Alexis Goujon

Alexis Goujon, Sebastian Neumayer, Michael Unser

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.

Alexis Goujon, Sebastian Neumayer, Pakshal Bohra et al.

The emergence of deep-learning-based methods to solve image-reconstruction problems has enabled a significant increase in reconstruction quality. Unfortunately, these new methods often lack reliability and explainability, and there is a growing interest to address these shortcomings while retaining the boost in performance. In this work, we tackle this issue by revisiting regularizers that are the sum of convex-ridge functions. The gradient of such regularizers is parameterized by a neural network that has a single hidden layer with increasing and learnable activation functions. This neural network is trained within a few minutes as a multistep Gaussian denoiser. The numerical experiments for denoising, CT, and MRI reconstruction show improvements over methods that offer similar reliability guarantees.

Stanislas Ducotterd, Alexis Goujon, Pakshal Bohra et al.

Lipschitz-constrained neural networks have several advantages over unconstrained ones and can be applied to a variety of problems, making them a topic of attention in the deep learning community. Unfortunately, it has been shown both theoretically and empirically that they perform poorly when equipped with ReLU activation functions. By contrast, neural networks with learnable 1-Lipschitz linear splines are known to be more expressive. In this paper, we show that such networks correspond to global optima of a constrained functional optimization problem that consists of the training of a neural network composed of 1-Lipschitz linear layers and 1-Lipschitz freeform activation functions with second-order total-variation regularization. Further, we propose an efficient method to train these neural networks. Our numerical experiments show that our trained networks compare favorably with existing 1-Lipschitz neural architectures.

Alexis Goujon, Arian Etemadi, Michael Unser

Many feedforward neural networks (NNs) generate continuous and piecewise-linear (CPWL) mappings. Specifically, they partition the input domain into regions on which the mapping is affine. The number of these so-called linear regions offers a natural metric to characterize the expressiveness of CPWL NNs. The precise determination of this quantity is often out of reach in practice, and bounds have been proposed for specific architectures, including for ReLU and Maxout NNs. In this work, we generalize these bounds to NNs with arbitrary and possibly multivariate CPWL activation functions. We first provide upper and lower bounds on the maximal number of linear regions of a CPWL NN given its depth, width, and the number of linear regions of its activation functions. Our results rely on the combinatorial structure of convex partitions and confirm the distinctive role of depth which, on its own, is able to exponentially increase the number of regions. We then introduce a complementary stochastic framework to estimate the average number of linear regions produced by a CPWL NN. Under reasonable assumptions, the expected density of linear regions along any 1D path is bounded by the product of depth, width, and a measure of activation complexity (up to a scaling factor). This yields an identical role to the three sources of expressiveness: no exponential growth with depth is observed anymore.

Sebastian Neumayer, Alexis Goujon, Pakshal Bohra et al.

Lipschitz-constrained neural networks have many applications in machine learning. Since designing and training expressive Lipschitz-constrained networks is very challenging, there is a need for improved methods and a better theoretical understanding. Unfortunately, it turns out that ReLU networks have provable disadvantages in this setting. Hence, we propose to use learnable spline activation functions with at least 3 linear regions instead. We prove that this choice is optimal among all component-wise $1$-Lipschitz activation functions in the sense that no other weight constrained architecture can approximate a larger class of functions. Additionally, this choice is at least as expressive as the recently introduced non component-wise Groupsort activation function for spectral-norm-constrained weights. Previously published numerical results support our theoretical findings.

Mehrsa Pourya, Alexis Goujon, Michael Unser

Regression is one of the core problems tackled in supervised learning. Rectified linear unit (ReLU) neural networks generate continuous and piecewise-linear (CPWL) mappings and are the state-of-the-art approach for solving regression problems. In this paper, we propose an alternative method that leverages the expressivity of CPWL functions. In contrast to deep neural networks, our CPWL parameterization guarantees stability and is interpretable. Our approach relies on the partitioning of the domain of the CPWL function by a Delaunay triangulation. The function values at the vertices of the triangulation are our learnable parameters and identify the CPWL function uniquely. Formulating the learning scheme as a variational problem, we use the Hessian total variation (HTV) as regularizer to favor CPWL functions with few affine pieces. In this way, we control the complexity of our model through a single hyperparameter. By developing a computational framework to compute the HTV of any CPWL function parameterized by a triangulation, we discretize the learning problem as the generalized least absolute shrinkage and selection operator (LASSO). Our experiments validate the usage of our method in low-dimensional scenarios.

Michael Unser, Alexis Goujon, Stanislas Ducotterd

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.

Nicolas Talabot, Olivier Clerc, Arda Cinar Demirtas et al.

Accurate 3D shape representation is essential in engineering applications such as design, optimization, and simulation. In practice, engineering workflows require structured, part-based representations, as objects are inherently designed as assemblies of distinct components. However, most existing methods either model shapes holistically or decompose them without predefined part structures, limiting their applicability in real-world design tasks. We propose PartSDF, a supervised implicit representation framework that explicitly models composite shapes with independent, controllable parts while maintaining shape consistency. Thanks to its simple but innovative architecture, PartSDF outperforms both supervised and unsupervised baselines in reconstruction and generation tasks. We further demonstrate its effectiveness as a structured shape prior for engineering applications, enabling precise control over individual components while preserving overall coherence. Code available at https://github.com/cvlab-epfl/PartSDF.