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.

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 $\ll$ 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 $>$ 1, 000. 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.

We introduce new smoothing estimators for complex signals on graphs, based on a recently studied Determinantal Point Process (DPP). These estimators are built from subsets of edges and nodes drawn according to this DPP, making up trees and unicycles, i.e., connected components containing exactly one cycle. We provide a Julia implementation of these estimators and study their performance when applied to a ranking problem.

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.

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 well parametrized form of regularized Laplacian matrices 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.

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.

Another facet of the elegant link between random processes on graphs and Laplacian-based numerical linear algebra is uncovered: based on random spanning forests, novel Monte-Carlo estimators for graph signal smoothing are proposed. These random forests are sampled efficiently via a variant of Wilson's algorithm -in time linear in the number of edges. The theoretical variance of the proposed estimators are analyzed, and their application to several problems are considered, such as Tikhonov denoising of graph signals or semi-supervised learning for node classification on graphs.

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.

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 Hr= (r^2−1)In+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.

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.

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.

BACKGROUND: Structural interaction frequency matrices between all genome loci are now experimentally achievable thanks to high-throughput chromosome conformation capture technologies. This ensues a new methodological challenge for computational biology which consists in objectively extracting from these data the structural motifs characteristic of genome organisation. RESULTS: We deployed the fast multi-scale community mining algorithm based on spectral graph wavelets to characterise the networks of intra-chromosomal interactions in human cell lines. We observed that there exist structural domains of all sizes up to chromosome length and demonstrated that the set of structural communities forms a hierarchy of chromosome segments. Hence, at all scales, chromosome folding predominantly involves interactions between neighbouring sites rather than the formation of links between distant loci. CONCLUSIONS: Multi-scale structural decomposition of human chromosomes provides an original framework to question structural organisation and its relationship to functional regulation across the scales. By construction the proposed methodology is independent of the precise assembly of the reference genome and is thus directly applicable to genomes whose assembly is not fully determined.

We build upon recent advances in graph signal processing to propose a faster spectral clustering algorithm. Indeed, classical spectral clustering is based on the computation of the first k eigenvectors of the similarity matrix' Laplacian, whose computation cost, even for sparse matrices, becomes prohibitive for large datasets. We show that we can estimate the spectral clustering distance matrix without computing these eigenvectors: by graph filtering random signals. Also, we take advantage of the stochasticity of these random vectors to estimate the number of clusters k. We compare our method to classical spectral clustering on synthetic data, and show that it reaches equal performance while being faster by a factor at least two for large datasets.

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.

Spatial information related to vegetation status and soil properties is needed in precision farming, especially early in the growing season. At these stages, vegetation has already emerged while soil is also visible in multispectral EO images. In this paper, linear spectral unmixing is applied to an 8 bands WorldView-2 image to extract information on both vegetation and soil acquired at in-season nitrogen sidedress stage in 2010 and 2011 for four corn fields located in the Montérégie region of Quebec, Canada. EO derived soil properties were strongly correlated to ECa. Correlation between dark soil abundance and ECareached R=0.9 and correlation between bright soil abundance and ECawas about R=-0.7. Vegetation abundance for combined data of several fields was better correlated to measured biomass than NDVI and SAVI. The possibility to get valuable soil and plant information from a single multispectral image offers an interesting cost reduction opportunity for precision farming applications.

Sensor deployments in large buildings allow the administrators to supervise the building infrastructure and identify abnormalities. Nevertheless, the numerous data streams reported by the increasing number of sensors overwhelm the building administrators. We propose a methodology that assists them to identify abnormal devices usages. The proposed method takes advantage of Ensemble Empirical Mode Decomposition (E-EMD) to uncover the patterns of power-draw signals, thereby enabling us to estimate the intrinsic inter-device correlations. By monitoring the devices correlations over time we compute the usual usage of the devices and report the devices that deviate from their normal usage. Our evaluation with 10 weeks of real data shows the efficiency of the proposed method to uncover the devices intrinsic relationships and detect peculiar events that require the administrators attention.

