Nicolas Tremblay

Nicolas Tremblay, Benjamin Ricaud, Filippo Maria Bianchi

We propose the Tikhonov layer, a graph neural network layer that is interpretable by design: once trained, its learned parameters directly reveal which node features and which aspects of the graph topology were leveraged for prediction. In practice, the layer's propagation matrix takes the closed-form $R = (p(L)+Q)^{-1} Q$, where $L$ is the normalized graph Laplacian, $Q = diag(q_1,...,q_n)$ a learnable diagonal matrix of positive node-importance scores, and $p(\cdot)$ a learnable polynomial. For any input feature $x$, the layer output $Rx$ is the exact minimizer of a generalized graph Tikhonov problem that trades off node-level data fidelity against a topology-driven regularization penalty. The learned pair $\{\{q_i\},p\}$ constitutes a built-in explanation: large $q_i$ indicates that node $i$'s own features drive the prediction, while small $q_i$ signals reliance on the local graph topology; the shape of $p$ reveals whether homophily, heterophily, or a band-pass response is exploited. Expressivity is preserved by routing complexity through a dedicated, arbitrarily deep Q-network that produces the importance scores, while the Tikhonov layer itself remains transparent. We prove that distinct node-importance matrices yield distinct propagation operators, structurally coupling the explanation to the computation. Additionally, the Tikhonov layer provides, in a single layer, a global receptive field, mitigating both oversmoothing and oversquashing. Experiments on standard graph classification benchmarks confirm that the model matches (and sometimes outperforms) opaque baselines while producing interpretable and faithful explanations.

Simon Barthelmé, Nicolas Tremblay, Pierre-Olivier Amblard

Discrete Determinantal Point Processes (DPPs) have a wide array of potential applications for subsampling datasets. They are however held back in some cases by the high cost of sampling. In the worst-case scenario, the sampling cost scales as O(n^3) where n is the number of elements of the ground set. A popular workaround to this prohibitive cost is to sample DPPs defined by low-rank kernels. In such cases, the cost of standard sampling algorithms scales as O(np^2 + nm^2) where m is the (average) number of samples of the DPP (usually m << n) and p the rank of the kernel used to define the DPP (m \leq p \leq n). The first term, O(np^2), comes from a SVD-like step. We focus here on the second term of this cost, O(nm^2), and show that it can be brought down to O(nm + m^3 log m) without loss on the sampling's exactness. In practice, we observe very substantial speedups compared to the classical algorithm as soon as n > 1000. The algorithm described here is a close variant of the standard algorithm for sampling continuous DPPs, and uses rejection sampling. In the specific case of projection DPPs, we also show that any additional sample can be drawn in time O(m^3 log m). Finally, an interesting by-product of the analysis is that a realisation from a DPP is typically contained in a subset of size O(m log m) formed using leverage score i.i.d. sampling.

Matthieu Cordonnier, Nicolas Keriven, Nicolas Tremblay et al.

We study the convergence of message passing graph neural networks on random graph models to their continuous counterpart 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.

Lauren Anderson, Lucas Chatelain, Nicolas Tremblay et al.

The mechanosensory system of teeth is currently believed to partly rely on Odontoblast cells stimulation by fluid flow through a porosity network extending through dentin. Visualizing the smallest sub-microscopic porosity vessels therefore requires the highest achievable resolution from confocal fluorescence microscopy, the current gold standard. This considerably limits the extent of the field of view to very small sample regions. To overcome this limitation, we tested different deep learning (DL) super-resolution (SR) models to allow faster experimental acquisitions of lower resolution images and restore optimal image quality by post-processing. Three supervised 2D SR models (RCAN, pix2pix, FSRCNN) and one unsupervised (CycleGAN) were applied to a unique set of experimentally paired high- and low-resolution confocal images acquired with different sampling schemes, resulting in a pixel size increase of x2, x4, x8. Model performance was quantified using a broad set of similarity and distribution-based image quality assessment (IQA) metrics, which yielded inconsistent results that mostly contradicted our visual perception. This raises the question of the relevance of such generic metrics to efficiently target the specific structure of dental porosity. To resolve this conflicting information, the generated SR images were segmented taking into account the specific scales and morphology of the porosity network and analysed by comparing connected components. Additionally, the capacity of the SR models to preserve 3D porosity connectivity throughout the confocal image stacks was evaluated using graph analysis. This biology-driven assessment allowed a far better mechanistic interpretation of SR performance, highlighting differences in model sensitivity to weak intensity features and the impact of non-linearity in image generation, which explains the failure of standard IQA metrics.

Lorenzo Dall'Amico, Romain Couillet, Nicolas Tremblay

This article unveils a new relation between the Nishimori temperature parametrizing a distribution P and the Bethe free energy on random Erdos-Renyi graphs with edge weights distributed according to P. Estimating the Nishimori temperature being a task of major importance in Bayesian inference problems, as a practical corollary of this new relation, a numerical method is proposed to accurately estimate the Nishimori temperature from the eigenvalues of the Bethe Hessian matrix of the weighted graph. The algorithm, in turn, is used to propose a new spectral method for node classification in weighted (possibly sparse) graphs. The superiority of the method over competing state-of-the-art approaches is demonstrated both through theoretical arguments and real-world data experiments.

