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We present a deep learning-based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers. Combining classical iterative multigrid solvers and convolutional neural networks (CNNs) via preconditioning, we obtain a learned neural solver that is faster and scales better than a standard multigrid solver. Our approach offers three main contributions over previous neural methods of this kind. First, we construct a multilevel U-Net-like encoder-solver CNN with an implicit layer on the coarsest grid of the U-Net, where convolution kernels are inverted. This alleviates the field of view problem in CNNs and allows better scalability. Second, we improve upon the previous CNN preconditioner in terms of the number of parameters, computation time, and convergence rates. Third, we propose a multiscale training approach that enables the network to scale to problems of previously unseen dimensions while still maintaining a reasonable training procedure. Our encoder-solver architecture can be used to generalize over different slowness models of various difficulties and is efficient at solving for many right-hand sides per slowness model. We demonstrate the benefits of our novel architecture with numerical experiments on a variety of heterogeneous two-dimensional problems at high wavenumbers., Submitted to SIAM Journal on Scientific Computing Copper Mountain Special Section on Multigrid Methods 2023

Inverse problems are mathematically ill-posed. Thus, given some (noisy) data, there is more than one solution that fits the data. In recent years, deep neural techniques that find the most appropriate solution, in the sense that it contains a-priori information, were developed. However, they suffer from several shortcomings. First, most techniques cannot guarantee that the solution fits the data at inference. Second, while the derivation of the techniques is inspired by the existence of a valid scalar regularization function, such techniques do not in practice rely on such a function, and therefore veer away from classical variational techniques. In this work we introduce a new family of neural regularizers for the solution of inverse problems. These regularizers are based on a variational formulation and are guaranteed to fit the data. We demonstrate their use on a number of highly ill-posed problems, from image deblurring to limited angle tomography.

We present an efficient matrix-free geometric multigrid method for the elastic Helmholtz equation, and a suitable discretization. Many discretization methods had been considered in the literature for the Helmholtz equations, as well as many solvers and preconditioners, some of which are adapted for the elastic version of the equation. However, there is very little work considering the reciprocity of discretization and a solver. In this work, we aim to bridge this gap. By choosing an appropriate stencil for re-discretization of the equation on the coarse grid, we develop a multigrid method that can be easily implemented as matrix-free, relying on stencils rather than sparse matrices. This is crucial for efficient implementation on modern hardware. Using two-grid local Fourier analysis, we validate the compatibility of our discretization with our solver, and tune a choice of weights for the stencil for which the convergence rate of the multigrid cycle is optimal. It results in a scalable multigrid preconditioner that can tackle large real-world 3D scenarios., Comment: 20 pages

Two main families of node feature augmentation schemes have been explored for enhancing GNNs: random features and spectral positional encoding. Surprisingly, however, there is still no clear understanding of the relation between these two augmentation schemes. Here we propose a novel family of positional encoding schemes which draws a link between the above two approaches and improves over both. The new approach, named Random Feature Propagation (RFP), is inspired by the power iteration method and its generalizations. It concatenates several intermediate steps of an iterative algorithm for computing the dominant eigenvectors of a propagation matrix, starting from random node features. Notably, these propagation steps are based on graph-dependent propagation operators that can be either predefined or learned. We explore the theoretical and empirical benefits of RFP. First, we provide theoretical justifications for using random features, for incorporating early propagation steps, and for using multiple random initializations. Then, we empirically demonstrate that RFP significantly outperforms both spectral PE and random features in multiple node classification and graph classification benchmarks., Comment: ICML 2023

Neural Representations have recently been shown to effectively reconstruct a wide range of signals from 3D meshes and shapes to images and videos. We show that, when adapted correctly, neural representations can be used to directly represent the weights of a pre-trained convolutional neural network, resulting in a Neural Representation for Neural Networks (NeRN). Inspired by coordinate inputs of previous neural representation methods, we assign a coordinate to each convolutional kernel in our network based on its position in the architecture, and optimize a predictor network to map coordinates to their corresponding weights. Similarly to the spatial smoothness of visual scenes, we show that incorporating a smoothness constraint over the original network's weights aids NeRN towards a better reconstruction. In addition, since slight perturbations in pre-trained model weights can result in a considerable accuracy loss, we employ techniques from the field of knowledge distillation to stabilize the learning process. We demonstrate the effectiveness of NeRN in reconstructing widely used architectures on CIFAR-10, CIFAR-100, and ImageNet. Finally, we present two applications using NeRN, demonstrating the capabilities of the learned representations.