A typical large building contains thousands of sensors, monitoring the HVAC system, lighting, and other operational sub-systems. With the increased push for operational efficiency, operators are relying more on historical data processing to uncover opportunities for energy-savings. However, they are overwhelmed with the deluge of data and seek more efficient ways to identify potential problems. In this paper, we present a new approach called the Strip, Bind and Search (SBS); a method for uncovering abnormal equipment behavior and in-concert usage patterns. SBS uncovers relationships between devices and constructs a model for their usage pattern relative to other devices. It then flags deviations from the model. We run SBS on a set of building sensor traces; each containing hundred sensors reporting data flows over 18 weeks from two separate buildings with fundamentally different infrastructures. We demonstrate that, in many cases, SBS uncovers misbehavior corresponding to inefficient device usage that leads to energy waste. The average waste uncovered is as high as 2500~kWh per device.

This study aims at using forward model simulations and ground-measurements (biophysical and spectral) to estimate chlorophyll concentration from hyperspectral data and imagery. Hence, intensive field campaigns were organized during the growing seasons of 2000, 2004, and 2005 in order to collect ground spectra and corresponding leaf chlorophyll content values, and crop growth status, as well as CASI (Compact Airborne Spectrographic Imager) hyperspectral images. Acquisition dates were planned to coincide with different phenological development stages, to monitor temporal changes in crop biophysical attributes. Field spectral measurements collected were used to test and evaluate several combined indices for chlorophyll determination using hyperspectral imagery. Several index combinations were investigated using both PROSPECT-SAILH canopy simulated spectra and field measured reflectances. The relationships between leaf chlorophyll content and combined optical indices showed similar trends for both PROSPECT-SAILH simulated data and ground measured datasets. The dataset used showed that crop type had a clear influence on the establishment of predictive equations as well as on their validation. In addition to generating different predictive equations, corn and wheat data yielded contrasting agreement between estimated and measured chlorophyll contents even for the same predictive algorithm. Among the set of indices tested in this study, index combinations MCARI/OSAVI, TCI/OSAVI, MTCI/MSAVI, and R-M/MSAVI were found relatively consistent and more stable as estimators of crop chlorophyll content.

The conventional pixel-oriented classification is the most commonly used approach in remote sensing for land use product extraction. The object-oriented classification based on the image segmentation is an alternative, which uses the pixel context, texture and shapes, in addition to their spectral characteristics. This paper reports on a comparative study between supervised pixel-oriented and object-oriented classifications in a precision agriculture context using three hyperspectral images. The images were acquired with the Compact Airborne Spectrographic Imager (CASI) sensor at three different altitudes, providing three different spatial resolutions: 1, 2 and 4 m. Pixel-oriented classifications were carried out using the maximum likelihood algorithm, and object-oriented classifications with a hierarchical segmentation and nearest neighbor classifier. The raw CASI data were transformed to absolute ground reflectance using calibration coefficients determined in the laboratory and the CAM5S radiative transfer code for atmospheric corrections. After segmentation, statistical comparison on the mean difference to neighbor objects confirmed that the segments had minimum mixing effects in respect to other segmentation levels and neighboring ground entities. After accuracy analysis on the classifications, the segmentation process allowed for the use of a spatially coarser hyperspectral image (4 m with kappa of 0.8268) to achieve better results than pixel oriented classification of spatially finer hyperspectral image (1 m with kappa of 0.7730), in the task of delineating agricultural classes.

This paper examines the use of simulated and measured canopy reflectance for chlorophyll estimation over crop canopies. Field spectral measurements were collected over corn and wheat canopies in different intensive field campaigns organized during the growing seasons of 2004 and 2005. They were used to test and evaluate several combined indices for chlorophyll determination using hyperspectral imagery (Compact Airborne Spectrographic Imager). Several index combinations were investigated using both PROSPECT–SAILH canopy simulated spectra and field-measured reflectances. The relationships between leaf chlorophyll content and combined optical indices have shown similar trends for both PROSPECT–SAILH simulated data and ground-measured data sets, which indicates that both spectral measurements and radiative transfer models hold comparable potential for the quantitative retrieval of crop foliar pigments. The data set used has shown that crop type had a clear influence on the establishment of predictive equations as well as on their validation. In addition to generating different predictive equations, corn and wheat data yielded contrasting agreement between estimated and measured chlorophyll contents even for the same predictive algorithm. Among the set of indices tested in this paper, index combinations like Modified Chlorophyll Absorption Ratio Index/Optimized Soil-Adjusted Vegetation Index (OSAVI), Triangular Chlorophyll Index/OSAVI, Moderate Resolution Imaging Spectrometer Terrestrial Chlorophyll Index/Improved Soil-Adjusted Vegetation Index (MSAVI), and Red-Edge Model/MSAVI seem to be relatively consistent and more stable as estimators of crop chlorophyll content.

