Matthew Hirn

Renming Liu, Semih Cantürk, Frederik Wenkel et al. · mila

Graph Neural Networks (GNNs) extend the success of neural networks to graph-structured data by accounting for their intrinsic geometry. While extensive research has been done on developing GNN models with superior performance according to a collection of graph representation learning benchmarks, it is currently not well understood what aspects of a given model are probed by them. For example, to what extent do they test the ability of a model to leverage graph structure vs. node features? Here, we develop a principled approach to taxonomize benchmarking datasets according to a $\textit{sensitivity profile}$ that is based on how much GNN performance changes due to a collection of graph perturbations. Our data-driven analysis provides a deeper understanding of which benchmarking data characteristics are leveraged by GNNs. Consequently, our taxonomy can aid in selection and development of adequate graph benchmarks, and better informed evaluation of future GNN methods. Finally, our approach and implementation in $\texttt{GTaxoGym}$ package are extendable to multiple graph prediction task types and future datasets.

Guillaume Huguet, Alexander Tong, Bastian Rieck et al. · mila

Diffusion condensation is a dynamic process that yields a sequence of multiscale data representations that aim to encode meaningful abstractions. It has proven effective for manifold learning, denoising, clustering, and visualization of high-dimensional data. Diffusion condensation is constructed as a time-inhomogeneous process where each step first computes and then applies a diffusion operator to the data. We theoretically analyze the convergence and evolution of this process from geometric, spectral, and topological perspectives. From a geometric perspective, we obtain convergence bounds based on the smallest transition probability and the radius of the data, whereas from a spectral perspective, our bounds are based on the eigenspectrum of the diffusion kernel. Our spectral results are of particular interest since most of the literature on data diffusion is focused on homogeneous processes. From a topological perspective, we show diffusion condensation generalizes centroid-based hierarchical clustering. We use this perspective to obtain a bound based on the number of data points, independent of their location. To understand the evolution of the data geometry beyond convergence, we use topological data analysis. We show that the condensation process itself defines an intrinsic condensation homology. We use this intrinsic topology as well as the ambient persistent homology of the condensation process to study how the data changes over diffusion time. We demonstrate both types of topological information in well-understood toy examples. Our work gives theoretical insights into the convergence of diffusion condensation, and shows that it provides a link between topological and geometric data analysis.

Joyce Chew, Matthew Hirn, Smita Krishnaswamy et al.

The scattering transform is a multilayered, wavelet-based transform initially introduced as a model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks' stability and invariance properties. Subsequently, there has been widespread interest in extending the success of CNNs to data sets with non-Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on geometric scattering as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, directed graphs, and on high-dimensional single-cell data.

Joyce Chew, Holly R. Steach, Siddharth Viswanath et al.

The manifold scattering transform is a deep feature extractor for data defined on a Riemannian manifold. It is one of the first examples of extending convolutional neural network-like operators to general manifolds. The initial work on this model focused primarily on its theoretical stability and invariance properties but did not provide methods for its numerical implementation except in the case of two-dimensional surfaces with predefined meshes. In this work, we present practical schemes, based on the theory of diffusion maps, for implementing the manifold scattering transform to datasets arising in naturalistic systems, such as single cell genetics, where the data is a high-dimensional point cloud modeled as lying on a low-dimensional manifold. We show that our methods are effective for signal classification and manifold classification tasks.

Renming Liu, Matthew Hirn, Arjun Krishnan

Node embedding is a powerful approach for representing the structural role of each node in a graph. $\textit{Node2vec}$ is a widely used method for node embedding that works by exploring the local neighborhoods via biased random walks on the graph. However, $\textit{node2vec}$ does not consider edge weights when computing walk biases. This intrinsic limitation prevents $\textit{node2vec}$ from leveraging all the information in weighted graphs and, in turn, limits its application to many real-world networks that are weighted and dense. Here, we naturally extend $\textit{node2vec}$ to $\textit{node2vec+}$ in a way that accounts for edge weights when calculating walk biases, but which reduces to $\textit{node2vec}$ in the cases of unweighted graphs or unbiased walks. We empirically show that $\textit{node2vec+}$ is more robust to additive noise than $\textit{node2vec}$ in weighted graphs using two synthetic datasets. We also demonstrate that $\textit{node2vec+}$ significantly outperforms $\textit{node2vec}$ on a commonly benchmarked multi-label dataset (Wikipedia). Furthermore, we test $\textit{node2vec+}$ against GCN and GraphSAGE using various challenging gene classification tasks on two protein-protein interaction networks. Despite some clear advantages of GCN and GraphSAGE, they show comparable performance with $\textit{node2vec+}$. Finally, $\textit{node2vec+}$ can be used as a general approach for generating biased random walks, benefiting all existing methods built on top of $\textit{node2vec}$. $\textit{Node2vec+}$ is implemented as part of $\texttt{PecanPy}$, which is available at https://github.com/krishnanlab/PecanPy .

