Projection-free Graph-based Classifier Learning using Gershgorin Disc Perfect Alignment

In semi-supervised graph-based binary classifier learning, a subset of known labels $\hat{x}_i$ are used to infer unknown labels, assuming that the label signal $x$ is smooth with respect to a similarity graph specified by a Laplacian matrix. When restricting labels $x_i$ to binary values, the problem is NP-hard. While a conventional semi-definite programming (SDP) relaxation can be solved in polynomial time using, for example, the alternating direction method of multipliers (ADMM), the complexity of iteratively projecting a candidate matrix $M$ onto the positive semi-definite (PSD) cone ($M \succeq 0$) remains high. In this paper, leveraging a recent linear algebraic theory called Gershgorin disc perfect alignment (GDPA), we propose a fast projection-free method by solving a sequence of linear programs (LP) instead. Specifically, we first recast the SDP relaxation to its SDP dual, where a feasible solution $H \succeq 0$ can be interpreted as a Laplacian matrix corresponding to a balanced signed graph sans the last node. To achieve graph balance, we split the last node into two that respectively contain the original positive and negative edges, resulting in a new Laplacian $\bar{H}$. We repose the SDP dual for solution $\bar{H}$, then replace the PSD cone constraint $\bar{H} \succeq 0$ with linear constraints derived from GDPA -- sufficient conditions to ensure $\bar{H}$ is PSD -- so that the optimization becomes an LP per iteration. Finally, we extract predicted labels from our converged LP solution $\bar{H}$. Experiments show that our algorithm enjoyed a $40\times$ speedup on average over the next fastest scheme while retaining comparable label prediction performance.

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.

Pre-demosaic Graph-based Light Field Image Compression

A plenoptic light field (LF) camera places an array of microlenses in front of an image sensor in order to separately capture different directional rays arriving at an image pixel. Using a conventional Bayer pattern, data captured at each pixel is a single color component (R, G or B). The sensed data then undergoes demosaicking (interpolation of RGB components per pixel) and conversion to an array of sub-aperture images (SAIs). In this paper, we propose a new LF image coding scheme based on graph lifting transform (GLT), where the acquired sensor data are coded in the original captured form without pre-processing. Specifically, we directly map raw sensed color data to the SAIs, resulting in sparsely distributed color pixels on 2D grids, and perform demosaicking at the receiver after decoding. To exploit spatial correlation among the sparse pixels, we propose a novel intra-prediction scheme, where the prediction kernel is determined according to the local gradient estimated from already coded neighboring pixel blocks. We then connect the pixels by forming a graph, modeling the prediction residuals statistically as a Gaussian Markov Random Field (GMRF). The optimal edge weights are computed via a graph learning method using a set of training SAIs. The residual data is encoded via low-complexity GLT. Experiments show that at high PSNRs -- important for archiving and instant storage scenarios -- our method outperformed significantly a conventional light field image coding scheme with demosaicking followed by High Efficiency Video Coding (HEVC).

Fast & Robust Image Interpolation using Gradient Graph Laplacian Regularizer

In the graph signal processing (GSP) literature, it has been shown that signal-dependent graph Laplacian regularizer (GLR) can efficiently promote piecewise constant (PWC) signal reconstruction for various image restoration tasks. However, for planar image patches, like total variation (TV), GLR may suffer from the well-known "staircase" effect. To remedy this problem, we generalize GLR to gradient graph Laplacian regularizer (GGLR) that provably promotes piecewise planar (PWP) signal reconstruction for the image interpolation problem -- a 2D grid with randomly missing pixels that requires completion. Specifically, we first construct two higher-order gradient graphs to connect local horizontal and vertical gradients. Each local gradient is estimated using structure tensor, which is robust using known pixels in a small neighborhood, mitigating the problem of larger noise variance when computing gradient of gradients. Moreover, unlike total generalized variation (TGV), GGLR retains the quadratic form of GLR, leading to an unconstrained quadratic programming (QP) problem per iteration that can be solved quickly using conjugate gradient (CG). We derive the means-square-error minimizing weight parameter for GGLR, trading off bias and variance of the signal estimate. Experiments show that GGLR outperformed competing schemes in interpolation quality for severely damaged images at a reduced complexity.

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.

Unrolling of Deep Graph Total Variation for Image Denoising

While deep learning (DL) architectures like convolutional neural networks (CNNs) have enabled effective solutions in image denoising, in general their implementations overly rely on training data, lack interpretability, and require tuning of a large parameter set. In this paper, we combine classical graph signal filtering with deep feature learning into a competitive hybrid design---one that utilizes interpretable analytical low-pass graph filters and employs 80% fewer network parameters than state-of-the-art DL denoising scheme DnCNN. Specifically, to construct a suitable similarity graph for graph spectral filtering, we first adopt a CNN to learn feature representations per pixel, and then compute feature distances to establish edge weights. Given a constructed graph, we next formulate a convex optimization problem for denoising using a graph total variation (GTV) prior. Via a $l_1$ graph Laplacian reformulation, we interpret its solution in an iterative procedure as a graph low-pass filter and derive its frequency response. For fast filter implementation, we realize this response using a Lanczos approximation. Experimental results show that in the case of statistical mistmatch, our algorithm outperformed DnCNN by up to 3dB in PSNR.

