In order to reduce the quadratic cost of matrix-vector multiplications in dense and attention layers, Monarch matrices have been recently introduced, achieving a sub-quadratic complexity. It consists in factorizing a matrix using fixed permutations and learned block diagonal matrices, at the price of a small performance drop. We propose a more general model where some permutations are learned. The optimization algorithm explores the space of permutations using a Straight-Through Estimator (STE) inspired by the support exploration algorithm designed for sparse support recovery. Our experimental results demonstrate performance improvement in the context of sparse matrix factorization and of end-to-end sparse learning.

Weighted least squares are often used to approximate log-likelihoods when solving inverse problems involving non-Gaussian noise as they are more appealing from an optimization perspective. Although a theoretical expression of the weights can be derived for specific noises, this may become intractable for more general noises. Moreover, such theoretical weights can be detrimental to the efficiency of optimization algorithms. To remedy these issues, we propose in this work to learn the weights from data so as to adapt to any general noise while maintaining the efficiency of optimization. The proposed pipeline combines a weight estimation module with an unrolled optimization algorithm. The weight estimation module and a few parameters of the unrolled algorithm are trained together in an end-to-end manner. We demonstrate the effectiveness of the proposed methodology in the context of Poisson image deconvolution.

The *straight-through estimator* (STE) is commonly used to optimize quantized neural networks, yet its contexts of effective performance are still unclear despite empirical successes. To make a step forward in this comprehension, we apply STE to a well-understood problem: *sparse support recovery*. We introduce the *Support Exploration Algorithm* (SEA), a novel algorithm promoting sparsity, and we analyze its performance in support recovery (a.k.a. model selection) problems. SEA explores more supports than the state-of-the-art, leading to superior performance in experiments, especially when the columns of $A$ are strongly coherent. The theoretical analysis considers recovery guarantees when the linear measurements matrix $A$ satisfies the *Restricted Isometry Property* (RIP). The sufficient conditions of recovery are comparable but more stringent than those of the state-of-the-art in sparse support recovery. Their significance lies mainly in their applicability to an instance of the STE.

Hyper-parameter tuning, and especially regularisation parameter estimation, is a challenging but essential task when solving inverse problems. The solution is obtained here through the minimization of a functional composed of a data fidelity term and a regularization term. Those terms are balanced through a (or several) regularisation parameter(s) whose estimation is made under an unrolled strategy together with the inverse problem solving. The resulting network is trained while incorporating information on the model through Maximum a Posteriori estimation which drastically decreases the amount of data needed for the training and results in better estimation results. The performances are demonstrated in a deconvolution context where the regularisation is performed in the wavelet domain.

In Nuclear Magnetic Resonance (NMR) spectroscopy, an efficient analysis and a relevant extraction of different molecule properties from a given chemical mixture are important tasks, especially when processing bidimensional NMR data. To that end, using a blind source separation approach based on a variational formulation seems to be a good strategy. However, the poor resolution of NMR spectra and their large dimension require a new and modern blind source separation method. In this work, we propose a new variational formulation for blind source separation (BSS) based on a β-divergence data fidelity term combined with sparsity promoting regularization functions. An application to 2D HSQC NMR experiments illustrates the interest and the effectiveness of the proposed method whether in simulated or real cases.

We address the problem of filtering out localized time-frequency components in signals. The problem is formulated as a minimization of a suitable quadratic form, that involves a data fidelity term on the short-time Fourier transform outside the support of the undesired component, and an energy penalization term inside the support. The minimization yields a linear system whose solution can be expressed in closed form using Gabor multipliers. We provide an analysis of the solution and its approximation by truncated eigenvalue expansion, and illustrate its performance on synthetic mixtures of audio signals. The proposed approach outperforms approaches that are routinely used in applications.

When using optimization methods with matrix variables in signal processing and machine learning, it is customary to assume some low-rank prior on the targeted solution. Nonnegative matrix factorization of spectrograms is a case in point in audio signal processing. However, this low-rank prior is not straightforwardly related to complex matrices obtained from a short-time Fourier - or discrete Gabor - transform (STFT), which is generally defined from and studied based on a modulation operator and a translation operator applied to a so-called window. This paper is a first study of the low-rankness property of time-frequency matrices. We characterize the set of signals with a rank- r (complex) STFT matrix in the case of a unit hop size and frequency step with few assumptions on the transform parameters. We discuss the scope of this result and its implications on low-rank approximations of STFT matrices.

