Exploring Structure-Adaptive Graph Learning for Robust Semi-Supervised Classification

Graph Convolutional Neural Networks (GCNNs) are generalizations of CNNs to graph-structured data, in which convolution is guided by the graph topology. In many cases where graphs are unavailable, existing methods manually construct graphs or learn task-driven adaptive graphs. In this paper, we propose Graph Learning Neural Networks (GLNNs), which exploit the optimization of graphs (the adjacency matrix in particular) from both data and tasks. Leveraging on spectral graph theory, we propose the objective of graph learning from a sparsity constraint, properties of a valid adjacency matrix as well as a graph Laplacian regularizer via maximum a posteriori estimation. The optimization objective is then integrated into the loss function of the GCNN, which adapts the graph topology to not only labels of a specific task but also the input data. Experimental results show that our proposed GLNN outperforms state-of-the-art approaches over widely adopted social network datasets and citation network datasets for semi-supervised classification

3D Dynamic Point Cloud Inpainting via Temporal Consistency on Graphs

With the development of 3D laser scanning techniques and depth sensors, 3D dynamic point clouds have attracted increasing attention as a representation of 3D objects in motion, enabling various applications such as 3D immersive tele-presence, gaming and navigation. However, dynamic point clouds usually exhibit holes of missing data, mainly due to the fast motion, the limitation of acquisition and complicated structure. Leveraging on graph signal processing tools, we represent irregular point clouds on graphs and propose a novel inpainting method exploiting both intra-frame self-similarity and inter-frame consistency in 3D dynamic point clouds. Specifically, for each missing region in every frame of the point cloud sequence, we search for its self-similar regions in the current frame and corresponding ones in adjacent frames as references. Then we formulate dynamic point cloud inpainting as an optimization problem based on the two types of references, which is regularized by a graph-signal smoothness prior. Experimental results show the proposed approach outperforms three competing methods significantly, both in objective and subjective quality.Comment: 7 pages, 5 figures, accepted by IEEE ICME 2020 at 2020.04.03. arXiv admin note: text overlap with arXiv:1810.0397

Radial symmetry of positive entire solutions of a fourth order elliptic equation with a singular nonlinearity

The necessary and sufficient conditions for a regular positive entire solution $u$ of the biharmonic equation: \begin{equation} \label{0.1} -\Delta^2 u=u^{-p} \;\; \mbox{in $\R^N \; (N \geq 3)$}, \;\; p>1 \end{equation} to be a radially symmetric solution are obtained via the moving plane method (MPM) of a system of equations. It is well-known that for any $a>0$, \eqref{0.1} admits a unique minimal positive entire radial solution ${\underline u}_a (r)$ and a family of non-minimal positive entire radial solutions $u_a (r)$ such that $u_a (0)={\underline u}_a (0)=a$ and $u_a (r) \geq {\underline u}_a (r)$ for $r \in (0, \infty)$. Moreover, the asymptotic behaviors of ${\underline u}_a (r)$ and $u_a (r)$ at $r=\infty$ are also known. We will see in this paper that the asymptotic behaviors similar to those of ${\underline u}_a (r)$ and $u_a (r)$ at $r=\infty$ can determine the radial symmetry of a general regular positive entire solution $u$ of \eqref{0.1}. The precisely asymptotic behaviors of $u (x)$ and $-\Delta u (x)$ at $|x|=\infty$ need to be established such that the moving-plane procedure can be started. We provide the necessary and sufficient conditions not only for a regular positive entire solution $u$ of \eqref{0.1} to be the minimal entire radial solution, but also for $u$ to be a non-minimal entire radial solution

Feature Preserving and Uniformity-controllable Point Cloud Simplification on Graph

With the development of 3D sensing technologies, point clouds have attracted increasing attention in a variety of applications for 3D object representation, such as autonomous driving, 3D immersive tele-presence and heritage reconstruction. However, it is challenging to process large-scale point clouds in terms of both computation time and storage due to the tremendous amounts of data. Hence, we propose a point cloud simplification algorithm, aiming to strike a balance between preserving sharp features and keeping uniform density during resampling. In particular, leveraging on graph spectral processing, we represent irregular point clouds naturally on graphs, and propose concise formulations of feature preservation and density uniformity based on graph filters. The problem of point cloud simplification is finally formulated as a trade-off between the two factors and efficiently solved by our proposed algorithm. Experimental results demonstrate the superiority of our method, as well as its efficient application in point cloud registration.Comment: 6 page