Hashem Ghanem, Nicolas Keriven, Nicolas Tremblay

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 that the graphlet kernel for the same, or better, accuracy.

Lorenzo Dall'Amico, Romain Couillet, Nicolas Tremblay

This article considers the problem of community detection in sparse dynamical graphs in which the community structure evolves over time. A fast spectral algorithm based on an extension of the Bethe-Hessian matrix is proposed, which benefits from the positive correlation in the class labels and in their temporal evolution and is designed to be applicable to any dynamical graph with a community structure. Under the dynamical degree-corrected stochastic block model, in the case of two classes of equal size, we demonstrate and support with extensive simulations that our proposed algorithm is capable of making non-trivial community reconstruction as soon as theoretically possible, thereby reaching the optimal detectability threshold and provably outperforming competing spectral methods.

Lorenzo Dall'Amico, Romain Couillet, Nicolas Tremblay

This article considers spectral community detection in the regime of sparse networks with heterogeneous degree distributions, for which we devise an algorithm to efficiently retrieve communities. Specifically, we demonstrate that a conveniently parametrized form of regularized Laplacian matrix can be used to perform spectral clustering in sparse networks, without suffering from its degree heterogeneity. Besides, we exhibit important connections between this proposed matrix and the now popular non-backtracking matrix, the Bethe-Hessian matrix, as well as the standard Laplacian matrix. Interestingly, as opposed to competitive methods, our proposed improved parametrization inherently accounts for the hardness of the classification problem. These findings are summarized under the form of an algorithm capable of both estimating the number of communities and achieving high-quality community reconstruction.

Lorenzo Dall'Amico, Romain Couillet, Nicolas Tremblay

Regularization of the classical Laplacian matrices was empirically shown to improve spectral clustering in sparse networks. It was observed that small regularizations are preferable, but this point was left as a heuristic argument. In this paper we formally determine a proper regularization which is intimately related to alternative state-of-the-art spectral techniques for sparse graphs.

Nicolas Tremblay, Andreas Loukas

Spectral clustering refers to a family of unsupervised learning algorithms that compute a spectral embedding of the original data based on the eigenvectors of a similarity graph. This non-linear transformation of the data is both the key of these algorithms' success and their Achilles heel: forming a graph and computing its dominant eigenvectors can indeed be computationally prohibitive when dealing with more that a few tens of thousands of points. In this paper, we review the principal research efforts aiming to reduce this computational cost. We focus on methods that come with a theoretical control on the clustering performance and incorporate some form of sampling in their operation. Such methods abound in the machine learning, numerical linear algebra, and graph signal processing literature and, amongst others, include Nyström-approximation, landmarks, coarsening, coresets, and compressive spectral clustering. We present the approximation guarantees available for each and discuss practical merits and limitations. Surprisingly, despite the breadth of the literature explored, we conclude that there is still a gap between theory and practice: the most scalable methods are only intuitively motivated or loosely controlled, whereas those that come with end-to-end guarantees rely on strong assumptions or enable a limited gain of computation time.

Lorenzo Dall'Amico, Romain Couillet, Nicolas Tremblay

Spectral clustering is one of the most popular, yet still incompletely understood, methods for community detection on graphs. This article studies spectral clustering based on the Bethe-Hessian matrix $H_r = (r^2-1)I_n + D-rA$ for sparse heterogeneous graphs (following the degree-corrected stochastic block model) in a two-class setting. For a specific value $r = ζ$, clustering is shown to be insensitive to the degree heterogeneity. We then study the behavior of the informative eigenvector of $H_ζ$ and, as a result, predict the clustering accuracy. The article concludes with an overview of the generalization to more than two classes along with extensive simulations on synthetic and real networks corroborating our findings.

Nicolas Tremblay, Simon Barthelmé, Pierre-Olivier Amblard

When faced with a data set too large to be processed all at once, an obvious solution is to retain only part of it. In practice this takes a wide variety of different forms, and among them "coresets" are especially appealing. A coreset is a (small) weighted sample of the original data that comes with the following guarantee: a cost function can be evaluated on the smaller set instead of the larger one, with low relative error. For some classes of problems, and via a careful choice of sampling distribution (based on the so-called "sensitivity" metric), iid random sampling has turned to be one of the most successful methods for building coresets efficiently. However, independent samples are sometimes overly redundant, and one could hope that enforcing diversity would lead to better performance. The difficulty lies in proving coreset properties in non-iid samples. We show that the coreset property holds for samples formed with determinantal point processes (DPP). DPPs are interesting because they are a rare example of repulsive point processes with tractable theoretical properties, enabling us to prove general coreset theorems. We apply our results to both the k-means and the linear regression problems, and give extensive empirical evidence that the small additional computational cost of DPP sampling comes with superior performance over its iid counterpart. Of independent interest, we also provide analytical formulas for the sensitivity in the linear regression and 1-means cases.

