Graph coarsening aims to diminish the size of a graph to lighten its memory footprint, and has numerous applications in graph signal processing and machine learning. It is usually defined using a reduction matrix and a lifting matrix, which, respectively, allows to project a graph signal from the original graph to the coarsened one and back. This results in a loss of information measured by the so-called Restricted Spectral Approximation (RSA). Most coarsening frameworks impose a fixed relationship between the reduction and lifting matrices, generally as pseudo-inverses of each other, and seek to define a coarsening that minimizes the RSA. In this paper, we remark that the roles of these two matrices are not entirely symmetric: indeed, putting constraints on the *lifting matrix alone* ensures the existence of important objects such as the coarsened graph's adjacency matrix or Laplacian. In light of this, in this paper, we introduce a more general notion of reduction matrix, that is *not* necessarily the pseudo-inverse of the lifting matrix. We establish a taxonomy of ``admissible'' families of reduction matrices, discuss the different properties that they must satisfy and whether they admit a closed-form description or not. We show that, for a *fixed* coarsening represented by a fixed lifting matrix, the RSA can be *further* reduced simply by modifying the reduction matrix. We explore different examples, including some based on a constrained optimization process of the RSA. Since this criterion has also been linked to the performance of Graph Neural Networks, we also illustrate the impact of this choices on different node classification tasks on coarsened graphs.

Graph coarsening aims to reduce the size of a large graph while preserving some of its key properties, which has been used in many applications to reduce computational load and memory footprint. For instance, in graph machine learning, training Graph Neural Networks (GNNs) on coarsened graphs leads to drastic savings in time and memory. However, GNNs rely on the Message-Passing (MP) paradigm, and classical spectral preservation guarantees for graph coarsening do not directly lead to theoretical guarantees when performing naive message-passing on the coarsened graph. In this work, we propose a new message-passing operation specific to coarsened graphs, which exhibit theoretical guarantees on the preservation of the propagated signal. Interestingly, and in a sharp departure from previous proposals, this operation on coarsened graphs is oriented, even when the original graph is undirected. We conduct node classification tasks on synthetic and real data and observe improved results compared to performing naive message-passing on the coarsened graph.

We study the convergence of message-passing graph neural networks on random graph models toward their continuous counterparts as the number of nodes tends to infinity. Until now, this convergence was only known for architectures with aggregation functions in the form of normalized means, or, equivalently, of an application of classical operators like the adjacency matrix or the graph Laplacian. We extend such results to a large class of aggregation functions, that encompasses all classically used message-passing graph neural networks, such as attention-based message-passing, max convolutional message-passing, (degree-normalized) convolutional message-passing, or moment-based aggregation message-passing. Under mild assumptions, we give non-asymptotic bounds with high probability to quantify this convergence. Our main result is based on the McDiarmid inequality. Interestingly, this result does not apply to the case where the aggregation is a coordinate-wise maximum. We treat this case separately and obtain a different convergence rate.

We aim to deepen the theoretical understanding of Graph Neural Networks (GNNs) on large graphs, with a focus on their expressive power. Existing analyses relate this notion to the graph isomorphism problem, which is mostly relevant for graphs of small sizes, or studied graph classification or regression tasks, while prediction tasks on \emph{nodes} are far more relevant on large graphs. Recently, several works showed that, on very general random graphs models, GNNs converge to certains functions as the number of nodes grows. In this paper, we provide a more complete and intuitive description of the function space generated by equivariant GNNs for node-tasks, through general notions of convergence that encompass several previous examples. We emphasize the role of input node features, and study the impact of \emph{node Positional Encodings} (PEs), a recent line of work that has been shown to yield state-of-the-art results in practice. Through the study of several examples of PEs on large random graphs, we extend previously known universality results to significantly more general models. Our theoretical results hint at some normalization tricks, which is shown numerically to have a positive impact on GNN generalization on synthetic and real data. Our proofs contain new concentration inequalities of independent interest.

Sparsity priors are commonly used in denoising and image reconstruction. For analysis-type priors, a dictionary defines a representation of signals that is likely to be sparse. In most situations, this dictionary is unknown, and is to be recovered from pairs of ground-truth signals and measurements, by minimizing the reconstruction error. This defines a hierarchical optimization problem, which can be cast as a bi-level optimization. Yet, this problem is unsolvable, as reconstructions and their derivativew.r.t.the dictionary have no closed-form expression. However, reconstructions can be iteratively computed using the Forward-Backward splitting (FB) algorithm. In this letter, we approximate reconstructions by the output of the FB algorithm, up to a linear transformation. Then, we leverage automatic differentiation to evaluate the gradient of this outputw.r.t.the dictionary, which we learn with projected gradient descent. Experiments show that our algorithm successfully learns the 1D Total Variation (TV) dictionary from piecewise constant signals. For the same case study, we propose to constrain our search to dictionaries of 0-centered columns, which removes undesired local minima and improves numerical stability.