Sarah McGuire Scullen, Ernst Röell, Elizabeth Munch et al.

For deep learning problems on graph-structured data, pooling layers are important for down sampling, reducing computational cost, and to minimize overfitting. We define a pooling layer, nervePool, for data structured as simplicial complexes, which are generalizations of graphs that include higher-dimensional simplices beyond vertices and edges; this structure allows for greater flexibility in modeling higher-order relationships. The proposed simplicial coarsening scheme is built upon partitions of vertices, which allow us to generate hierarchical representations of simplicial complexes, collapsing information in a learned fashion. NervePool builds on the learned vertex cluster assignments and extends to coarsening of higher dimensional simplices in a deterministic fashion. While in practice the pooling operations are computed via a series of matrix operations, the topological motivation is a set-theoretic construction based on unions of stars of simplices and the nerve complex.

Frederik Wenkel, Yimeng Min, Matthew Hirn et al.

Geometric deep learning has made great strides towards generalizing the design of structure-aware neural networks from traditional domains to non-Euclidean ones, giving rise to graph neural networks (GNN) that can be applied to graph-structured data arising in, e.g., social networks, biochemistry, and material science. Graph convolutional networks (GCNs) in particular, inspired by their Euclidean counterparts, have been successful in processing graph data by extracting structure-aware features. However, current GNN models are often constrained by various phenomena that limit their expressive power and ability to generalize to more complex graph datasets. Most models essentially rely on low-pass filtering of graph signals via local averaging operations, leading to oversmoothing. Moreover, to avoid severe oversmoothing, most popular GCN-style networks tend to be shallow, with narrow receptive fields, leading to underreaching. Here, we propose a hybrid GNN framework that combines traditional GCN filters with band-pass filters defined via geometric scattering. We further introduce an attention framework that allows the model to locally attend over combined information from different filters at the node level. Our theoretical results establish the complementary benefits of the scattering filters to leverage structural information from the graph, while our experiments show the benefits of our method on various learning tasks.

Renming Liu, Semih Cantürk, Frederik Wenkel et al.

Graph neural networks (GNNs) have attracted much attention due to their ability to leverage the intrinsic geometries of the underlying data. Although many different types of GNN models have been developed, with many benchmarking procedures to demonstrate the superiority of one GNN model over the others, there is a lack of systematic understanding of the underlying benchmarking datasets, and what aspects of the model are being tested. Here, we provide a principled approach to taxonomize graph benchmarking datasets by carefully designing a collection of graph perturbations to probe the essential data characteristics that GNN models leverage to perform predictions. Our data-driven taxonomization of graph datasets provides a new understanding of critical dataset characteristics that will enable better model evaluation and the development of more specialized GNN models.

Michael Perlmutter, Jieqian He, Mark Iwen et al.

The scattering transform is a wavelet-based model of Convolutional Neural Networks originally introduced by S. Mallat. Mallat's analysis shows that this network has desirable stability and invariance guarantees and therefore helps explain the observation that the filters learned by early layers of a Convolutional Neural Network typically resemble wavelets. Our aim is to understand what sort of filters should be used in the later layers of the network. Towards this end, we propose a two-layer hybrid scattering transform. In our first layer, we convolve the input signal with a wavelet filter transform to promote sparsity, and, in the second layer, we convolve with a Gabor filter to leverage the sparsity created by the first layer. We show that these measurements characterize information about signals with isolated singularities. We also show that the Gabor measurements used in the second layer can be used to synthesize sparse signals such as those produced by the first layer.

Jieqian He, Matthew Hirn

We provide a new model for texture synthesis based on a multiscale, multilayer feature extractor. Within the model, textures are represented by a set of statistics computed from ReLU wavelet coefficients at different layers, scales and orientations. A new image is synthesized by matching the target statistics via an iterative projection algorithm. We explain the necessity of the different types of pre-defined wavelet filters used in our model and the advantages of multilayer structures for image synthesis. We demonstrate the power of our model by generating samples of high quality textures and providing insights into deep representations for texture images.

Xitong Zhang, Yixuan He, Nathan Brugnone et al.

