initialized by FBP reconstructions.

Matlab implementation available athttp://isucsp.github.io/imgRecSrc/npg.html.

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PG Methods

PNPG with . ; b/ D .2; 1=4/.AT (Auslender and Teboulle 2006) implemented in thetemplates for first-order conic solvers (TFOCS) package(Becker et al. 2011) with a periodic restart every 200 iterations(tuned for its best performance) and our proximal mapping.SPIRAL, when possible.

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(a) radio-isotope concentration (b) attenuation map

Figure 3: (a) True emission image and (b) density map.

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RSE=3.09 %

(a) FBP

RSE=0.66 %

(b) `1

Figure 4: Reconstructions of the emission concentration map for expectedtotal annihilation photon count (SNR) equal to 108.

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RSE=0.66 %

(a) `1

RSE=0.22 %

(b) TV

Figure 5: Comparison of the two sparsity regularizations.

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10�1

0 20 40 60 80 100 120 140

Time/s

ATPNPG .n D1/

PNPG .n D m D 0/PNPG .n D m D 4/

(a) `1, 1T E.y/ D 108

10�4

10�3

10�2

10�1

0 10 20 30 40 50 60

Time/s

SPIRALAT

PNPG .n D1/PNPG .n D m D 0/PNPG .n D m D 4/

(b) TV, 1T E.y/ D 107

Figure 6: Centered objectives as functions of CPU time.

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PNPG Algorithm Applications References References Conclusion References

10�1

0 20 40 60 80 100 120 140

Time/s

ATPNPG .n D1/

PNPG .n D m D 0/PNPG .n D m D 4/

(a) `1, 1T E.y/ D 108

10�4

10�3

10�2

10�1

0 10 20 30 40 50 60

Time/s

SPIRALAT

PNPG .n D1/PNPG .n D m D 0/PNPG .n D m D 4/

(b) TV, 1T E.y/ D 107

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Linear Model with Gaussian Noise

L.x/ D 1

2ky �ˆxk22

wherey 2 RN is the measurement vector andthe elements of the sensing matrix ˆ are independent,identically distributed (i.i.d.), drawn from the standard normaldistribution.

We select the `1-norm sparsifying signal penalty with linear map:

.x/ D ‰Tx:

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−0.5

0 200 400 600 800 1000

Figure 7: True signal.

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Comments I

More methods available for comparison:

sparse reconstruction by separable approximation (SpaRSA)(Wright et al. 2009),generalized forward-backward (GFB) (Raguet et al. 2013),primal-dual splitting (PDS) (Condat 2013).

Select the regularization parameter u as

u D 10aU; U , ‰TrL.0/ 1

where a is an integer selected from the interval Œ�9;�1� and Uis an upper bound on u of interest.

Choose the nonnegativity convex set:

C D RpC:

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Comments II

If we remove the convex-set constraint by setting C D Rp,PNPG iteration reduces to the Nesterov’s proximal gradientiteration with adaptive step size that imposes signal sparsityonly in the analysis form (termed NPGS).

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−0.5

0 200 400 600 800 1000

RSE=0.011 %

(a) PNPG

−0.5

0 200 400 600 800 1000

RSE=0.23 %

(b) NPGS

Figure 8: PNPG and NPGS reconstructions for N=p D 0:34.

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10−8

10−6

10−4

10−2

0 10 20 30 40 50 60 70 80 90

time/s

(a) a D �5;N=p D 0:34

10−8

10−6

10−4

10−2

0 10 20 30 40 50 60 70 80 90

time/s

SPIRALSpaRSA (cont.)

PNPGPNPG (cont.)

GFBPDSAT

(b) a D �6;N=p D 0:49

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10−8

10−6

10−4

10−2

0 10 20 30 40 50 60 70 80time/s

(a) a D �4;N=p D 0:34

10−8

10−6

10−4

10−2

0 10 20 30 40 50 60 70 80time/s

PNPGPNPG (cont.)

GFBPDSAT

(b) a D �5;N=p D 0:49

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10−8

10−6

10−4

10−2

0 5 10 15 20 25 30time/s

(a) a D �3;N=p D 0:34

10−8

10−6

10−4

10−2

0 5 10 15 20time/s

PNPGPNPG (cont.)

GFBPDSAT

(b) a D �4;N=p D 0:49

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Publications

R. G. and A. D. (May 2016), Projected Nesterov’sproximal-gradient algorithm for sparse signal reconstructionwith a convex constraint, version 5. arXiv: 1502.02613[stat.CO].

R. G. and A. D., “Projected Nesterov’s proximal-gradientsignal recovery from compressive Poisson measurements,”Proc. Asilomar Conf. Signals, Syst. Comput., Pacific Grove,CA, Nov. 2015, pp. 1490–1495.

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Preliminary Work

R. G. and A. D., “A fast proximal gradient algorithm forreconstructing nonnegative signals with sparse transformcoefficients,” Proc. Asilomar Conf. Signals, Syst. Comput.,Pacific Grove, CA, Nov. 2014, pp. 1662–1667.

