arXiv subjects

Pierre Vandergheynst

Publications and source records attributed to Pierre Vandergheynst.

At least 55 records · Page 3Linked to original sources

arXiv2016

UNLocBoX: A MATLAB convex optimization toolbox for proximal-splitting methods

Convex optimization is an essential tool for machine learning, as many of its problems can be formulated as minimization problems of specific objective functions. While there is a large variety of algorithms available to solve convex problems, we can argue that it becomes more and more important to focus on efficient, scalable methods that can deal with big data. When the objective function can be written as a sum of "simple" terms, proximal splitting methods are a good choice. UNLocBoX is a MATLAB library that implements many of these methods, designed to solve convex optimization problems of the form $\min_{x \in \mathbb{R}^N} \sum_{n=1}^K f_n(x).$ It contains the most recent solvers such as FISTA, Douglas-Rachford, SDMM as well a primal dual techniques such as Chambolle-Pock and forward-backward-forward. It also includes an extensive list of common proximal operators that can be combined, allowing for a quick implementation of a large variety of convex problems.

cs.LG↗

arXiv2016

Structured Sequence Modeling with Graph Convolutional Recurrent Networks

This paper introduces Graph Convolutional Recurrent Network (GCRN), a deep learning model able to predict structured sequences of data. Precisely, GCRN is a generalization of classical recurrent neural networks (RNN) to data structured by an arbitrary graph. Such structured sequences can represent series of frames in videos, spatio-temporal measurements on a network of sensors, or random walks on a vocabulary graph for natural language modeling. The proposed model combines convolutional neural networks (CNN) on graphs to identify spatial structures and RNN to find dynamic patterns. We study two possible architectures of GCRN, and apply the models to two practical problems: predicting moving MNIST data, and modeling natural language with the Penn Treebank dataset. Experiments show that exploiting simultaneously graph spatial and dynamic information about data can improve both precision and learning speed.

stat.ML↗

arXiv2016

Source Localization on Graphs via l1 Recovery and Spectral Graph Theory

We cast the problem of source localization on graphs as the simultaneous problem of sparse recovery and diffusion kernel learning. An l1 regularization term enforces the sparsity constraint while we recover the sources of diffusion from a single snapshot of the diffusion process. The diffusion kernel is estimated by assuming the process to be as generic as the standard heat diffusion. We show with synthetic data that we can concomitantly learn the diffusion kernel and the sources, given an estimated initialization. We validate our model with cholera mortality and atmospheric tracer diffusion data, showing also that the accuracy of the solution depends on the construction of the graph from the data points.

cs.LG↗

arXiv2016

Multilinear Low-Rank Tensors on Graphs & Applications

We propose a new framework for the analysis of low-rank tensors which lies at the intersection of spectral graph theory and signal processing. As a first step, we present a new graph based low-rank decomposition which approximates the classical low-rank SVD for matrices and multi-linear SVD for tensors. Then, building on this novel decomposition we construct a general class of convex optimization problems for approximately solving low-rank tensor inverse problems, such as tensor Robust PCA. The whole framework is named as 'Multilinear Low-rank tensors on Graphs (MLRTG)'. Our theoretical analysis shows: 1) MLRTG stands on the notion of approximate stationarity of multi-dimensional signals on graphs and 2) the approximation error depends on the eigen gaps of the graphs. We demonstrate applications for a wide variety of 4 artificial and 12 real tensor datasets, such as EEG, FMRI, BCI, surveillance videos and hyperspectral images. Generalization of the tensor concepts to non-euclidean domain, orders of magnitude speed-up, low-memory requirement and significantly enhanced performance at low SNR are the key aspects of our framework.

