Maximal monotone operator theory is about to turn (or just has turned) fifty. I intend to briefly survey the history of the subject. I shall try to explain why maximal monotone operators are both interesting and important—culminating with a description of the remarkable progress made during the past decade. 1

In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenable to solution by current methods but they feature Fenchel-Moreau-Rockafellar dual problems that can be solved by forward-backward splitting. The proposed algorithm produces simultaneously a sequence converging weakly to a dual solution, and a sequence converging strongly to the primal solution. Our framework is shown to capture and extend several existing duality-based signal recovery methods and to be applicable to a variety of new problems beyond their scope.

Abstract—We propose and assess the performance of new imaging techniques for radio interferometry that rely on the versatility of the compressed sensing framework to account for prior information on the signals. The present manuscript represents a summary of recent work [1]. I. RADIO INTERFEROMETRY Visibility measurement: Radio interferometry is a powerful technique for aperture synthesis in astronomy [2]. Thanks to interferometric techniques, radio telescope arrays synthesize the aperture of a unique telescope of the same size as the maximum projected baseline, i.e. the maximum projected distance between two telescopes on the plane perpendicular to the pointing direction of the instrument. The portion of the celestial sphere accessible to the instrument around the pointing direction tracked during observation defines the original signal or image to be recovered. We consider a standard interferometer with a so-called illumination function limiting the field of view to a small and finite patch of the celestial sphere identified to a planar patch: P ⊂ R 2. The signal and the illumination function thus respectively appear as functions I(⃗p) and A(⃗p) of a vector ⃗p ∈ P with an origin at the pointing direction of the array. At each instant of observation, each telescope pair identified by an index b measures a complex visibility yb ∈ C corresponding to the value of the Fourier transform of the image multiplied by the illumination function AI at a single spatial frequency ⃗ub. One has yb = ÂI (⃗ub) with ⃗ub = ⃗ B ⊥ b

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Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the Shannon-Nyquist theory requires. Many images can be sparsely approximated in expansions of suitable frames as wavelets, curvelets, wave atoms and others. Generally, wavelets represent point-like features while curvelets represent line-like features well. For a suitable recovery of images, we propose models that contain weighted sparsity constraints in two different frames. Given the incomplete measurements f = Φu + ɛ with the measurement matrix Φ ∈ R K×N, K<<N, we consider a jointly sparsity-constrained optimization problem of the form argmin{‖ΛcΨcu‖1 + ‖ΛwΨwu‖1 + u 1 2‖f − Φu‖22}. Here Ψcand Ψw are the transform matrices corresponding to the two frames, and the diagonal matrices Λc, Λw contain the weights for the frame coefficients. We present efficient iteration methods to solve the optimization problem, based on Alternating Split Bregman algorithms. The convergence of the proposed iteration schemes will be proved by showing that they can be understood as special cases of the Douglas-Rachford Split algorithm. Numerical experiments for compressed sensing based Fourier-domain random imaging show good performances of the proposed curvelet-wavelet regularized split Bregman (CWSpB) methods,whereweparticularlyuseacombination of wavelet and curvelet coefficients as sparsity constraints.

This short note studies a variation of the Compressed Sensing paradigm introduced recently by Vaswani et al., i.e. the recovery of sparse signals from a certain number of linear measurements when the signal support is partially known. The reconstruction method is based on a convex minimization program coined innovative Basis Pursuit DeNoise (or i BPDN). Under the common ℓ2-fidelity constraint made on the available measurements, this optimization promotes the (ℓ1) sparsity of the candidate signal over the complement of this known part. In particular, this paper extends the results of Vaswani et al. to the cases of compressible signals and noisy measurements. Our proof relies on a small adaption of the results of Candes in 2008 for characterizing the stability of the Basis Pursuit DeNoise (BPDN) program. We emphasize also an interesting link between our method and the recent work of Davenport et al. on the δ-stable embeddings and the cancel-then-recover strategy applied to our problem. For both approaches, reconstructions are indeed stabilized when the sensing matrix respects the Restricted Isometry Property for the same sparsity order. We conclude by sketching an easy numerical method relying on monotone operator splitting and proximal methods that iteratively solves i BPDN.

During the last five years, several convex optimization algorithms have been proposed for solving inverse problems. Most of the time, they allow us to minimize a criterion composed of two terms one of which permits to “stabilize ” the solution. Different choices are possible for the so-called regularization term, which plays a prominent role for solving ill-posed problems. While a total variation regularization introduces staircase effects, a wavelet regularization may bring other kinds of visual artefacts. A compromise can be envisaged combining these regularization functions. In the context of Poisson data, we propose in this paper an algorithm to achieve the minimization of the associated (possibly constrained) convex optimization problem. 1.

We address the problem of learning classifiers using several kernel functions. On the contrary to many contributions in the field of learning from different sources of information using kernels, we here do not assume that the kernels used are positive definite. The learning problem that we are interested in involves a misclassification loss term and a regularization term that is expressed by means of a mixed norm. The use of a mixed norm allows us to enforce some sparsity structure, a particular case of which is, for instance, the Group Lasso. We solve the convex problem by employing proximal minimization algorithms, which can be viewed as refined versions of gradient descent procedures capable of naturally dealing with nondifferentiability. A numerical simulation on a UCI dataset shows the modularity of our approach. 1.

maximal monotone operators