R. G. and A. D. (Mar. 2015), Reconstruction of nonnegativesparse signals using accelerated proximal-gradient algorithms,version 3. arXiv: 1502.02613v3 [stat.CO].

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Conclusion

Developed a fast framework for reconstructing signals that aresparse in a transform domain and belong to a closed convex setby employing a projected proximal-gradient scheme withNesterov’s acceleration, restart and adaptive step size.Applied the proposed framework to construct the firstNesterov-accelerated Poisson compressed-sensingreconstruction algorithm.Derived convergence-rate upper-bound that accounts forinexactness of the proximal operator.Proved convergence of iterates.Our PNPG approach is computationally efficient compared withthe state-of-the-art.

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References I

A. Auslender and M. Teboulle, “Interior gradient and proximal methods forconvex and conic optimization,” SIAM J. Optim., vol. 16, no. 3, pp. 697–725,2006.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm forlinear inverse problems,” SIAM J. Imag. Sci., vol. 2, no. 1, pp. 183–202, 2009.

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrainedtotal variation image denoising and deblurring problems,” IEEE Trans. ImageProcess., vol. 18, no. 11, pp. 2419–2434, 2009.

S. R. Becker, E. J. Candès, and M. C. Grant, “Templates for convex coneproblems with applications to sparse signal recovery,” Math. Program. Comp.,vol. 3, no. 3, pp. 165–218, 2011. [Online]. Available: http://cvxr.com/tfocs.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image deblurringwith Poisson data: From cells to galaxies,” Inverse Prob., vol. 25, no. 12,pp. 123006-1–123006-26, 2009.

S. Bonettini, I. Loris, F. Porta, and M. Prato, “Variable metric inexactline-search-based methods for nonsmooth optimization,” SIAM J. Optim., vol.26, no. 2, pp. 891–921, 2016.

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References II

S. Bonettini, F. Porta, and V. Ruggiero, “A variable metric forward-backwardmethod with extrapolation,” SIAM J. Sci. Comput., vol. 38, no. 4,A2558–A2584, 2016.

A. Chambolle and C. Dossal, “On the convergence of the iterates of the ‘fastiterative shrinkage/thresholding algorithm’,” J. Optim. Theory Appl., vol. 166,no. 3, pp. 968–982, 2015.

L. Condat, “A primal–dual splitting method for convex optimization involvingLipschitzian, proximable and linear composite terms,” J. Optim. Theory Appl.,vol. 158, no. 2, pp. 460–479, 2013.

J. A. Fessler, Image reconstruction toolbox, [Online]. Available:http://www.eecs.umich.edu/~fessler/code (visited on 08/23/2016).

Z. T. Harmany, R. F. Marcia, and R. M. Willett, “This is SPIRAL-TAP:Sparse Poisson intensity reconstruction algorithms—theory and practice,”IEEE Trans. Image Process., vol. 21, no. 3, pp. 1084–1096, Mar. 2012.

Y. Nesterov, “A method of solving a convex programming problem withconvergence rate O.1=k2/,” Sov. Math. Dokl., vol. 27, 1983, pp. 372–376.

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References III

B. O‘Donoghue and E. Candès, “Adaptive restart for accelerated gradientschemes,” Found. Comput. Math., vol. 15, no. 3, pp. 715–732, 2015.

J. M. Ollinger and J. A. Fessler, “Positron-emission tomography,” IEEESignal Process. Mag., vol. 14, no. 1, pp. 43–55, 1997.

B. T. Polyak, “Some methods of speeding up the convergence of iterationmethods,” USSR Comput. Math. Math. Phys., vol. 4, no. 5, pp. 1–17, 1964.

B. T. Polyak, Introduction to Optimization. New York: Optimization Software,1987.

J. L. Prince and J. M. Links, Medical Imaging Signals and Systems, 2nd ed.Upper Saddle River, NJ: Pearson, 2015.

H. Raguet, J. Fadili, and G. Peyré, “A generalized forward-backward splitting,”SIAM J. Imag. Sci., vol. 6, no. 3, pp. 1199–1226, 2013.

R. T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton Univ. Press,1970.

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References IV

S. Salzo. (May 2016), The variable metric forward-backward splittingalgorithm under mild differentiability assumptions, arXiv: 1605.00952[math.OC].

S. Villa, S. Salzo, L. Baldassarre, and A. Verri, “Accelerated and inexactforward-backward algorithms,” SIAM J. Optim., vol. 23, no. 3,pp. 1607–1633, 2013.

R. M. Willett, M. F. Duarte, M. A. Davenport, and R. G. Baraniuk, “Sparsityand structure in hyperspectral imaging: Sensing, reconstruction, and targetdetection,” IEEE Signal Process. Mag., vol. 31, no. 1, pp. 116–126, Jan. 2014.

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstructionby separable approximation,” IEEE Trans. Signal Process., vol. 57, no. 7,pp. 2479–2493, 2009.