Graph Neural Networks (GNNs) are prominent in handling sparse and unstructured data efficiently and effectively. Specifically, GNNs were shown to be highly effective for node classification tasks, where labelled information is available for only a fraction of the nodes. Typically, the optimization process, through the objective function, considers only labelled nodes while ignoring the rest. In this paper, we propose novel objective terms for the training of GNNs for node classification, aiming to exploit all the available data and improve accuracy. Our first term seeks to maximize the mutual information between node and label features, considering both labelled and unlabelled nodes in the optimization process. Our second term promotes anisotropic smoothness in the prediction maps. Lastly, we propose a cross-validating gradients approach to enhance the learning from labelled data. Our proposed objectives are general and can be applied to various GNNs and require no architectural modifications. Extensive experiments demonstrate our approach using popular GNNs like GCN, GAT and GCNII, reading a consistent and significant accuracy improvement on 10 real-world node classification datasets.

Graph Neural Networks (GNNs) are limited in their propagation operators. In many cases, these operators often contain non-negative elements only and are shared across channels, limiting the expressiveness of GNNs. Moreover, some GNNs suffer from over-smoothing, limiting their depth. On the other hand, Convolutional Neural Networks (CNNs) can learn diverse propagation filters, and phenomena like over-smoothing are typically not apparent in CNNs. In this paper, we bridge these gaps by incorporating trainable channel-wise weighting factors $\omega$ to learn and mix multiple smoothing and sharpening propagation operators at each layer. Our generic method is called $\omega$GNN, and is easy to implement. We study two variants: $\omega$GCN and $\omega$GAT. For $\omega$GCN, we theoretically analyse its behaviour and the impact of $\omega$ on the obtained node features. Our experiments confirm these findings, demonstrating and explaining how both variants do not over-smooth. Additionally, we experiment with 15 real-world datasets on node- and graph-classification tasks, where our $\omega$GCN and $\omega$GAT perform on par with state-of-the-art methods., Comment: Accepted to the International Conference on Machine Learning (ICML) 2023

Graph Convolutional Networks (GCNs), similarly to Convolutional Neural Networks (CNNs), are typically based on two main operations - spatial and point-wise convolutions. In the context of GCNs, differently from CNNs, a pre-determined spatial operator based on the graph Laplacian is often chosen, allowing only the point-wise operations to be learnt. However, learning a meaningful spatial operator is critical for developing more expressive GCNs for improved performance. In this paper we propose pathGCN, a novel approach to learn the spatial operator from random paths on the graph. We analyze the convergence of our method and its difference from existing GCNs. Furthermore, we discuss several options of combining our learnt spatial operator with point-wise convolutions. Our extensive experiments on numerous datasets suggest that by properly learning both the spatial and point-wise convolutions, phenomena like over-smoothing can be inherently avoided, and new state-of-the-art performance is achieved., ICML 2022

Convolutional Neural Networks (CNNs) are known for requiring extensive computational resources, and quantization is among the best and most common methods for compressing them. While aggressive quantization (i.e., less than 4-bits) performs well for classification, it may cause severe performance degradation in image-to-image tasks such as semantic segmentation and depth estimation. In this paper, we propose Wavelet Compressed Convolution (WCC) -- a novel approach for high-resolution activation maps compression integrated with point-wise convolutions, which are the main computational cost of modern architectures. To this end, we use an efficient and hardware-friendly Haar-wavelet transform, known for its effectiveness in image compression, and define the convolution on the compressed activation map. We experiment with various tasks that benefit from high-resolution input. By combining WCC with light quantization, we achieve compression rates equivalent to 1-4bit activation quantization with relatively small and much more graceful degradation in performance. Our code is available at https://github.com/BGUCompSci/WaveletCompressedConvolution.

