Controlled Learning of Pointwise Nonlinearities in Neural-Network-Like Architectures

We present a general variational framework for the training of freeform nonlinearities in layered computational architectures subject to some slope constraints. The regularization that we add to the traditional training loss penalizes the second-order total variation of each trainable activation. The slope constraints allow us to impose properties such as 1-Lipschitz stability, firm non-expansiveness, and monotonicity/invertibility. These properties are crucial to ensure the proper functioning of certain classes of signal-processing algorithms (e.g., plug-and-play schemes, unrolled proximal gradient, invertible flows). We prove that the global optimum of the stated constrained-optimization problem is achieved with nonlinearities that are adaptive nonuniform linear splines. We then show how to solve the resulting function-optimization problem numerically by representing the nonlinearities in a suitable (nonuniform) B-spline basis. Finally, we illustrate the use of our framework with the data-driven design of (weakly) convex regularizers for the denoising of images and the resolution of inverse problems.

Parseval Convolution Operators and Neural Networks

We first establish a kernel theorem that characterizes all linear shift-invariant (LSI) operators acting on discrete multicomponent signals. This result naturally leads to the identification of the Parseval convolution operators as the class of energy-preserving filterbanks. We then present a constructive approach for the design/specification of such filterbanks via the chaining of elementary Parseval modules, each of which being parameterized by an orthogonal matrix or a 1-tight frame. Our analysis is complemented with explicit formulas for the Lipschitz constant of all the components of a convolutional neural network (CNN), which gives us a handle on their stability. Finally, we demonstrate the usage of those tools with the design of a CNN-based algorithm for the iterative reconstruction of biomedical images. Our algorithm falls within the plug-and-play framework for the resolution of inverse problems. It yields better-quality results than the sparsity-based methods used in compressed sensing, while offering essentially the same convergence and robustness guarantees.

Self-Calibrated Variance-Stabilizing Transformations for Real-World Image Denoising

Supervised deep learning has become the method of choice for image denoising. It involves the training of neural networks on large datasets composed of pairs of noisy and clean images. However, the necessity of training data that are specific to the targeted application constrains the widespread use of denoising networks. Recently, several approaches have been developed to overcome this difficulty by whether artificially generating realistic clean/noisy image pairs, or training exclusively on noisy images. In this paper, we show that, contrary to popular belief, denoising networks specialized in the removal of Gaussian noise can be efficiently leveraged in favor of real-world image denoising, even without additional training. For this to happen, an appropriate variance-stabilizing transform (VST) has to be applied beforehand. We propose an algorithm termed Noise2VST for the learning of such a model-free VST. Our approach requires only the input noisy image and an off-the-shelf Gaussian denoiser. We demonstrate through extensive experiments the efficiency and superiority of Noise2VST in comparison to existing methods trained in the absence of specific clean/noisy pairs.

Iteratively Refined Image Reconstruction with Learned Attentive Regularizers

We propose a regularization scheme for image reconstruction that leverages the power of deep learning while hinging on classic sparsity-promoting models. Many deep-learning-based models are hard to interpret and cumbersome to analyze theoretically. In contrast, our scheme is interpretable because it corresponds to the minimization of a series of convex problems. For each problem in the series, a mask is generated based on the previous solution to refine the regularization strength spatially. In this way, the model becomes progressively attentive to the image structure. For the underlying update operator, we prove the existence of a fixed point. As a special case, we investigate a mask generator for which the fixed-point iterations converge to a critical point of an explicit energy functional. In our experiments, we match the performance of state-of-the-art learned variational models for the solution of inverse problems. Additionally, we offer a promising balance between interpretability, theoretical guarantees, reliability, and performance.

Random ReLU Neural Networks as Non-Gaussian Processes

We consider a large class of shallow neural networks with randomly initialized parameters and rectified linear unit activation functions. We prove that these random neural networks are well-defined non-Gaussian processes. As a by-product, we demonstrate that these networks are solutions to stochastic differential equations driven by impulsive white noise (combinations of random Dirac measures). These processes are parameterized by the law of the weights and biases as well as the density of activation thresholds in each bounded region of the input domain. We prove that these processes are isotropic and wide-sense self-similar with Hurst exponent $3/2$. We also derive a remarkably simple closed-form expression for their autocovariance function. Our results are fundamentally different from prior work in that we consider a non-asymptotic viewpoint: The number of neurons in each bounded region of the input domain (i.e., the width) is itself a random variable with a Poisson law with mean proportional to the density parameter. Finally, we show that, under suitable hypotheses, as the expected width tends to infinity, these processes can converge in law not only to Gaussian processes, but also to non-Gaussian processes depending on the law of the weights. Our asymptotic results provide a new take on several classical results (wide networks converge to Gaussian processes) as well as some new ones (wide networks can converge to non-Gaussian processes).

