There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Research in graph signal processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper, we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing, along with a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas. We then summarize recent advances in developing basic GSP tools, including methods for sampling, filtering, or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning.

Graph Signal Processing: Overview, Challenges, and Applications

Recent deep networks that directly handle points in a point set, e.g., PointNet, have been state-of-the-art for supervised learning tasks on point clouds such as classification and segmentation. In this work, a novel end-to-end deep auto-encoder is proposed to address unsupervised learning challenges on point clouds. On the encoder side, a graph-based enhancement is enforced to promote local structures on top of PointNet. Then, a novel folding-based decoder deforms a canonical 2D grid onto the underlying 3D object surface of a point cloud, achieving low reconstruction errors even for objects with delicate structures. The proposed decoder only uses about 7% parameters of a decoder with fully-connected neural networks, yet leads to a more discriminative representation that achieves higher linear SVM classification accuracy than the benchmark. In addition, the proposed decoder structure is shown, in theory, to be a generic architecture that is able to reconstruct an arbitrary point cloud from a 2D grid. Our code is available at http://www.merl.com/research/license#FoldingNet

/pdf/foldingnet-point-cloud-auto-encoder-via-deep-grid-2qgev9oz4q.pdf

FoldingNet: Point Cloud Auto-Encoder via Deep Grid Deformation

We propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that perfect recovery is possible for graph signals bandlimited under the graph Fourier transform. The sampled signal coefficients form a new graph signal, whose corresponding graph structure preserves the first-order difference of the original graph signal. For general graphs, an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise; for graphs whose graph Fourier transforms are frames with maximal robustness to erasures as well as for Erdős-Renyi graphs, random sampling leads to perfect recovery with high probability. We further establish the connection to the sampling theory of finite discrete-time signal processing and previous work on signal recovery on graphs. To handle full-band graph signals, we propose a graph filter bank based on sampling theory on graphs. Finally, we apply the proposed sampling theory to semi-supervised classification of online blogs and digit images, where we achieve similar or better performance with fewer labeled samples compared to previous work.

/pdf/discrete-signal-processing-on-graphs-sampling-theory-54wtrrynr1.pdf

Discrete Signal Processing on Graphs: Sampling Theory

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.

Learning Laplacian Matrix in Smooth Graph Signal Representations

The construction of a meaningful graph topology plays a crucial role in the effective representation, processing, analysis and visualization of structured data. When a natural choice of the graph is not readily available from the data sets, it is thus desirable to infer or learn a graph topology from the data. In this tutorial overview, we survey solutions to the problem of graph learning, including classical viewpoints from statistics and physics, and more recent approaches that adopt a graph signal processing (GSP) perspective. We further emphasize the conceptual similarities and differences between classical and GSP-based graph inference methods, and highlight the potential advantage of the latter in a number of theoretical and practical scenarios. We conclude with several open issues and challenges that are keys to the design of future signal processing and machine learning algorithms for learning graphs from data.

Learning graphs from data: A signal representation perspective

This paper addresses the problem of compression of 3D point cloud sequences that are characterized by moving 3D positions and color attributes. As temporally successive point cloud frames are similar, motion estimation is key to effective compression of these sequences. It however remains a challenging problem as the point cloud frames have varying numbers of points without explicit correspondence information. We represent the time-varying geometry of these sequences with a set of graphs, and consider 3D positions and color attributes of the points clouds as signals on the vertices of the graphs. We then cast motion estimation as a feature matching problem between successive graphs. The motion is estimated on a sparse set of representative vertices using new spectral graph wavelet descriptors. A dense motion field is eventually interpolated by solving a graph-based regularization problem. The estimated motion is finally used for removing the temporal redundancy in the predictive coding of the 3D positions and the color characteristics of the point cloud sequences. Experimental results demonstrate that our method is able to accurately estimate the motion between consecutive frames. Moreover, motion estimation is shown to bring significant improvement in terms of the overall compression performance of the sequence. To the best of our knowledge, this is the first paper that exploits both the spatial correlation inside each frame (through the graph) and the temporal correlation between the frames (through the motion estimation) to compress the color and the geometry of 3D point cloud sequences in an efficient way.

Graph-based compression of dynamic 3D point cloud sequences

The construction of a meaningful graph topology plays a crucial role in the effective representation, processing, analysis, and visualization of structured data. When a natural choice of the graph is not readily available from the data sets, it is thus desirable to infer or learn a graph topology from the data. In this article, we survey solutions to the problem of graph learning, including classical viewpoints from statistics and physics, and more recent approaches that adopt a graph signal processing (GSP) perspective. We further emphasize the conceptual similarities and differences between classical and GSP-based graph-inference methods and highlight the potential advantage of the latter in a number of theoretical and practical scenarios. We conclude with several open issues and challenges that are keys to the design of future signal processing and machine-learning algorithms for learning graphs from data.

Learning Graphs From Data: A Signal Representation Perspective

This paper addresses the problem of compression of 3D point cloud sequences that are characterized by moving 3D positions and color attributes. As temporally successive point cloud frames share some similarities, motion estimation is key to effective compression of these sequences. It, however, remains a challenging problem as the point cloud frames have varying numbers of points without explicit correspondence information. We represent the time-varying geometry of these sequences with a set of graphs, and consider 3D positions and color attributes of the point clouds as signals on the vertices of the graphs. We then cast motion estimation as a feature-matching problem between successive graphs. The motion is estimated on a sparse set of representative vertices using new spectral graph wavelet descriptors. A dense motion field is eventually interpolated by solving a graph-based regularization problem. The estimated motion is finally used for removing the temporal redundancy in the predictive coding of the 3D positions and the color characteristics of the point cloud sequences. Experimental results demonstrate that our method is able to accurately estimate the motion between consecutive frames. Moreover, motion estimation is shown to bring a significant improvement in terms of the overall compression performance of the sequence. To the best of our knowledge, this is the first paper that exploits both the spatial correlation inside each frame (through the graph) and the temporal correlation between the frames (through the motion estimation) to compress the color and the geometry of 3D point cloud sequences in an efficient way.

/pdf/graph-based-compression-of-dynamic-3d-point-cloud-sequences-1q4h35lp4p.pdf

Dorina Thanou

Papers

Learning Laplacian Matrix in Smooth Graph Signal Representations

Learning graphs from data: A signal representation perspective

Graph-based compression of dynamic 3D point cloud sequences

Learning Graphs From Data: A Signal Representation Perspective

Graph-Based Compression of Dynamic 3D Point Cloud Sequences