We propose a novel quasi-Newton method for solving the sparse inverse covariance estimation problem also known as the graphical least absolute shrinkage and selection operator (GLASSO). This problem is often solved using a second-order quadratic approximation. However, in such algorithms the Hessian term is complex and computationally expensive to handle. Therefore, our method uses the inverse of the Hessian as a preconditioner to simplify and approximate the quadratic element at the cost of a more complex \(\ell_1\) element. The variables of the resulting preconditioned problem are coupled only by the \(\ell_1\) sub-derivative of each other, which can be guessed with minimal cost using the gradient itself, allowing the algorithm to be parallelized and implemented efficiently on GPU hardware accelerators. Numerical results on synthetic and real data demonstrate that our method is competitive with other state-of-the-art approaches.

We propose methods for making inferences on the fairness and accuracy of a given classifier, using only aggregate population statistics. This is necessary when it is impossible to obtain individual classification data, for instance when there is no access to the classifier or to a representative individual-level validation set. We study fairness with respect to the equalized odds criterion, which we generalize to multiclass classification. We propose a measure of unfairness with respect to this criterion, which quantifies the fraction of the population that is treated unfairly. We then show how inferences on the unfairness and error of a given classifier can be obtained using only aggregate label statistics such as the rate of prediction of each label in each sub-population, as well as the true rate of each label. We derive inference procedures for binary classifiers and for multiclass classifiers, for the case where confusion matrices in each sub-population are known, and for the significantly more challenging case where they are unknown. We report experiments on data sets representing diverse applications, which demonstrate the effectiveness and the wide range of possible uses of the proposed methodology.

Graph neural networks are increasingly becoming the go-to approach in various fields such as computer vision, computational biology and chemistry, where data are naturally explained by graphs. However, unlike traditional convolutional neural networks, deep graph networks do not necessarily yield better performance than shallow graph networks. This behavior usually stems from the over-smoothing phenomenon. In this work, we propose a family of architectures to control this behavior by design. Our networks are motivated by numerical methods for solving Partial Differential Equations (PDEs) on manifolds, and as such, their behavior can be explained by similar analysis. Moreover, as we demonstrate using an extensive set of experiments, our PDE-motivated networks can generalize and be effective for various types of problems from different fields. Our architectures obtain better or on par with the current state-of-the-art results for problems that are typically approached using different architectures., NeurIPS 2021

Diffraction tomography is a noninvasive technique that estimates the refractive indices of unknown objects and involves an inverse-scattering problem governed by the wave equation. Recent works have shown the benefit of nonlinear models of wave propagation that account for multiple scattering and reflections. In particular, the Lippmann-Schwinger~(LiS) model defines an inverse problem to simulate the wave propagation. Although accurate, this model is hard to solve when the samples are highly contrasted or have a large physical size. In this work, we introduce instead a Helmholtz-based nonlinear model for inverse scattering. To solve the corresponding inverse problem, we propose a robust and efficient multigrid-based solver. Moreover, we show that our method is a suitable alternative to the LiS model, especially for strongly scattering objects. Numerical experiments on simulated and real data demonstrate the effectiveness of the Helmholtz model, as well as the efficiency of the proposed multigrid method., 12 pages,13 figures, 2 tables

Quantized neural networks (QNNs) are among the main approaches for deploying deep neural networks on low resource edge devices. Training QNNs using different levels of precision throughout the network (dynamic quantization) typically achieves superior trade-offs between performance and computational load. However, optimizing the different precision levels of QNNs can be complicated, as the values of the bit allocations are discrete and difficult to differentiate for. Also, adequately accounting for the dependencies between the bit allocation of different layers is not straight-forward. To meet these challenges, in this work we propose GradFreeBits: a novel joint optimization scheme for training dynamic QNNs, which alternates between gradient-based optimization for the weights, and gradient-free optimization for the bit allocation. Our method achieves better or on par performance with current state of the art low precision neural networks on CIFAR10/100 and ImageNet classification. Furthermore, our approach can be extended to a variety of other applications involving neural networks used in conjunction with parameters which are difficult to optimize for.