cs.CV↗

arXiv2016

Compressive PCA for Low-Rank Matrices on Graphs

We introduce a novel framework for an approxi- mate recovery of data matrices which are low-rank on graphs, from sampled measurements. The rows and columns of such matrices belong to the span of the first few eigenvectors of the graphs constructed between their rows and columns. We leverage this property to recover the non-linear low-rank structures efficiently from sampled data measurements, with a low cost (linear in n). First, a Resrtricted Isometry Property (RIP) condition is introduced for efficient uniform sampling of the rows and columns of such matrices based on the cumulative coherence of graph eigenvectors. Secondly, a state-of-the-art fast low-rank recovery method is suggested for the sampled data. Finally, several efficient, parallel and parameter-free decoders are presented along with their theoretical analysis for decoding the low-rank and cluster indicators for the full data matrix. Thus, we overcome the computational limitations of the standard linear low-rank recovery methods for big datasets. Our method can also be seen as a major step towards efficient recovery of non- linear low-rank structures. For a matrix of size n X p, on a single core machine, our method gains a speed up of $p^2/k$ over Robust Principal Component Analysis (RPCA), where k << p is the subspace dimension. Numerically, we can recover a low-rank matrix of size 10304 X 1000, 100 times faster than Robust PCA.

cs.LG↗

arXiv2016

Locating Temporal Functional Dynamics of Visual Short-Term Memory Binding using Graph Modular Dirichlet Energy

Visual short-term memory binding tasks are a promising early marker for Alzheimer's disease (AD). To uncover functional deficits of AD in these tasks it is meaningful to first study unimpaired brain function. Electroencephalogram recordings were obtained from encoding and maintenance periods of tasks performed by healthy young volunteers. We probe the task's transient physiological underpinnings by contrasting shape only (Shape) and shape-colour binding (Bind) conditions, displayed in the left and right sides of the screen, separately. Particularly, we introduce and implement a novel technique named Modular Dirichlet Energy (MDE) which allows robust and flexible analysis of the functional network with unprecedented temporal precision. We find that connectivity in the Bind condition is less integrated with the global network than in the Shape condition in occipital and frontal modules during the encoding period of the right screen condition. Using MDE we are able to discern driving effects in the occipital module between 100-140ms, coinciding with the P100 visually evoked potential, followed by a driving effect in the frontal module between 140-180ms, suggesting that the differences found constitute an information processing difference between these modules. This provides temporally precise information over a heterogeneous population in promising tasks for the detection of AD.

q-bio.NC↗

arXiv2016

Towards stationary time-vertex signal processing

Graph-based methods for signal processing have shown promise for the analysis of data exhibiting irregular structure, such as those found in social, transportation, and sensor networks. Yet, though these systems are often dynamic, state-of-the-art methods for signal processing on graphs ignore the dimension of time, treating successive graph signals independently or taking a global average. To address this shortcoming, this paper considers the statistical analysis of time-varying graph signals. We introduce a novel definition of joint (time-vertex) stationarity, which generalizes the classical definition of time stationarity and the more recent definition appropriate for graphs. Joint stationarity gives rise to a scalable Wiener optimization framework for joint denoising, semi-supervised learning, or more generally inversing a linear operator, that is provably optimal. Experimental results on real weather data demonstrate that taking into account graph and time dimensions jointly can yield significant accuracy improvements in the reconstruction effort.

cs.LG↗

arXiv2016

Low-Rank Matrices on Graphs: Generalized Recovery & Applications

Many real world datasets subsume a linear or non-linear low-rank structure in a very low-dimensional space. Unfortunately, one often has very little or no information about the geometry of the space, resulting in a highly under-determined recovery problem. Under certain circumstances, state-of-the-art algorithms provide an exact recovery for linear low-rank structures but at the expense of highly inscalable algorithms which use nuclear norm. However, the case of non-linear structures remains unresolved. We revisit the problem of low-rank recovery from a totally different perspective, involving graphs which encode pairwise similarity between the data samples and features. Surprisingly, our analysis confirms that it is possible to recover many approximate linear and non-linear low-rank structures with recovery guarantees with a set of highly scalable and efficient algorithms. We call such data matrices as \textit{Low-Rank matrices on graphs} and show that many real world datasets satisfy this assumption approximately due to underlying stationarity. Our detailed theoretical and experimental analysis unveils the power of the simple, yet very novel recovery framework \textit{Fast Robust PCA on Graphs}

cs.CV↗

arXiv2016

Compressive Spectral Clustering

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.

