For first-order optimization of non-convex functions with Lipschitz-continuous gradient and Hessian, the best-known complexity for reaching an ε-approximation of a stationary point is 𝒪(ε^-7/4). The existing algorithms achieving this bound are based on momentum, but are always complemented with safeguard mechanisms that erase the accumulated momentum if a certain condition is violated. Whether such momentum-control mechanisms are fundamentally necessary has remained an open question. We show that randomizing the parameters enables one to achieve this complexity in expectation when using momentum without any of such mechanisms, and we improve the numerical constant factor of the bound in the case of a large enough number of iterations. From an analysis perspective, we do so by leveraging the continuized method, which interprets the algorithm as a realization of a continuous-time stochastic differential equation (SDE) involving a Poisson process. We show that this SDE converges in probability to the Heavy Ball ordinary differential equation when the stepsize goes to zero, paralleling the behavior of more classical instances of Nesterov momentum.

This paper introduces novel theoretical approximation bounds for the output of quantized neural networks, with a focus on convolutional neural networks (CNN). By considering layerwise parametrization and focusing on the quantization of weights, we provide bounds that gain several orders of magnitude compared to state-of-the-art results on classical deep convolutional neural networks such as MobileNetV2 or ResNets. These gains are achieved by improving the behaviour of the approximation bounds with respect to the depth parameter, which has the most impact on the approximation error induced by quantization. To complement our theoretical result, we provide a numerical exploration of our bounds on MobileNetV2 and ResNets.

We introduce a new adaptive step-size strategy for convex optimization with stochastic gradient that exploits the local geometry of the objective function only by means of a first-order stochastic oracle and without any hyper-parameter tuning. The method comes from a theoretically-grounded adaptation of the Adaptive Gradient Descent Without Descent method to the stochastic setting. We prove the convergence of stochastic gradient descent with our step-size under various assumptions, and we show that it empirically competes against tuned baselines.

Empirically, it has been observed that adding momentum to Stochastic Gradient Descent (SGD) accelerates the convergence of the algorithm.However, the literature has been rather pessimistic, even in the case of convex functions, about the possibility of theoretically proving this observation.We investigate the possibility of obtaining accelerated convergence of the Stochastic Nesterov Accelerated Gradient (SNAG), a momentum-based version of SGD, when minimizing a sum of functions in a convex setting. We demonstrate that the average correlation between gradients allows to verify the strong growth condition, which is the key ingredient to obtain acceleration with SNAG.Numerical experiments, both in linear regression and deep neural network optimization, confirm in practice our theoretical results.

To address the problem of separating images into a structure and a texture component, we introduce a joint structure-texture model. Instead of considering two separate regularizations for each component, we consider a joint structure-texture model regularization function that takes both components as inputs. This allows for the regularization to take into account the shared information between the two components. We present evidence that shows a performance gain compared to separate regularization models. To implement the joint regularization, we adapt the plug-and-play framework to our setting, using deep neural networks. We train the corresponding deep prior on a randomly generated synthetic dataset of examples of this model. In the context of image decomposition, we show that while trained on synthetic datasets, our plug-and-play method generalizes well to natural images. Furthermore, we show that this framework permits us to leverage the structure-texture decompositions to solve inverse imaging problems such as inpainting.

In this paper, we propose a restart scheme for FISTA (Fast Iterative Shrinking-Threshold Algorithm) [6]. This method which is a generalisation of Nesterov’s accelerated gradient algorithm [23] is widely used in the field of large-scale convex optimization problems as it ensures a quadratic decrease o( 1/k^2) of the error for convex functions [3, 12]. When considering a function that satisfies stronger assumptions such as strong convexity or quadratic growth, several methods provide faster convergence rates by taking advantage of this geometry property, including FISTA restart schemes. In particular, the schemes that provide the fastest theoretical convergence rates rely on the growth parameter of the function, which is generally difficult to estimate. Recent works [1, 2] show that restarting FISTA can ensure a fast convergence for functions having a quadratic growth without requiring any knowledge on the growth parameter. We improve these restart schemes by providing a better asymptotical convergence rate and by requiring a lower computational cost. We illustrate our theoretical results with some numerical examples.