The prevalence of graph-based data has spurred the rapid development of graph neural networks (GNNs) and related machine learning algorithms. Yet, despite the many datasets naturally modeled as directed graphs, including citation, website, and traffic networks, the vast majority of this research focuses on undirected graphs. In this paper, we propose MagNet, a spectral GNN for directed graphs based on a complex Hermitian matrix known as the magnetic Laplacian. This matrix encodes undirected geometric structure in the magnitude of its entries and directional information in their phase. A "charge" parameter attunes spectral information to variation among directed cycles. We apply our network to a variety of directed graph node classification and link prediction tasks showing that MagNet performs well on all tasks and that its performance exceeds all other methods on a majority of such tasks. The underlying principles of MagNet are such that it can be adapted to other spectral GNN architectures.

Paul Sinz, Michael W. Swift, Xavier Brumwell et al.

The dream of machine learning in materials science is for a model to learn the underlying physics of an atomic system, allowing it to move beyond interpolation of the training set to the prediction of properties that were not present in the original training data. In addition to advances in machine learning architectures and training techniques, achieving this ambitious goal requires a method to convert a 3D atomic system into a feature representation that preserves rotational and translational symmetry, smoothness under small perturbations, and invariance under re-ordering. The atomic orbital wavelet scattering transform preserves these symmetries by construction, and has achieved great success as a featurization method for machine learning energy prediction. Both in small molecules and in the bulk amorphous $\text{Li}_α\text{Si}$ system, machine learning models using wavelet scattering coefficients as features have demonstrated a comparable accuracy to Density Functional Theory at a small fraction of the computational cost. In this work, we test the generalizability of our $\text{Li}_α\text{Si}$ energy predictor to properties that were not included in the training set, such as elastic constants and migration barriers. We demonstrate that statistical feature selection methods can reduce over-fitting and lead to remarkable accuracy in these extrapolation tasks.

Michael Perlmutter, Alexander Tong, Feng Gao et al.

The scattering transform is a multilayered wavelet-based deep learning architecture that acts as a model of convolutional neural networks. Recently, several works have introduced generalizations of the scattering transform for non-Euclidean settings such as graphs. Our work builds upon these constructions by introducing windowed and non-windowed geometric scattering transforms for graphs based upon a very general class of asymmetric wavelets. We show that these asymmetric graph scattering transforms have many of the same theoretical guarantees as their symmetric counterparts. As a result, the proposed construction unifies and extends known theoretical results for many of the existing graph scattering architectures. In doing so, this work helps bridge the gap between geometric scattering and other graph neural networks by introducing a large family of networks with provable stability and invariance guarantees. These results lay the groundwork for future deep learning architectures for graph-structured data that have learned filters and also provably have desirable theoretical properties.

Michael Perlmutter, Feng Gao, Guy Wolf et al.

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric scattering transform on manifolds. Similar to the Euclidean scattering transform, the geometric scattering transform is based on a cascade of wavelet filters and pointwise nonlinearities. It is invariant to local isometries and stable to certain types of diffeomorphisms. Empirical results demonstrate its utility on several geometric learning tasks. Our results generalize the deformation stability and local translation invariance of Euclidean scattering, and demonstrate the importance of linking the used filter structures to the underlying geometry of the data.

Michael Perlmutter, Jieqian He, Matthew Hirn

We present a machine learning model for the analysis of randomly generated discrete signals, modeled as the points of an inhomogeneous, compound Poisson point process. Like the wavelet scattering transform introduced by Mallat, our construction is naturally invariant to translations and reflections, but it decouples the roles of scale and frequency, replacing wavelets with Gabor-type measurements. We show that, with suitable nonlinearities, our measurements distinguish Poisson point processes from common self-similar processes, and separate different types of Poisson point processes.

Michael Perlmutter, Guy Wolf, Matthew Hirn

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of the success of convolutional neural networks (ConvNets) in image data analysis and other tasks. Inspired by recent interest in geometric deep learning, which aims to generalize ConvNets to manifold and graph-structured domains, we generalize the scattering transform to compact manifolds. Similar to the Euclidean scattering transform, our geometric scattering transform is based on a cascade of designed filters and pointwise nonlinearities, which enables rigorous analysis of the feature extraction provided by scattering layers. Our main focus here is on theoretical understanding of this geometric scattering network, while setting aside implementation aspects, although we remark that application of similar transforms to graph data analysis has been studied recently in related work. Our results establish conditions under which geometric scattering provides localized isometry invariant descriptions of manifold signals, which are also stable to families of diffeomorphisms formulated in intrinsic manifolds terms. These results not only generalize the deformation stability and local roto-translation invariance of Euclidean scattering, but also demonstrate the importance of linking the used filter structures (e.g., in geometric deep learning) to the underlying manifold geometry, or the data geometry it represents.

Xavier Brumwell, Paul Sinz, Kwang Jin Kim et al.