Banach Space Optimality of Neural Architectures With Multivariate Nonlinearities

We investigate the variational optimality (specifically, the Banach space optimality) of a large class of neural architectures with multivariate nonlinearities/activation functions. To that end, we construct a new family of Banach spaces defined via a regularization operator and the $k$-plane transform. We prove a representer theorem that states that the solution sets to learning problems posed over these Banach spaces are completely characterized by neural architectures with multivariate nonlinearities. These optimal architectures have skip connections and are tightly connected to orthogonal weight normalization and multi-index models, both of which have received considerable interest in the neural network community. Our framework is compatible with a number of classical nonlinearities including the rectified linear unit (ReLU) activation function, the norm activation function, and the radial basis functions found in the theory of thin-plate/polyharmonic splines. We also show that the underlying spaces are special instances of reproducing kernel Banach spaces and variation spaces. Our results shed light on the regularity of functions learned by neural networks trained on data, particularly with multivariate nonlinearities, and provide new theoretical motivation for several architectural choices found in practice.

Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms

We propose to learn non-convex regularizers with a prescribed upper bound on their weak-convexity modulus. Such regularizers give rise to variational denoisers that minimize a convex energy. They rely on few parameters (less than 15,000) and offer a signal-processing interpretation as they mimic handcrafted sparsity-promoting regularizers. Through numerical experiments, we show that such denoisers outperform convex-regularization methods as well as the popular BM3D denoiser. Additionally, the learned regularizer can be deployed to solve inverse problems with iterative schemes that provably converge. For both CT and MRI reconstruction, the regularizer generalizes well and offers an excellent tradeoff between performance, number of parameters, guarantees, and interpretability when compared to other data-driven approaches.

Optical Diffraction Tomography Meets Fluorescence Localization Microscopy

We show that structural information can be extracted from single molecule localization microscopy (SMLM) data. More precisely, we reinterpret SMLM data as the measures of a phaseless optical diffraction tomography system for which the illumination sources are fluorophores within the sample. Building upon this model, we propose a joint optimization framework to estimate both the refractive index map and the position of fluorescent molecules from the sole SMLM frames.

A Mini-Batch Quasi-Newton Proximal Method for Constrained Total-Variation Nonlinear Image Reconstruction

Over the years, computational imaging with accurate nonlinear physical models has drawn considerable interest due to its ability to achieve high-quality reconstructions. However, such nonlinear models are computationally demanding. A popular choice for solving the corresponding inverse problems is accelerated stochastic proximal methods (ASPMs), with the caveat that each iteration is expensive. To overcome this issue, we propose a mini-batch quasi-Newton proximal method (BQNPM) tailored to image-reconstruction problems with total-variation regularization. It involves an efficient approach that computes a weighted proximal mapping at a cost similar to that of the proximal mapping in ASPMs. However, BQNPM requires fewer iterations than ASPMs to converge. We assess the performance of BQNPM on three-dimensional inverse-scattering problems with linear and nonlinear physical models. Our results on simulated and real data show the effectiveness and efficiency of BQNPM,

On the Effect of Initialization: The Scaling Path of 2-Layer Neural Networks

In supervised learning, the regularization path is sometimes used as a convenient theoretical proxy for the optimization path of gradient descent initialized with zero. In this paper, we study a modification of the regularization path for infinite-width 2-layer ReLU neural networks with non-zero initial distribution of the weights at different scales. By exploiting a link with unbalanced optimal transport theory, we show that, despite the non-convexity of the 2-layer network training, this problem admits an infinite dimensional convex counterpart. We formulate the corresponding functional optimization problem and investigate its main properties. In particular, we show that as the scale of the initialization ranges between $0$ and $+\infty$, the associated path interpolates continuously between the so-called kernel and rich regimes. The numerical experiments confirm that, in our setting, the scaling path and the final states of the optimization path behave similarly even beyond these extreme points.