We analyze graph smoothing with mean aggregation, where each node successively receives the average of the features of its neighbors. Indeed, it has quickly been observed that Graph Neural Networks (GNNs), which generally follow some variant of Message-Passing (MP) with repeated aggregation, may be subject to the oversmoothing phenomenon: by performing too many rounds of MP, the node features tend to converge to a non-informative limit. In the case of mean aggregation, for connected graphs, the node features become constant across the whole graph. At the other end of the spectrum, it is intuitively obvious that some MP rounds are necessary, but existing analyses do not exhibit both phenomena at once: beneficial ``finite'' smoothing and oversmoothing in the limit. In this paper, we consider simplified linear GNNs, and rigorously analyze two examples for which a finite number of mean aggregation steps provably improves the learning performance, before oversmoothing kicks in. We consider a latent space random graph model, where node features are partial observations of the latent variables and the graph contains pairwise relationships between them. We show that graph smoothing restores some of the lost information, up to a certain point, by two phenomena: graph smoothing shrinks non-principal directions in the data faster than principal ones, which is useful for regression, and shrinks nodes within communities faster than they collapse together, which improves classification.

The graphlet kernel is a classical method in graph classification. It however suffers from a high computation cost due to the isomorphism test it includes. As a generic proxy, and in general at the cost of losing some information, this test can be efficiently replaced by a user-defined mapping that computes various graph characteristics. In this paper, we propose to leverage kernel random features within the graphlet framework, and establish a theoretical link with a mean kernel metric. If this method can still be prohibitively costly for usual random features, we then incorporate optical random features that can be computed in constant time. Experiments show that the resulting algorithm is orders of magnitude faster than the graphlet kernel for the same, or better, accuracy.

We study the approximation power of Graph Neural Networks (GNNs) on latent position random graphs. In the large graph limit, GNNs are known to converge to certain ``continuous'' models known as c-GNNs, which directly enables a study of their approximation power on random graph models. In the absence of input node features however, just as GNNs are limited by the Weisfeiler-Lehman isomorphism test, c-GNNs will be severely limited on simple random graph models. For instance, they will fail to distinguish the communities of a well-separated Stochastic Block Model (SBM) with constant degree function. Thus, we consider recently proposed architectures that augment GNNs with unique node identifiers, referred to as Structural GNNs here (SGNNs). We study the convergence of SGNNs to their continuous counterpart (c-SGNNs) in the large random graph limit, under new conditions on the node identifiers. We then show that c-SGNNs are strictly more powerful than c-GNNs in the continuous limit, and prove their universality on several random graph models of interest, including most SBMs and a large class of random geometric graphs. Our results cover both permutation-invariant and permutation-equivariant architectures.

We study properties of Graph Convolutional Networks (GCNs) by analyzing their behavior on standard models of random graphs, where nodes are represented by random latent variables and edges are drawn according to a similarity kernel. This allows us to overcome the difficulties of dealing with discrete notions such as isomorphisms on very large graphs, by considering instead more natural geometric aspects. We first study the convergence of GCNs to their continuous counterpart as the number of nodes grows. Our results are fully non-asymptotic and are valid for relatively sparse graphs with an average degree that grows logarithmically with the number of nodes. We then analyze the stability of GCNs to small deformations of the random graph model. In contrast to previous studies of stability in discrete settings, our continuous setup allows us to provide more intuitive deformation-based metrics for understanding stability, which have proven useful for explaining the success of convolutional representations on Euclidean domains.

Sparse regularization is a central technique for both machine learning (to achieve supervised features selection or unsupervised mixture learning) and imaging sciences (to achieve super-resolution). Existing performance guaranties assume a separation of the spikes based on an ad-hoc (usually Euclidean) minimum distance condition, which ignores the geometry of the problem. In this article, we study the BLASSO (i.e. the off-the-grid version of l1 LASSO regularization) and show that the Fisher-Rao distance is the natural way to ensure and quantify support recovery, since it preserves the invariance of the problem under reparameterization. We prove that under mild regularity and curvature conditions, stable support identification is achieved even in the presence of randomized sub-sampled observations (which is the case in compressed sensing or learning scenario). On deconvolution problems, which are translation invariant, this generalizes to the multi-dimensional setting existing results of the literature. For more complex translation-varying problems, such as Laplace transform inversion, this gives the first geometry-aware guarantees for sparse recovery.

Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or \emph{equivariant} (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we prove new universality theorems. More precisely, we consider networks with a single hidden layer, obtained by summing channels formed by applying an equivariant linear operator, a pointwise non-linearity, and either an invariant or equivariant linear output layer. Recently, Maron et al. (2019) showed that by allowing higher-order tensorization inside the network, universal invariant GNNs can be obtained. As a first contribution, we propose an alternative proof of this result, which relies on the Stone-Weierstrass theorem for algebra of real-valued functions. Our main contribution is then an extension of this result to the \emph{equivariant} case, which appears in many practical applications but has been less studied from a theoretical point of view. The proof relies on a new generalized Stone-Weierstrass theorem for algebra of equivariant functions, which is of independent interest. Additionally, unlike many previous works that consider a fixed number of nodes, our results show that a GNN defined by a single set of parameters can approximate uniformly well a function defined on graphs of varying size.

In this paper, we address the problem of high-dimensional k-means clustering in a large-scale setting, i.e. for datasets that comprise a large number of items. Sketching techniques have already been used to deal with this “large-scale” issue, by compressing the whole dataset into a single vector of random nonlinear generalized moments from which the k centroids are then retrieved efficiently. However, this approach usually scales quadratically with the dimension; to cope with high-dimensional datasets, we show how to use fast structured random matrices to compute the sketching operator efficiently. This yields significant speed-ups and memory savings for high-dimensional data, while the clustering results are shown to be much more stable, both on artificial and real datasets.

We propose a new blind source separation algorithm based on mixtures of α-stable distributions. Complex symmetric α-stable distributions have been recently showed to better model audio signals in the time-frequency domain than classical Gaussian distributions thanks to their larger dynamic range. However, inference with these models is notoriously hard to perform because their probability density functions do not have a closed-form expression in general. Here, we introduce a novel method for estimating mixtures of α-stable distributions based on characteristic function matching. We apply this to the blind estimation of binary masks in individual frequency bands from multichannel convolutive audio mixtures. We show that the proposed method yields better separation performance than Gaussian-based binary-masking methods.

The Lloyd-Max algorithm is a classical approach to perform K-means clustering. Unfortunately, its cost becomes prohibitive as the training dataset grows large. We propose a compressive version of K-means (CKM), that estimates cluster centers from a sketch, i.e. from a drastically compressed representation of the training dataset. We demonstrate empirically that CKM performs similarly to Lloyd-Max, for a sketch size proportional to the number of centroids times the ambient dimension, and independent of the size of the original dataset. Given the sketch, the computational complexity of CKM is also independent of the size of the dataset. Unlike Lloyd-Max which requires several replicates, we further demonstrate that CKM is almost insensitive to initialization. For a large dataset of 107data points, we show that CKM can run two orders of magnitude faster than five replicates of Lloyd-Max, with similar clustering performance on artificial data. Finally, CKM achieves lower classification errors on handwritten digits classification.

Learning parameters from voluminous data can be prohibitive in terms of memory and computational requirements. We propose a "compressive learning" framework where we first sketch the data by computing random generalized moments of the underlying probability distribution, then estimate mixture model parameters from the sketch using an iterative algorithm analogous to greedy sparse signal recovery. We exemplify our framework with the sketched estimation of Gaussian Mixture Models (GMMs). We experimentally show that our approach yields results comparable to the classical Expectation-Maximization (EM) technique while requiring significantly less memory and fewer computations when the number of database elements is large. We report large-scale experiments in speaker verification, where our approach makes it possible to fully exploit a corpus of 1000 hours of speech signal to learn a universal background model at scales computationally inaccessible to EM.

Automatic music transcription (AMT) can be performed by deriving a pitch-time representation through decomposition of a spectrogram with a dictionary of pitch-labelled atoms. Typically, non-negative matrix factorisation (NMF) methods are used to decompose magnitude spectrograms. One atom is often used to represent each note. However, the spectrum of a note may change over time. Previous research considered this variability using different atoms to model specific parts of a note, or large dictionaries comprised of datapoints from the spectrograms of full notes. In this paper, the use of subspace modelling of note spectra is explored, with group sparsity employed as a means of coupling activations of related atoms into a pitched subspace. Stepwise and gradient-based methods for non-negative group sparse decompositions are proposed. Finally, a group sparse NMF approach is used to tune a generic harmonic subspace dictionary, leading to improved NMF-based AMT results.