We consider the problem of recovering an unknown low-dimensional vector from noisy, underdetermined observations. We focus on the Generalized Projected Gradient Descent (GPGD) framework, which unifies traditional sparse recovery methods and modern approaches using learned deep projective priors. We extend previous convergence results to robustness to model and projection errors. We use these theoretical results to explore ways to better control stability and robustness constants. To reduce recovery errors due to measurement noise, we consider generalized back-projection strategies to adapt GPGD to structured noise, such as sparse outliers. To improve the stability of GPGD, we propose a normalized idempotent regularization for the learning of deep projective priors. We provide numerical experiments in the context of sparse recovery and image inverse problems, highlighting the trade-offs between identifiability and stability that can be achieved with such methods.

In convex optimization, continuous-time counterparts have been a fruitful tool for analyzing momentum algorithms. Fewer such examples are available when the function to minimize is non-convex. In several cases, discrepancies arise between the existing discrete-time results, namely those obtained for momentum algorithms, and their continuous-time counterparts, with the latter typically yielding stronger guarantees. We argue that the continuized framework (Even et al., 2021), mixing continuous and discrete components, can tighten the gap between known continuous and discrete results. This framework relies on computations akin to standard Lyapunov analyses, from which are deduced convergence bounds for an algorithm that can be written as a Nesterov momentum algorithm with stochastic parameters. In this work, we extend the range of applicability of the continuized framework, e.g. by allowing it to handle non-smooth Lyapunov functions. We then strengthen its trajectory-wise guarantees for linear convergence rate, deriving finite time bounds with high probability and asymptotic almost sure bounds. We apply this framework to the non-convex class of strongly quasar convex functions. Adapting continuous-time results that have weaker discrete equivalents to the continuized method, we improve by a constant factor the known convergence rate, and relax the existing assumptions on the set of minimizers.

In this work, we propose a parameter-free and efficient method to tackle the structure-texture image decomposition problem. In particular, we present a neural network LPR-NET based on the unrolling of the Low Patch Rank model. On the one hand, this allows us to automatically learn parameters from data, and on the other hand to be computationally faster while obtaining qualitatively similar results compared to traditional iterative model-based methods. Moreover, despite being trained on synthetic images, numerical experiments show the ability of our network to generalize well when applied to natural images.

We consider a combined restarting and adaptive backtracking strategy for the popular Fast Iterative Shrinking-Thresholding Algorithm frequently employed for accelerating the convergence speed of large-scale structured convex optimization problems. Several variants of FISTA enjoy a provable linear convergence rate for the function values $F(x_n)$ of the form $\mathcal{O}( e^{-K\sqrt{\mu/L}~n})$ under the prior knowledge of problem conditioning, i.e. of the ratio between the (\L ojasiewicz) parameter $\mu$ determining the growth of the objective function and the Lipschitz constant $L$ of its smooth component. These parameters are nonetheless hard to estimate in many practical cases. Recent works address the problem by estimating either parameter via suitable adaptive strategies. In our work both parameters can be estimated at the same time by means of an algorithmic restarting scheme where, at each restart, a non-monotone estimation of $L$ is performed. For this scheme, theoretical convergence results are proved, showing that a $\mathcal{O}( e^{-K\sqrt{\mu/L}n})$ convergence speed can still be achieved along with quantitative estimates of the conditioning. The resulting Free-FISTA algorithm is therefore parameter-free. Several numerical results are reported to confirm the practical interest of its use in many exemplar problems.

In this work, we are interested in gradient sparse plus low patch-rank signal recovery for image structure-texture decomposition, the structure being gradient-sparse and the texture low patch-rank. Based upon theoretical results of sparse plus low-rank matrix recovery, we propose an algorithm to automatically tune the regularization parameters of our model depending on the content of the image. This permits to provide an improved localized version of gradient sparse plus low patch- rank decomposition. This algorithm is validated by experiments on synthetic and real images.

We consider the problem of recovering off-the-grid spikes from linear measurements. The state of the art Over-Parametrized Continuous Orthogonal Matching Pursuit (OP-COMP) with Projected Gradient Descent (PGD) successfully recovers those signals. In most cases, the main computational cost lies in a unique global descent on all parameters (positions and amplitudes). In this paper, we propose to improve this algorithm by accelerating this descent step. We introduce a new algorithm, based on Block Coordinate Descent, that takes advantages of the sparse structure of the problem. Based on qualitative theoretical results, this algorithm shows improvement in calculation times in realistic synthetic microscopy experiments.