Local Frequency Interpretation and Non-Local Self-Similarity on Graph for Point Cloud Inpainting

As 3D scanning devices and depth sensors mature, point clouds have attracted increasing attention as a format for 3D object representation, with applications in various fields such as tele-presence, navigation and heritage reconstruction. However, point clouds usually exhibit holes of missing data, mainly due to the limitation of acquisition techniques and complicated structure. Further, point clouds are defined on irregular non-Euclidean domains, which is challenging to address especially with conventional signal processing tools. Hence, leveraging on recent advances in graph signal processing, we propose an efficient point cloud inpainting method, exploiting both the local smoothness and the non-local self-similarity in point clouds. Specifically, we first propose a frequency interpretation in graph nodal domain, based on which we introduce the local graph-signal smoothness prior in order to describe the local smoothness of point clouds. Secondly, we explore the characteristics of non-local self-similarity, by globally searching for the most similar area to the missing region. The similarity metric between two areas is defined based on the direct component and the anisotropic graph total variation of normals in each area. Finally, we formulate the hole-filling step as an optimization problem based on the selected most similar area and regularized by the graph-signal smoothness prior. Besides, we propose voxelization and automatic hole detection methods for the point cloud prior to inpainting. Experimental results show that the proposed approach outperforms four competing methods significantly, both in objective and subjective quality.Comment: 11 pages, 11 figures, submitted to IEEE Transactions on Image Processing at 2018.09.0

On Delaunay solutions of a biharmonic elliptic equation with critical exponent

We are interested in the qualitative properties of positive entire solutions $u \in C^4 (\mathbb{R}^n \backslash \{0\})$ of the equation \begin{equation} \label{0.0} \Delta^2 u=u^{\frac{n+4}{n-4}} \;\;\mbox{in $\mathbb{R}^n \backslash \{0\}$ and 0 is a non-removable singularity of $u(x)$}. \end{equation} It is known from [Theorem 4.2] that any positive entire solution $u$ of \eqref{0.0} is radially symmetric with respect to $x=0$, i.e. $u(x)=u(|x|)$, and equation \eqref{0.0} also admits a special positive entire solution $u_s (x)=\Big(\frac{n^2 (n-4)^2}{16} \Big)^{\frac{n-4}{8}} |x|^{-\frac{n-4}{2}}$. We first show that $u-u_s$ changes signs infinitely many times in $(0, \infty)$ for any positive singular entire solution $u \not \equiv u_s$ in $\mathbb{R}^N \backslash \{0\}$ of \eqref{0.0}. Moreover, equation \eqref{0.0} admits a positive entire singular solution $u(x) \; (=u(|x|)$ such that the scalar curvature of the conformal metric with conformal factor $u^{\frac{4}{n-4}}$ is positive and $v(t):=e^{\frac{n-4}{2} t} u(e^t)$ is $2T$-periodic with suitably large $T$. It is still open that $v(t):=e^{\frac{n-4}{2} t} u(e^t)$ is periodic for any positive entire solution $u(x)$ of \eqref{0.0}.Comment: 21 pages; comments welcom

Joint Learning of Graph Representation and Node Features in Graph Convolutional Neural Networks

Graph Convolutional Neural Networks (GCNNs) extend classical CNNs to graph data domain, such as brain networks, social networks and 3D point clouds. It is critical to identify an appropriate graph for the subsequent graph convolution. Existing methods manually construct or learn one fixed graph for all the layers of a GCNN. In order to adapt to the underlying structure of node features in different layers, we propose dynamic learning of graphs and node features jointly in GCNNs. In particular, we cast the graph optimization problem as distance metric learning to capture pairwise similarities of features in each layer. We deploy the Mahalanobis distance metric and further decompose the metric matrix into a low-dimensional matrix, which converts graph learning to the optimization of a low-dimensional matrix for efficient implementation. Extensive experiments on point clouds and citation network datasets demonstrate the superiority of the proposed method in terms of both accuracies and robustness