This study aims at using forward model simulations and ground-measurements (biophysical and spectral) to estimate chlorophyll concentration from hyperspectral data and imagery. Its specific objectives were: (i) to evaluate various combinations of indices as estimators of chlorophyll content from simulated spectra (PROSPECT and SAILH); (ii) to establish chlorophyll predictive equations using spectral indices determined from field spectra and corresponding chlorophyll concentrations; (iii) to assess the effect of crop type (corn and wheat) on these relationships; and (iv) to validate and compare the indices' prediction capability using hyperspectral images and ground truth measurements. Hence, intensive field campaigns were organized during the growing seasons of 2000, 2004, and 2005 in order to collect ground spectra and corresponding leaf chlorophyll content values as well as crop growth measures. The relationships between leaf chlorophyll content and combined optical indices have shown similar trends for both PROSPECT- SAILH simulated data and ground measured datasets, indicating that both spectral measurements and radiative transfer models hold comparable potential for quantitative retrieval of crop foliar pigments. The dataset used showed that crop type had a clear influence on the establishment of predictive equations as well as on their validation. Moreover, corn and wheat data have led to contrasting agreement between estimated and measured chlorophyll contents even for the same predictive algorithm. Indices TCARI/TRDVI and TCI/TRDVI seem to be relatively consistent and more stable as estimators of crop chlorophyll content.

Soil electrical conductivity (SEC) surveys have recently become available for the characterization of soil patterns within fields. The question arises as to whether or not SEC maps can be used to drive site-specific nitrogen (N) fertilizer applications during the growing season. Corn was grown with N fertilization treatments ranging from 0 to 150 kg N/ha at topdressing. CASI airborne images were collected using a multispectral mode of operation, with 1 m spatial resolution and computed to provide an Optimized Soil-Adjusted Vegetation Index (OSAVI). Low SEC were related to high OSAVI. There were neither effects of N fertilization on OSAVI nor interaction with SEC properties. It is concluded that (1) N application at topdressing may not impact corn growth and (2) that variable rate N application can hardly be based on SEC patterns.

This work provides a description of the investigations conducted to assess the detection of chlorophyll fluorescence from hyperspectral CASI data. The viability of retrieval of solar-induced fluorescence through airborne imaging spectrometer measurements of radiance of targets under natural illumination is studied. A method based on in-filling of fluorescence signals in atmospheric oxygen absorption lines is applied to study sites of corn crop grown under different stress conditions due to variation in nitrogen treatment. Results of the relationships found between measurements of laser-induced fluorescence and chlorophyll concentration at the ground level with the in-filling of the 762 nm oxygen band and optical indices calculated from CASI imagery R/sub 685//R/sub 655/, derivative D/sub 730//D/sub 706/, and the double-peak derivative f/spl bsol/reflectance index Dpi (D/sub 688//spl middot/D/sub 710/ )/ D/sub 697//sup 2/ are presented.

In this study, impacts of vector quantization compression on prediction of leaf chlorophyll content of crops for the application to precision agriculture were evaluated. The compression algorithm tested in this paper is called successive approximation multi-stage vector quantization (SAMVQ). The hyperspectral data used were acquired by CASI over corn fields at L' Acadie experimental farm (Agriculture and Agri-Food Canada) during the summer of 2000. Nine zones in the corn fields with different fertilization levels (no fertilization, intermediate fertilization, and over-fertilization) were used to evaluate the difference between the leaf chlorophyll contents obtained from original and reconstructed reflectance data cubes. The root mean square errors (RMSEs) and the correlations between the chlorophyll content derived from the original data cube and that derived from the reconstructed data cubes were calculated in the nine zones. The spatial variability of chlorophyll content in the nine zones was also examined for the images of chlorophyll content created from the original and reconstructed reflectance data cubes. The results show that the chlorophyll content image created from the reconstructed reflectance data cube corresponding to SAMVQ with a compression ratio of 20 maintains good agreement with that derived from the original reflectance data cube in the nine zones in terms of estimated pigment mean and spatial variability. As a result, SAMVQ with a compression ratio of 20 is considered acceptable for the retrieval of crop chlorophyll content from CASI hyperspectral data for agriculture corn crops.