Single Image Super Resolution (SISR) methods aim to recover the clean images in high resolution from low resolution observations.A family of patch-based approaches have received considerable attention and development. The minimum mean square error (MMSE) methodis a powerful image restoration method that uses a probability model on the patches of images. This paper proposes an algorithm to learn a jointgeneralized Gaussian mixture model (GGMM) from a pair of the low resolution patches and the corresponding high resolution patches fromthe reference data. We then reconstruct the high resolution image based on the MMSE method. Our numerical evaluations indicate that theMMSE-GGMM method competes with other state of the art methods.

Despite the rapid development of computational hardware, the treatment of large and high dimensional data sets is still a challenging problem. The contribution of this paper to the topic is twofold. First, we propose a Gaussian mixture model in conjunction with a reduction of the dimensionality of the data in each component of the model by principal component analysis, which we call PCA-GMM. To learn the (low dimensional) parameters of the mixture model we propose an EM algorithm whose M-step requires the solution of constrained optimization problems. Fortunately, these constrained problems do not depend on the usually large number of samples and can be solved efficiently by an (inertial) proximal alternating linearized minimization algorithm. Second, we apply our PCA-GMM for the superresolution of 2D and 3D material images based on the approach of Sandeep and Jacob. Numerical results confirm the moderate influence of the dimensionality reduction on the overall superresolution result.

The pansharpening problem amounts to fusing a high-resolution panchromatic image with a low-resolution multispectral image so as to obtain a high-resolution multispectral image. Therefore, the preservation of the spatial resolution of the panchromatic image and the spectral resolution of the multispectral image is of key importance for the pansharpening problem. To cope with it, we propose a new method based on a bidiscriminator in a generative adversarial network (GAN) framework. The first discriminator is optimized to preserve textures of images by taking as input the luminance and the near-infrared band of images, and the second discriminator preserves the color by comparing the chroma components Cb and Cr. Thus, this method allows to train two discriminators, each one with a different and complementary task. Moreover, to enhance these aspects, the proposed method based on bidiscriminator, and called MDSSC-GAN SAM, considers a spatial and a spectral constraint in the loss function of the generator. We show the advantages of this new method on experiments carried out on Pléiades and World View 3 satellite images.

The expected patch log-likelihood algorithm (EPLL) and its extensions have shown good perfor-mances for image denoising. The prior model used by EPLL is usually a Gaussian mixture model (GMM) estimated from a database of image patches. Classical mixture model estimation meth-ods face computational issues as the high dimensionality of the problem requires training on large datasets. In this work, we adapt a compressive statistical learning framework to carry out the GMM estimation. With this method, called sketching, we estimate models from a compressive representa-tion (the sketch) of the training patches. The cost of estimating the prior from the sketch no longer depends on the number of items in the original large database. To accelerate further the estimation, we add another dimension reduction technique (low-rank modeling of the covariance matrices) to the compressing learning framework. To demonstrate the advantages of our method, we test it on real large-scale data. We show that we can produce denoising performances similar to performances obtained with models estimated from the original training database using GMM priors learned from the sketch with improved execution times.

We consider the problem of recovering off-the-grid spikes from Fourier measurements. Successful methods such as sliding Frank-Wolfe and continuous orthogonal matching pursuit (OMP) iteratively add spikes to the solution then perform a costly (when the number of spikes is large) descent on all parameters at each iteration. In 2D, it was shown that performing a projected gradient descent (PGD) from a gridded over-parametrized initialization was faster than continuous orthogonal matching pursuit. In this paper, we propose an off-the-grid over-parametrized initialization of the PGD based on OMP that permits to fully avoid grids and gives faster results in 3D.

We propose to detect defects in old movies, as the first step of a larger framework of old movies restoration by inpainting techniques. The specificity of our work is to learn a film restorer's expertise from a pair of sequences, composed of a movie with defects, and the same movie which was semi-automatically restored with the help of a specialized software. In order to detect those defects with minimal human interaction and further reduce the time spent for a restoration, we feed a U-Net with consecutive defective frames as input to detect the unexpected variations of pixel intensity over space and time. Since the output of the network is a mask of defect location, we first have to create the dataset of mask frames on the basis of restored frames from the software used by the film restorer, instead of classical synthetic ground truth, which is not available. These masks are estimated by computing the absolute difference between restored frames and defectuous frames, combined with thresholding and morphological closing. Our network succeeds in automatically detecting real defects with more precision than the manual selection with an all-encompassing shape, including some the expert restorer could have missed for lack of time.