In the field of 3D image recovery, huge amounts of data need to be processed. Parallel optimization methods are then of main interest since they allow to overcome memory limitation issues, while benefiting from the intrinsic acceleration provided by recent multicore computing architectures. In this context, we propose a Block Parallel Majorize-Minimize Memory Gradient (BP3MG) algorithm for solving large scale optimization problems. This algorithm combines a block coordinate strategy with an efficient parallel update. The proposed method is applied to a 3D microscopy image restoration problem involving a depth-variant blur, where it is shown to lead to significant computational time savings with respect to a sequential approach.

This paper proposes a novel approach for echo-like multiple removal in two-dimensional seismic images. It is based on constrained adaptive filtering associated with geometric wavelets. Approximate templates of multiple reflections are assumed to be available and they are matched to multiple reflections throughout estimated finite impulse response filters. The problem is formulated under a constrained convex optimization form where the data of interest and filters are estimated jointly. Proximal approaches are used to perform the minimization of the derived criterion. The effectiveness of the proposed approach is demonstrated with various noise levels on realistic simulated data and on field seismic data.

In this letter, the problem of nonnegative tensor decompositions is addressed. Classically, this problem is carried out using iterative (either alternating or global) deterministic optimization algorithms. Here, a rather different stochastic approach is suggested. In addition, the ever-increasing volume of data requires the development of new and more efficient approaches to be able to process “Big data” tensors to extract relevant information. The stochastic algorithm outlined here comes within this framework. Both flexible and easy to implement, it is designed to solve the problem of the CP (Candecomp/Parafac) decomposition of huge nonnegative 3-way tensors while simultaneously enabling to handle possible missing data.

Random and structured noise both affect seismic data, hiding the reflections of interest (primaries) that carry meaningful geophysical interpretation. When the structured noise is composed of multiple reflections, its adaptive cancellation is obtained through time-varying filtering, compensating inaccuracies in given approximate templates. The under-determined problem can then be formulated as a convex optimization one, providing estimates of both filters and primaries. Within this framework, the criterion to be minimized mainly consists of two parts: a data fidelity term and hard constraints modeling a priori information. This formulation may avoid, or at least facilitate, some parameter determination tasks, usually difficult to perform in inverse problems. Not only classical constraints, such as sparsity, are considered here, but also constraints expressed through hyperplanes, onto which the projection is easy to compute. The latter constraints lead to improved performance by further constraining the space of geophysically sound solutions.

This paper deals with noise parameter estimation from a single image under Poisson-Gaussian noise statistics. The problem is formulated within a mixed discrete-continuous optimization framework. The proposed approach jointly estimates the signal of interest and the noise parameters. This is achieved by introducing an adjustable regularization term inside an optimized criterion, together with a data fidelity error measure. The optimal solution is sought iteratively by alternating the minimization of a label field and of a noise parameter vector. Noise parameters are updated at each iteration using an Expectation-Maximization approach. The proposed algorithm is inspired from a spatial regularization approach for vector quantization. We illustrate the usefulness of our approach on macroconfocal images. The identified noise parameters are applied to a denoising algorithm, so yielding a complete denoising scheme.

In this paper we are interested in regularizing hyperparameter estimation by maximum likelihood in inverse problems with wavelet regularization. One parameter per subband will be estimated by gradient ascent algorithm. We have to face with two main difficulties: i) sampling the a posteriori image distribution to compute the gradient; ii) choosing a suited step-size to ensure good convergence properties. We first show that introducing an auxiliary variable makes the sampling feasible using classical Metropolis-Hastings algorithm and Gibbs sampler. Secondly, we propose an adaptive step-size selection and a line-search strategy to improve the gradient-based method. Good performances of the proposed approach are demonstrated on both synthetic and real data.

Both random and structured perturbations affect seismic data. Their removal, to unveil meaningful geophysical information, requires additional priors. Seismic multiples are one form of structured perturbations related to wave-field bouncing. In this paper, we model these undesired signals through a time-varying filtering process accounting for inaccuracies in amplitude, time-shift and average frequency of available templates. We recast the problem of jointly estimating the filters and the signal of interest (primary) in a new convex variational formulation, allowing the incorporation of knowledge about the noise statistics. By making some physically plausible assumptions about the slow time variations of the filters, and by adopting a potential promoting the sparsity of the primary in a wavelet frame, we design a primal-dual algorithm which yields good performance in the provided simulation examples.