Simon Barthelmé, Pierre-Olivier Amblard, Nicolas Tremblay

Determinantal Point Processes (DPPs) are popular models for point processes with repulsion. They appear in numerous contexts, from physics to graph theory, and display appealing theoretical properties. On the more practical side of things, since DPPs tend to select sets of points that are some distance apart (repulsion), they have been advocated as a way of producing random subsets with high diversity. DPPs come in two variants: fixed-size and varying-size. A sample from a varying-size DPP is a subset of random cardinality, while in fixed-size "$k$-DPPs" the cardinality is fixed. The latter makes more sense in many applications, but unfortunately their computational properties are less attractive, since, among other things, inclusion probabilities are harder to compute. In this work we show that as the size of the ground set grows, $k$-DPPs and DPPs become equivalent, meaning that their inclusion probabilities converge. As a by-product, we obtain saddlepoint formulas for inclusion probabilities in $k$-DPPs. These turn out to be extremely accurate, and suffer less from numerical difficulties than exact methods do. Our results also suggest that $k$-DPPs and DPPs also have equivalent maximum likelihood estimators. Finally, we obtain results on asymptotic approximations of elementary symmetric polynomials which may be of independent interest.

Nicolas Tremblay, Simon Barthelme, Pierre-Olivier Amblard

In this technical report, we discuss several sampling algorithms for Determinantal Point Processes (DPP). DPPs have recently gained a broad interest in the machine learning and statistics literature as random point processes with negative correlation, i.e., ones that can generate a "diverse" sample from a set of items. They are parametrized by a matrix $\mathbf{L}$, called $L$-ensemble, that encodes the correlations between items. The standard sampling algorithm is separated in three phases: 1/~eigendecomposition of $\mathbf{L}$, 2/~an eigenvector sampling phase where $\mathbf{L}$'s eigenvectors are sampled independently via a Bernoulli variable parametrized by their associated eigenvalue, 3/~a Gram-Schmidt-type orthogonalisation procedure of the sampled eigenvectors. In a naive implementation, the computational cost of the third step is on average $\mathcal{O}(Nμ^3)$ where $μ$ is the average number of samples of the DPP. We give an algorithm which runs in $\mathcal{O}(Nμ^2)$ and is extremely simple to implement. If memory is a constraint, we also describe a dual variant with reduced memory costs. In addition, we discuss implementation details often missing in the literature.

Nicolas Tremblay, Simon Barthelme, Pierre-Olivier Amblard

We consider the problem of sampling k-bandlimited graph signals, ie, linear combinations of the first k graph Fourier modes. We know that a set of k nodes embedding all k-bandlimited signals always exists, thereby enabling their perfect reconstruction after sampling. Unfortunately, to exhibit such a set, one needs to partially diagonalize the graph Laplacian, which becomes prohibitive at large scale. We propose a novel strategy based on determinantal point processes that side-steps partial diagonalisation and enables reconstruction with only O(k) samples. While doing so, we exhibit a new general algorithm to sample determinantal process, faster than the state-of-the-art algorithm by an order k.

Nicolas Tremblay, Pierre-Olivier Amblard, Simon Barthelmé

We present a new random sampling strategy for k-bandlimited signals defined on graphs, based on determinantal point processes (DPP). For small graphs, ie, in cases where the spectrum of the graph is accessible, we exhibit a DPP sampling scheme that enables perfect recovery of bandlimited signals. For large graphs, ie, in cases where the graph's spectrum is not accessible, we investigate, both theoretically and empirically, a sub-optimal but much faster DPP based on loop-erased random walks on the graph. Preliminary experiments show promising results especially in cases where the number of measurements should stay as small as possible and for graphs that have a strong community structure. Our sampling scheme is efficient and can be applied to graphs with up to $10^6$ nodes.

Nicolas Keriven, Nicolas Tremblay, Yann Traonmilin et al.

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 cen-troids 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 10^7 data 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.

Nicolas Tremblay, Gilles Puy, Remi Gribonval et al.

Spectral clustering has become a popular technique due to its high performance in many contexts. It comprises three main steps: create a similarity graph between N objects to cluster, compute the first k eigenvectors of its Laplacian matrix to define a feature vector for each object, and run k-means on these features to separate objects into k classes. Each of these three steps becomes computationally intensive for large N and/or k. We propose to speed up the last two steps based on recent results in the emerging field of graph signal processing: graph filtering of random signals, and random sampling of bandlimited graph signals. We prove that our method, with a gain in computation time that can reach several orders of magnitude, is in fact an approximation of spectral clustering, for which we are able to control the error. We test the performance of our method on artificial and real-world network data.

Gilles Puy, Nicolas Tremblay, Rémi Gribonval et al.

We study the problem of sampling k-bandlimited signals on graphs. We propose two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure, and its performance depends on a parameter called the graph coherence. On the contrary, the second strategy is adaptive but yields optimal results. Indeed, no more than O(k log(k)) measurements are sufficient to ensure an accurate and stable recovery of all k-bandlimited signals. This second strategy is based on a careful choice of the sampling distribution, which can be estimated quickly. Then, we propose a computationally efficient decoder to reconstruct k-bandlimited signals from their samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we conduct several experiments to test these techniques.