Feature Graph Learning for 3D Point Cloud Denoising

Identifying an appropriate underlying graph kernel that reflects pairwise similarities is critical in many recent graph spectral signal restoration schemes, including image denoising, dequantization, and contrast enhancement. Existing graph learning algorithms compute the most likely entries of a properly defined graph Laplacian matrix $\mathbf{L}$, but require a large number of signal observations $\mathbf{z}$'s for a stable estimate. In this work, we assume instead the availability of a relevant feature vector $\mathbf{f}_i$ per node $i$, from which we compute an optimal feature graph via optimization of a feature metric. Specifically, we alternately optimize the diagonal and off-diagonal entries of a Mahalanobis distance matrix $\mathbf{M}$ by minimizing the graph Laplacian regularizer (GLR) $\mathbf{z}^{\top} \mathbf{L} \mathbf{z}$, where edge weight is $w_{i,j} = \exp\{-(\mathbf{f}_i - \mathbf{f}_j)^{\top} \mathbf{M} (\mathbf{f}_i - \mathbf{f}_j) \}$, given a single observation $\mathbf{z}$. We optimize diagonal entries via proximal gradient (PG), where we constrain $\mathbf{M}$ to be positive definite (PD) via linear inequalities derived from the Gershgorin circle theorem. To optimize off-diagonal entries, we design a block descent algorithm that iteratively optimizes one row and column of $\mathbf{M}$. To keep $\mathbf{M}$ PD, we constrain the Schur complement of sub-matrix $\mathbf{M}_{2,2}$ of $\mathbf{M}$ to be PD when optimizing via PG. Our algorithm mitigates full eigen-decomposition of $\mathbf{M}$, thus ensuring fast computation speed even when feature vector $\mathbf{f}_i$ has high dimension. To validate its usefulness, we apply our feature graph learning algorithm to the problem of 3D point cloud denoising, resulting in state-of-the-art performance compared to competing schemes in extensive experiments

RGCNN: Regularized Graph CNN for Point Cloud Segmentation

Point cloud, an efficient 3D object representation, has become popular with the development of depth sensing and 3D laser scanning techniques. It has attracted attention in various applications such as 3D tele-presence, navigation for unmanned vehicles and heritage reconstruction. The understanding of point clouds, such as point cloud segmentation, is crucial in exploiting the informative value of point clouds for such applications. Due to the irregularity of the data format, previous deep learning works often convert point clouds to regular 3D voxel grids or collections of images before feeding them into neural networks, which leads to voluminous data and quantization artifacts. In this paper, we instead propose a regularized graph convolutional neural network (RGCNN) that directly consumes point clouds. Leveraging on spectral graph theory, we treat features of points in a point cloud as signals on graph, and define the convolution over graph by Chebyshev polynomial approximation. In particular, we update the graph Laplacian matrix that describes the connectivity of features in each layer according to the corresponding learned features, which adaptively captures the structure of dynamic graphs. Further, we deploy a graph-signal smoothness prior in the loss function, thus regularizing the learning process. Experimental results on the ShapeNet part dataset show that the proposed approach significantly reduces the computational complexity while achieving competitive performance with the state of the art. Also, experiments show RGCNN is much more robust to both noise and point cloud density in comparison with other methods. We further apply RGCNN to point cloud classification and achieve competitive results on ModelNet40 dataset

Exploring Hypergraph Representation on Face Anti-spoofing Beyond 2D Attacks

Face anti-spoofing plays a crucial role in protecting face recognition systems from various attacks. Previous model-based and deep learning approaches achieve satisfactory performance for 2D face spoofs, but remain limited for more advanced 3D attacks such as vivid masks. In this paper, we address 3D face anti-spoofing via the proposed Hypergraph Convolutional Neural Networks (HGCNN). Firstly, we construct a computation-efficient and posture-invariant face representation with only a few key points on hypergraphs. The hypergraph representation is then fed into the designed HGCNN with hypergraph convolution for feature extraction, while the depth auxiliary is also exploited for 3D mask anti-spoofing. Further, we build a 3D face attack database with color, depth and infrared light information to overcome the deficiency of 3D face anti-spoofing data. Experiments show that our method achieves the state-of-the-art performance over widely used 3D and 2D databases as well as the proposed one under various tests