Graph Convolutional Networks (GCNs) are widely used in a variety of applications, and can be seen as an unstructured version of standard Convolutional Neural Networks (CNNs). As in CNNs, the computational cost of GCNs for large input graphs (such as large point clouds or meshes) can be high and inhibit the use of these networks, especially in environments with low computational resources. To ease these costs, quantization can be applied to GCNs. However, aggressive quantization of the feature maps can lead to a significant degradation in performance. On a different note, Haar wavelet transforms are known to be one of the most effective and efficient approaches to compress signals. Therefore, instead of applying aggressive quantization to feature maps, we propose to utilize Haar wavelet compression and light quantization to reduce the computations and the bandwidth involved with the network. We demonstrate that this approach surpasses aggressive feature quantization by a significant margin, for a variety of problems ranging from node classification to point cloud classification and part and semantic segmentation.

We present a multigrid approach that combats the quadratic growth of the number of parameters with respect to the number of channels in standard convolutional neural networks (CNNs). It has been shown that there is a redundancy in standard CNNs, as networks with much sparser convolution operators can yield similar performance to full networks. The sparsity patterns that lead to such behavior, however, are typically random, hampering hardware efficiency. In this work, we present a multigrid-in-channels approach for building CNN architectures that achieves full coupling of the channels, and whose number of parameters is linearly proportional to the width of the network. To this end, we replace each convolution layer in a generic CNN with a multilevel layer consisting of structured (i.e., grouped) convolutions. Our examples from supervised image classification show that applying this strategy to residual networks and MobileNetV2 considerably reduces the number of parameters without negatively affecting accuracy. Therefore, we can widen networks without dramatically increasing the number of parameters or operations., This paper has been withdrawn by the authors. This paper has been superseded by arXiv:2011.09128

Graph Convolutional Networks (GCNs) have shown to be effective in handling unordered data like point clouds and meshes. In this work we propose novel approaches for graph convolution, pooling and unpooling, inspired from finite differences and algebraic multigrid frameworks. We form a parameterized convolution kernel based on discretized differential operators, leveraging the graph mass, gradient and Laplacian. This way, the parameterization does not depend on the graph structure, only on the meaning of the network convolutions as differential operators. To allow hierarchical representations of the input, we propose pooling and unpooling operations that are based on algebraic multigrid methods, which are mainly used to solve partial differential equations on unstructured grids. To motivate and explain our method, we compare it to standard convolutional neural networks, and show their similarities and relations in the case of a regular grid. Our proposed method is demonstrated in various experiments like classification and part-segmentation, achieving on par or better than state of the art results. We also analyze the computational cost of our method compared to other GCNs.

Convolutional Neural Networks (CNNs) filter the input data using spatial convolution operators with compact stencils. Commonly, the convolution operators couple features from all channels, which leads to immense computational cost in the training of and prediction with CNNs. To improve the efficiency of CNNs, we introduce lean convolution operators that reduce the number of parameters and computational complexity, and can be used in a wide range of existing CNNs. Here, we exemplify their use in residual networks (ResNets), which have been very reliable for a few years now and analyzed intensively. In our experiments on three image classification problems, the proposed LeanResNet yields results that are comparable to other recently proposed reduced architectures using similar number of parameters.

Convolutional Neural Networks (CNNs) filter the input data using a series of spatial convolution operators with compactly supported stencils and point-wise nonlinearities. Commonly, the convolution operators couple features from all channels. For wide networks, this leads to immense computational cost in the training of and prediction with CNNs. In this paper, we present novel ways to parameterize the convolution more efficiently, aiming to decrease the number of parameters in CNNs and their computational complexity. We propose new architectures that use a sparser coupling between the channels and thereby reduce both the number of trainable weights and the computational cost of the CNN. Our architectures arise as new types of residual neural network (ResNet) that can be seen as discretizations of a Partial Differential Equations (PDEs) and thus have predictable theoretical properties. Our first architecture involves a convolution operator with a special sparsity structure, and is applicable to a large class of CNNs. Next, we present an architecture that can be seen as a discretization of a diffusion reaction PDE, and use it with three different convolution operators. We outline in our experiments that the proposed architectures, although considerably reducing the number of trainable weights, yield comparable accuracy to existing CNNs that are fully coupled in the channel dimension., Comment: 10 pages, 2 figures