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Adaptive Step Size

1 if no step-size backtracking events or increase attempts for nconsecutive iterations, start with a larger step size

x̌.i/ D ˇ.i�1/

�(increase attempt)

where � 2 .0; 1/ is a step-size adaptation parameter;otherwise start with

x̌.i/ D ˇ.i�1/I2 (backtracking search) select

ˇ.i/ D � ti x̌.i/ (2)

where ti � 0 is the smallest integer such that (2) satisfies themajorization condition (2); backtracking event corresponds toti > 0.

3 if max�ˇ.i/; ˇ.i�1/

�< x̌.i/, increase n by a nonnegative

integer m: n nCm: back

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Restart

Whenever f�x.i/

�> f

�x.i�1/

�or xx.i/ 2 C n domL, we set

� .i�1/ D 1 (restart)

and refer to this action as function restart (O‘Donoghue and Candès2015) or domain restart respectively.

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Inner Convergence Criteria

TV: x.i;j / � x.i;j�1/

2� �

pı.i�1/ (3a)

`1: max� s.i;j / �‰Tx.i;j /

2; s.i;j / � s.i;j�1/

��

‰T �x.i�1/ � x.i�2/� 2

where j is the inner-iteration index,x.i;j / is the iterate of x in the j th inner iteration step withinthe ith step of the (outer) PNPG iteration, and

� 2 .0; 1/is the convergence tuning constant chosen to trade off theaccuracy and speed of the inner iterations and providesufficiently accurate solutions to the proximal mapping.

63 / 71

Definition (Inexact Proximal Operator (Villa et al. 2013))

We say that x is an approximation of proxur a with "-precision,denoted by

x Ñ" proxur a

a � xu2 @ "2

Note: This definition implies

kx � proxur ak22 � "2:back

64 / 71

Relationship with FISTA I

PNPG can be thought of as a generalized fast iterativeshrinkage-thresholding algorithm (FISTA) (Beck and Teboulle2009a) that accommodates

convex constraints,more general NLLs,‘ and (increasing) adaptive step size

thanks to this step-size adaptation, PNPG does not requireLipschitz continuity of the gradient of the NLL.

‘FISTA has been developed for the linear Gaussian model.65 / 71

Relationship with FISTA II

Need B.i/ to derive theoretical guarantee for convergencespeed of the PNPG iteration.In contrast with PNPG, FISTA has a non-increasing step sizeˇ.i/, which allows for setting

B.i/ D 1for all i :‖

� .i/ D 1

q1C 4�� .i�1/�2�:

A simpler version of FISTA is

� .i/ D 1

2C � .i�1/ D i C 1

for i � 1, which corresponds to . ; b/ D .2; 0/.back

‖Y. Nesterov, “A method of solving a convex programming problem withconvergence rate O.1=k2/,” Sov. Math. Dokl., vol. 27, 1983, pp. 372–376.

66 / 71

Relationship with AT

(Auslender and Teboulle 2006):

� .i/ D 1

q1C 4�� .i�1/�2�

xx.i/ D�1 � 1

� .i/

�x.i�1/ C 1

� .i/zx.i�1/

zx.i/ D prox�.i/ˇ .i/ur

�zx.i�1/ � � .i/ˇ.i/rL�xx.i/��

x.i/ D�1 � 1

� .i/

�x.i�1/ C 1

� .i/zx.i/

Other variants with infinite memory are available at (Becker et al.2011).

67 / 71

Heavy-ball Method

(Polyak 1964; Polyak 1987):

x.i/ D proxˇ .i/ur

�x.i�1/ � ˇ.i/rL�x.i�1/��C‚.i/�x.i�1/ � x.i�2/�:

68 / 71

Glossary I

ADMM alternating direction method of multipliers. 21

BB Barzilai-Borwein. 37

CPU central processing unit. 45

CT computed tomography. 6

FBP filtered backprojection. 40

FISTA fast iterative shrinkage-thresholding algorithm. 66, 67

GFB generalized forward-backward. 49

GLM generalized linear model. 17

i.i.d. independent, identically distributed. 47

IRT Image Reconstruction Toolbox. 38

MRI magnetic resonance imaging. 6

70 / 71

Glossary IINLL negative log-likelihood. 10, 11, 26, 27, 31, 32, 37, 66

NPGS Nesterov’s proximal-gradient sparse. 50, 51PDS primal-dual splitting. 49PET positron emission tomography. 6, 35, 38PG proximal-gradient. 9, 16, 22, 23, 28, 40

PNPG projected Nesterov’s proximal-gradient. 23, 24, 27, 31,32, 41, 51, 57, 64, 66, 67

RSE relative square error. 39SpaRSA sparse reconstruction by separable approximation. 49SPECT single photon emission computed tomography. 6SPIRAL sparse Poisson-intensity reconstruction algorithm. 37,

41TFOCS templates for first-order conic solvers. 41

TV total-variation. 2171 / 71

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