Signed Graph Metric Learning via Gershgorin Disc Alignment

Given a convex and differentiable objective $Q(\M)$ for a real, symmetric matrix $\M$ in the positive definite (PD) cone---used to compute Mahalanobis distances---we propose a fast general metric learning framework that is entirely projection-free. We first assume that $\M$ resides in a space $\cS$ of generalized graph Laplacian matrices (graph metric matrices) corresponding to balanced signed graphs. Unlike low-rank metric matrices common in the literature, $\cS$ includes the important diagonal-only matrices as a special case. The key theorem to circumvent full eigen-decomposition and enable fast metric matrix optimization is Gershgorin disc alignment (GDA): given graph metric matrix $\M \in \cS$ and diagonal matrix $\S$, where $S_{ii} = 1/v_i$ and $\v$ is the first eigenvector of $\M$, we prove that Gershgorin disc left-ends of similar transform $\B = \S \M \S^{-1}$ are perfectly aligned at the smallest eigenvalue $\lambda_{\min}$. Using this theorem, we replace the PD cone constraint in the metric learning problem with tightest possible linear constraints per iteration, so that the alternating optimization of the diagonal / off-diagonal terms in $\M$ can be solved efficiently as linear programs via Frank-Wolfe iterations. We update $\v$ using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as matrix entries in $\M$ are optimized successively. Experiments show that our graph metric optimization is significantly faster than cone-projection methods, and produces competitive binary classification performance.

Sampling on Graphs: From Theory to Applications

The study of sampling signals on graphs, with the goal of building an analog of sampling for standard signals in the time and spatial domains, has attracted considerable attention recently. Beyond adding to the growing theory on graph signal processing (GSP), sampling on graphs has various promising applications. In this article, we review current progress on sampling over graphs focusing on theory and potential applications. Most methodologies used in graph signal sampling are designed to parallel those used in sampling for standard signals, however, sampling theory for graph signals significantly differs from that for Shannon--Nyquist and shift invariant signals. This is due in part to the fact that the definitions of several important properties, such as shift invariance and bandlimitedness, are different in GSP systems. Throughout, we discuss similarities and differences between standard and graph sampling and highlight open problems and challenges.

Graph Metric Learning via Gershgorin Disc Alignment

We propose a fast general projection-free metric learning framework, where the minimization objective $\min_{\textbf{M} \in \mathcal{S}} Q(\textbf{M})$ is a convex differentiable function of the metric matrix $\textbf{M}$, and $\textbf{M}$ resides in the set $\mathcal{S}$ of generalized graph Laplacian matrices for connected graphs with positive edge weights and node degrees. Unlike low-rank metric matrices common in the literature, $\mathcal{S}$ includes the important positive-diagonal-only matrices as a special case in the limit. The key idea for fast optimization is to rewrite the positive definite cone constraint in $\mathcal{S}$ as signal-adaptive linear constraints via Gershgorin disc alignment, so that the alternating optimization of the diagonal and off-diagonal terms in $\textbf{M}$ can be solved efficiently as linear programs via Frank-Wolfe iterations. We prove that the Gershgorin discs can be aligned perfectly using the first eigenvector $\textbf{v}$ of $\textbf{M}$, which we update iteratively using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as diagonal / off-diagonal terms are optimized. Experiments show that our efficiently computed graph metric matrices outperform metrics learned using competing methods in terms of classification tasks.

Robust Deep Graph Based Learning for Binary Classification

Convolutional neural network (CNN)-based feature learning has become state of the art, since given sufficient training data, CNN can significantly outperform traditional methods for various classification tasks. However, feature learning becomes more difficult if some training labels are noisy. With traditional regularization techniques, CNN often overfits to the noisy training labels, resulting in sub-par classification performance. In this paper, we propose a robust binary classifier, based on CNNs, to learn deep metric functions, which are then used to construct an optimal underlying graph structure used to clean noisy labels via graph Laplacian regularization (GLR). GLR is posed as a convex maximum a posteriori (MAP) problem solved via convex quadratic programming (QP). To penalize samples around the decision boundary, we propose two regularized loss functions for semi-supervised learning. The binary classification experiments on three datasets, varying in number and type of features, demonstrate that given a noisy training dataset, our proposed networks outperform several state-of-the-art classifiers, including label-noise robust support vector machine, CNNs with three different robust loss functions, model-based GLR, and dynamic graph CNN classifiers.