Learning Sparse Graph Laplacian with K Eigenvector Prior via Iterative GLASSO and Projection

Learning a suitable graph is an important precursor to many graph signal processing (GSP) pipelines, such as graph spectral signal compression and denoising. Previous graph learning algorithms either i) make some assumptions on connectivity (e.g., graph sparsity), or ii) make simple graph edge assumptions such as positive edges only. In this paper, given an empirical covariance matrix $\bar{C}$ computed from data as input, we consider a structural assumption on the graph Laplacian matrix $L$: the first $K$ eigenvectors of $L$ are pre-selected, e.g., based on domain-specific criteria, such as computation requirement, and the remaining eigenvectors are then learned from data. One example use case is image coding, where the first eigenvector is pre-chosen to be constant, regardless of available observed data. We first prove that the subspace of symmetric positive semi-definite (PSD) matrices $H_{u}^+$ with the first $K$ eigenvectors being $\{u_k\}$ in a defined Hilbert space is a convex cone. We then construct an operator to project a given positive definite (PD) matrix $L$ to $H_{u}^+$, inspired by the Gram-Schmidt procedure. Finally, we design an efficient hybrid graphical lasso/projection algorithm to compute the most suitable graph Laplacian matrix $L^* \in H_{u}^+$ given $\bar{C}$. Experimental results show that given the first $K$ eigenvectors as a prior, our algorithm outperforms competing graph learning schemes using a variety of graph comparison metrics

Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment

3D point cloud (PC) -- a collection of discrete geometric samples of a physical object's surface -- is typically large in size, which entails expensive subsequent operations like viewpoint image rendering and object recognition. Leveraging on recent advances in graph sampling, we propose a fast PC sub-sampling algorithm that reduces its size while preserving the overall object shape. Specifically, to articulate a sampling objective, we first assume a super-resolution (SR) method based on feature graph Laplacian regularization (FGLR) that reconstructs the original high-resolution PC, given 3D points chosen by a sampling matrix \H. We prove that minimizing a worst-case SR reconstruction error is equivalent to maximizing the smallest eigenvalue $\lambda_{\min}$ of a matrix \H^{\top} \H + \mu \cL, where \cL is a symmetric, positive semi-definite matrix computed from the neighborhood graph connecting the 3D points. Instead, for fast computation we maximize a lower bound \lambda^-_{\min}(\H^{\top} \H + \mu \cL) via selection of \H in three steps. Interpreting \cL as a generalized graph Laplacian matrix corresponding to an unbalanced signed graph \cG, we first approximate \cG with a balanced graph \cG_B with the corresponding generalized graph Laplacian matrix \cL_B. Second, leveraging on a recent theorem called Gershgorin disc perfect alignment (GDPA), we perform a similarity transform \cL_p = \S \cL_B \S^{-1} so that Gershgorin disc left-ends of \cL_p are all aligned at the same value \lambda_{\min}(\cL_B). Finally, we perform PC sub-sampling on \cG_B using a graph sampling algorithm to maximize \lambda^-_{\min}(\H^{\top} \H + \mu \cL_p) in roughly linear time. Experimental results show that 3D points chosen by our algorithm outperformed competing schemes both numerically and visually in SR reconstruction quality

Graph Signal Processing for Geometric Data and Beyond: Theory and Applications

Geometric data acquired from real-world scenes, e.g, 2D depth images, 3D point clouds, and 4D dynamic point clouds, have found a wide range of applications including immersive telepresence, autonomous driving, surveillance, etc. Due to irregular sampling patterns of most geometric data, traditional image/video processing methodologies are limited, while Graph Signal Processing (GSP) -- a fast-developing field in the signal processing community -- enables processing signals that reside on irregular domains and plays a critical role in numerous applications of geometric data from low-level processing to high-level analysis. To further advance the research in this field, we provide the first timely and comprehensive overview of GSP methodologies for geometric data in a unified manner by bridging the connections between geometric data and graphs, among the various geometric data modalities, and with spectral/nodal graph filtering techniques. We also discuss the recently developed Graph Neural Networks (GNNs) and interpret the operation of these networks from the perspective of GSP. We conclude with a brief discussion of open problems and challenges.Comment: 16 pages, 7 figure