A general machine learning architecture is introduced that uses wavelet scattering coefficients of an inputted three dimensional signal as features. Solid harmonic wavelet scattering transforms of three dimensional signals were previously introduced in a machine learning framework for the regression of properties of small organic molecules. Here this approach is extended for general steerable wavelets which are equivariant to translations and rotations, resulting in a sparse model of the target function. The scattering coefficients inherit from the wavelets invariance to translations and rotations. As an illustration of this approach a linear regression model is learned for the formation energy of amorphous lithium-silicon material states trained over a database generated using plane-wave Density Functional Theory methods. State-of-the-art results are produced as compared to other machine learning approaches over similarly generated databases.

Feng Gao, Guy Wolf, Matthew Hirn

We explore the generalization of scattering transforms from traditional (e.g., image or audio) signals to graph data, analogous to the generalization of ConvNets in geometric deep learning, and the utility of extracted graph features in graph data analysis. In particular, we focus on the capacity of these features to retain informative variability and relations in the data (e.g., between individual graphs, or in aggregate), while relating our construction to previous theoretical results that establish the stability of similar transforms to families of graph deformations. We demonstrate the application the our geometric scattering features in graph classification of social network data, and in data exploration of biochemistry data.

Michael Eickenberg, Georgios Exarchakis, Matthew Hirn et al.

We present a machine learning algorithm for the prediction of molecule properties inspired by ideas from density functional theory. Using Gaussian-type orbital functions, we create surrogate electronic densities of the molecule from which we compute invariant "solid harmonic scattering coefficients" that account for different types of interactions at different scales. Multi-linear regressions of various physical properties of molecules are computed from these invariant coefficients. Numerical experiments show that these regressions have near state of the art performance, even with relatively few training examples. Predictions over small sets of scattering coefficients can reach a DFT precision while being interpretable.

Adam Gustafson, Matthew Hirn, Kitty Mohammed et al.

One means of fitting functions to high-dimensional data is by providing smoothness constraints. Recently, the following smooth function approximation problem was proposed: given a finite set $E \subset \mathbb{R}^d$ and a function $f: E \rightarrow \mathbb{R}$, interpolate the given information with a function $\widehat{f} \in \dot{C}^{1, 1}(\mathbb{R}^d)$ (the class of first-order differentiable functions with Lipschitz gradients) such that $\widehat{f}(a) = f(a)$ for all $a \in E$, and the value of $\mathrm{Lip}(\nabla \widehat{f})$ is minimal. An algorithm is provided that constructs such an approximating function $\widehat{f}$ and estimates the optimal Lipschitz constant $\mathrm{Lip}(\nabla \widehat{f})$ in the noiseless setting. We address statistical aspects of reconstructing the approximating function $\widehat{f}$ from a closely-related class $C^{1, 1}(\mathbb{R}^d)$ given samples from noisy data. We observe independent and identically distributed samples $y(a) = f(a) + ξ(a)$ for $a \in E$, where $ξ(a)$ is a noise term and the set $E \subset \mathbb{R}^d$ is fixed and known. We obtain uniform bounds relating the empirical risk and true risk over the class $\mathcal{F}_{\widetilde{M}} = \{f \in C^{1, 1}(\mathbb{R}^d) \mid \mathrm{Lip}(\nabla f) \leq \widetilde{M}\}$, where the quantity $\widetilde{M}$ grows with the number of samples at a rate governed by the metric entropy of the class $C^{1, 1}(\mathbb{R}^d)$. Finally, we provide an implementation using Vaidya's algorithm, supporting our results via numerical experiments on simulated data.

Matthew Hirn, Stéphane Mallat, Nicolas Poilvert

We introduce multiscale invariant dictionaries to estimate quantum chemical energies of organic molecules, from training databases. Molecular energies are invariant to isometric atomic displacements, and are Lipschitz continuous to molecular deformations. Similarly to density functional theory (DFT), the molecule is represented by an electronic density function. A multiscale invariant dictionary is calculated with wavelet scattering invariants. It cascades a first wavelet transform which separates scales, with a second wavelet transform which computes interactions across scales. Sparse scattering regressions give state of the art results over two databases of organic planar molecules. On these databases, the regression error is of the order of the error produced by DFT codes, but at a fraction of the computational cost.

Matthew Hirn, Nicolas Poilvert, Stéphane Mallat

We present a novel approach to the regression of quantum mechanical energies based on a scattering transform of an intermediate electron density representation. A scattering transform is a deep convolution network computed with a cascade of multiscale wavelet transforms. It possesses appropriate invariant and stability properties for quantum energy regression. This new framework removes fundamental limitations of Coulomb matrix based energy regressions, and numerical experiments give state-of-the-art accuracy over planar molecules.