Regularization approaches have demonstrated their effectiveness for solving ill-posed problems. However, in the context of variational restoration methods, a challenging question remains, namely how to find a good regularizer. While total variation introduces staircase effects, wavelet-domain regularization brings other artefacts, e.g., ringing. However, a tradeoff can be made by introducing a hybrid regularization including several terms not necessarily acting in the same domain (e.g., spatial and wavelet transform domains). While this approach was shown to provide good results for solving deconvolution problems in the presence of additive Gaussian noise, an important issue is to efficiently deal with this hybrid regularization for more general noise models. To solve this problem, we adopt a convex optimization framework where the criterion to be minimized is split in the sum of more than two terms. For spatial domain regularization, isotropic or anisotropic total variation definitions using various gradient filters are considered. An accelerated version of the Parallel Proximal Algorithm is proposed to perform the minimization. Some difficulties in the computation of the proximity operators involved in this algorithm are also addressed in this paper. Numerical experiments performed in the context of Poisson data recovery, show the good behavior of the algorithm as well as promising results concerning the use of hybrid regularization techniques.

Quantization, defined as the act of attributing a finite number of grey-levels to an image, is an essential task in image acquisition and coding. It is also intricately linked to various image analysis tasks, such as denoising and segmentation. In this paper, we investigate quantization combined with regularity constraints, a little-studied area which is of interest, in particular, when quantizing in the presence of noise or other acquisition artifacts. We present an optimization approach to the problem involving a novel two-step, iterative, flexible, joint quantizing-regularization method featuring both convex and combinatorial optimization techniques. We show that when using a small number of grey-levels, our approach can yield better quality images in terms of SNR, with lower entropy, than conventional optimal quantization methods.

Iterative optimization algorithms such as the forward-backward and Douglas-Rachford algorithms have gained much popularity since they provide efficient solutions to a wide class of non-smooth convex minimization problems arising in signal/image recovery. However, when images are degraded by a convolution operator and a Poisson noise, a particular attention must be paid to the associated minimization problem. To solve it, we propose a new optimization method which consists of two nested iterative steps. The effectiveness of the proposed method is demonstrated via numerical comparisons.

The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be nonsmooth and g is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas–Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of g is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively.

Dual-tree wavelet transforms have recently gained popularity since they provide low-redundancy directional analyses of images. In our recent work, dyadic real dual-tree decompositions have been extended to the M-band case, so adding much flexibility to this analysis tool. In this work, we propose to further extend this framework on two fronts by considering (i) biorthogonal and (ii) complex M-band dual-tree decompositions. Denoising results are finally provided to demonstrate the validity of the proposed design rules.

Dual-tree wavelet decompositions have recently gained much popularity, mainly due to their ability to provide an accurate directional analysis of images combined with a reduced redundancy. When the decomposition of a random process is performed-which occurs in particular when an additive noise is corrupting the signal to be analyzed-it is useful to characterize the statistical properties of the dual-tree wavelet coefficients of this process. As dual-tree decompositions constitute over-complete frame expansions, correlation structures are introduced among the coefficients, even when a white noise is analyzed. In this paper, we show that it is possible to provide an accurate description of the covariance properties of the dual-tree coefficients of a wide-sense-stationary process. The expressions of the (cross-) covariance sequences of the coefficients are derived in the one- and two-dimensional cases. Asymptotic results are also provided, allowing to predict the behavior of the second-order moments for large lag values or at coarse resolution. In addition, the cross-correlations between the primal and dual wavelets, which play a primary role in our theoretical analysis, are calculated for a number of classical wavelet families. Simulation results are finally provided to validate these results.

We propose a new estimator for image denoising using a 2D dual-tree M-band wavelet transform. Our work extends existing block-based wavelet thresholding methods by exploiting simultaneously coefficients in the two M-band wavelet trees. The contributions of this paper are two-fold. Firstly, we perform a statistical analysis of the noise in the considered redundant decomposition. Secondly, we propose an efficient method to remove the noise. Our approach relies on an extension of Stein's formula which allows us to take into account the specific correlations of the noise components. Simulation results are then presented to validate the proposed method

We propose a two-dimensional generalization to the M-band case of the dual-tree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the resulting directional analysis. We also revisit the necessary pre-processing stage in the M-band case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposed M-band decomposition is demonstrated via denoising comparisons on several image types (natural, texture, seismics), with various M-band wavelets and thresholding strategies. Significant improvements in terms of both overall noise reduction and direction preservation are observed.

We propose a 2D generalization to the M-band case of the dual-tree structure (initially proposed by N.G. Kingsbury and further investigated by I.W. Selesnick) based on a Hilbert pair of wavelets. We particularly address the construction of the dual basis and the resulting directional analysis. We revisit the necessary preprocessing stage in the M-band case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposed M-band decomposition is demonstrated via image denoising comparisons.