cs.DS↗

arXiv2016

Random sampling of bandlimited signals on graphs

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.

cs.SI↗

arXiv2016

A Multiscale Pyramid Transform for Graph Signals

Multiscale transforms designed to process analog and discrete-time signals and images cannot be directly applied to analyze high-dimensional data residing on the vertices of a weighted graph, as they do not capture the intrinsic geometric structure of the underlying graph data domain. In this paper, we adapt the Laplacian pyramid transform for signals on Euclidean domains so that it can be used to analyze high-dimensional data residing on the vertices of a weighted graph. Our approach is to study existing methods and develop new methods for the four fundamental operations of graph downsampling, graph reduction, and filtering and interpolation of signals on graphs. Equipped with appropriate notions of these operations, we leverage the basic multiscale constructs and intuitions from classical signal processing to generate a transform that yields both a multiresolution of graphs and an associated multiresolution of a graph signal on the underlying sequence of graphs.

cs.IT↗

arXiv2016

GSPBOX: A toolbox for signal processing on graphs

This document introduces the Graph Signal Processing Toolbox (GSPBox) a framework that can be used to tackle graph related problems with a signal processing approach. It explains the structure and the organization of this software. It also contains a general description of the important modules.

cs.IT↗

arXiv2016

Graph Based Sinogram Denoising for Tomographic Reconstructions

Limited data and low dose constraints are common problems in a variety of tomographic reconstruction paradigms which lead to noisy and incomplete data. Over the past few years sinogram denoising has become an essential pre-processing step for low dose Computed Tomographic (CT) reconstructions. We propose a novel sinogram denoising algorithm inspired by the modern field of signal processing on graphs. Graph based methods often perform better than standard filtering operations since they can exploit the signal structure. This makes the sinogram an ideal candidate for graph based denoising since it generally has a piecewise smooth structure. We test our method with a variety of phantoms and different reconstruction methods. Our numerical study shows that the proposed algorithm improves the performance of analytical filtered back-projection (FBP) and iterative methods ART (Kaczmarz) and SIRT (Cimmino).We observed that graph denoised sinogram always minimizes the error measure and improves the accuracy of the solution as compared to regular reconstructions.

cs.CV↗

arXiv2016

Global and Local Uncertainty Principles for Signals on Graphs

Uncertainty principles such as Heisenberg's provide limits on the time-frequency concentration of a signal, and constitute an important theoretical tool for designing and evaluating linear signal transforms. Generalizations of such principles to the graph setting can inform dictionary design for graph signals, lead to algorithms for reconstructing missing information from graph signals via sparse representations, and yield new graph analysis tools. While previous work has focused on generalizing notions of spreads of a graph signal in the vertex and graph spectral domains, our approach is to generalize the methods of Lieb in order to develop uncertainty principles that provide limits on the concentration of the analysis coefficients of any graph signal under a dictionary transform whose atoms are jointly localized in the vertex and graph spectral domains. One challenge we highlight is that due to the inhomogeneity of the underlying graph data domain, the local structure in a single small region of the graph can drastically affect the uncertainty bounds for signals concentrated in different regions of the graph, limiting the information provided by global uncertainty principles. Accordingly, we suggest a new way to incorporate a notion of locality, and develop local uncertainty principles that bound the concentration of the analysis coefficients of each atom of a localized graph spectral filter frame in terms of quantities that depend on the local structure of the graph around the center vertex of the given atom. Finally, we demonstrate how our proposed local uncertainty measures can improve the random sampling of graph signals.

stat.ML↗

arXiv2016

Learning Laplacian Matrix in Smooth Graph Signal Representations

The construction of a meaningful graph plays a crucial role in the success of many graph-based representations and algorithms for handling structured data, especially in the emerging field of graph signal processing. However, a meaningful graph is not always readily available from the data, nor easy to define depending on the application domain. In particular, it is often desirable in graph signal processing applications that a graph is chosen such that the data admit certain regularity or smoothness on the graph. In this paper, we address the problem of learning graph Laplacians, which is equivalent to learning graph topologies, such that the input data form graph signals with smooth variations on the resulting topology. To this end, we adopt a factor analysis model for the graph signals and impose a Gaussian probabilistic prior on the latent variables that control these signals. We show that the Gaussian prior leads to an efficient representation that favors the smoothness property of the graph signals. We then propose an algorithm for learning graphs that enforces such property and is based on minimizing the variations of the signals on the learned graph. Experiments on both synthetic and real world data demonstrate that the proposed graph learning framework can efficiently infer meaningful graph topologies from signal observations under the smoothness prior.

cs.LG↗

arXiv2016

Principal Patterns on Graphs: Discovering Coherent Structures in Datasets

Graphs are now ubiquitous in almost every field of research. Recently, new research areas devoted to the analysis of graphs and data associated to their vertices have emerged. Focusing on dynamical processes, we propose a fast, robust and scalable framework for retrieving and analyzing recurring patterns of activity on graphs. Our method relies on a novel type of multilayer graph that encodes the spreading or propagation of events between successive time steps. We demonstrate the versatility of our method by applying it on three different real-world examples. Firstly, we study how rumor spreads on a social network. Secondly, we reveal congestion patterns of pedestrians in a train station. Finally, we show how patterns of audio playlists can be used in a recommender system. In each example, relevant information previously hidden in the data is extracted in a very efficient manner, emphasizing the scalability of our method. With a parallel implementation scaling linearly with the size of the dataset, our framework easily handles millions of nodes on a single commodity server.

cs.SI↗

arXiv2016

Fast Robust PCA on Graphs

Mining useful clusters from high dimensional data has received significant attention of the computer vision and pattern recognition community in the recent years. Linear and non-linear dimensionality reduction has played an important role to overcome the curse of dimensionality. However, often such methods are accompanied with three different problems: high computational complexity (usually associated with the nuclear norm minimization), non-convexity (for matrix factorization methods) and susceptibility to gross corruptions in the data. In this paper we propose a principal component analysis (PCA) based solution that overcomes these three issues and approximates a low-rank recovery method for high dimensional datasets. We target the low-rank recovery by enforcing two types of graph smoothness assumptions, one on the data samples and the other on the features by designing a convex optimization problem. The resulting algorithm is fast, efficient and scalable for huge datasets with O(nlog(n)) computational complexity in the number of data samples. It is also robust to gross corruptions in the dataset as well as to the model parameters. Clustering experiments on 7 benchmark datasets with different types of corruptions and background separation experiments on 3 video datasets show that our proposed model outperforms 10 state-of-the-art dimensionality reduction models. Our theoretical analysis proves that the proposed model is able to recover approximate low-rank representations with a bounded error for clusterable data.

cs.CV↗

arXiv2016

Song Recommendation with Non-Negative Matrix Factorization and Graph Total Variation

This work formulates a novel song recommender system as a matrix completion problem that benefits from collaborative filtering through Non-negative Matrix Factorization (NMF) and content-based filtering via total variation (TV) on graphs. The graphs encode both playlist proximity information and song similarity, using a rich combination of audio, meta-data and social features. As we demonstrate, our hybrid recommendation system is very versatile and incorporates several well-known methods while outperforming them. Particularly, we show on real-world data that our model overcomes w.r.t. two evaluation metrics the recommendation of models solely based on low-rank information, graph-based information or a combination of both.

stat.ML↗