Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment

Graph sampling set selection, where a subset of nodes are chosen to collect samples to reconstruct a smooth graph signal, is a fundamental problem in graph signal processing (GSP). Previous works employ an unbiased least-squares (LS) signal reconstruction scheme and select samples via expensive extreme eigenvector computation. Instead, we assume a biased graph Laplacian regularization (GLR) based scheme that solves a system of linear equations for reconstruction. We then choose samples to minimize the condition number of the coefficient matrix---specifically, maximize the smallest eigenvalue $\lambda_{\min}$. Circumventing explicit eigenvalue computation, we maximize instead the lower bound of $\lambda_{\min}$, designated by the smallest left-end of all Gershgorin discs of the matrix. To achieve this efficiently, we first convert the optimization to a dual problem, where we minimize the number of samples needed to align all Gershgorin disc left-ends at a chosen lower-bound target $T$. Algebraically, the dual problem amounts to optimizing two disc operations: i) shifting of disc centers due to sampling, and ii) scaling of disc radii due to a similarity transformation of the matrix. We further reinterpret the dual as an intuitive disc coverage problem bearing strong resemblance to the famous NP-hard set cover (SC) problem. The reinterpretation enables us to derive a fast approximation scheme from a known SC error-bounded approximation algorithm. We find an appropriate target $T$ efficiently via binary search. Extensive simulation experiments show that our disc-based sampling algorithm runs substantially faster than existing sampling schemes and outperforms other eigen-decomposition-free sampling schemes in reconstruction error.Comment: Very fast deterministic graph sampling set selection algorithm without explicit eigen-decompositio

Sparse Graphical Designs via Linear Programming

Graphical designs are a framework for sampling and numerical integration of functions on graphs. In this note, we introduce a method to address the trade-off between graphical design sparsity and accuracy. We show how to obtain sparse graphical designs via linear programming and design objective functions that aim to maximize their accuracy. We showcase our approach using yellow taxicab data from New York City

Model-Free Prediction of Adversarial Drop Points in 3D Point Clouds

Adversarial attacks pose serious challenges for deep neural network (DNN)-based analysis of various input signals. In the case of 3D point clouds, methods have been developed to identify points that play a key role in the network decision, and these become crucial in generating existing adversarial attacks. For example, a saliency map approach is a popular method for identifying adversarial drop points, whose removal would significantly impact the network decision. Generally, methods for identifying adversarial points rely on the deep model itself in order to determine which points are critically important for the model's decision. This paper aims to provide a novel viewpoint on this problem, in which adversarial points can be predicted independently of the model. To this end, we define 14 point cloud features and use multiple linear regression to examine whether these features can be used for model-free adversarial point prediction, and which combination of features is best suited for this purpose. Experiments show that a suitable combination of features is able to predict adversarial points of three different networks -- PointNet, PointNet++, and DGCNN -- significantly better than a random guess. The results also provide further insight into DNNs for point cloud analysis, by showing which features play key roles in their decision-making process.Comment: 10 pages, 6 figure

Evolutionary Models for Signal Enhancement and Approximation

This thesis deals with nature-inspired evolution processes for the purpose of signal enhancement and approximation. The focus lies on mathematical models which originate from the description of swarm behaviour. We extend existing approaches and show the potential of swarming processes as a modelling tool in image processing. In our work, we discuss the use cases of grey scale quantisation, contrast enhancement, line detection, and coherence enhancement. Furthermore, we propose a new and purely repulsive model of swarming that turns out to describe a specific type of backward diffusion process. It is remarkable that our model provides extensive stability guarantees which even support the utilisation of standard numerics. In experiments, we demonstrate its applicability to global and local contrast enhancement of digital images. In addition, we study the problem of one-dimensional signal approximation with limited resources using an adaptive sampling approach including tonal optimisation. We suggest a direct energy minimisation strategy and validate its efficacy in experiments. Moreover, we show that our approximation model can outperform a method recently proposed by Dar and Bruckstein

Sparse Gaussian chain graphs with the spike-and-slab LASSO: Algorithms and asymptotics

The Gaussian chain graph model simultaneously parametrizes (i) the direct effects of $p$ predictors on $q$ correlated outcomes and (ii) the residual partial covariance between pair of outcomes. We introduce a new method for fitting sparse Gaussian chain graph models with spike-and-slab LASSO (SSL) priors. We develop an Expectation-Conditional Maximization algorithm to obtain sparse estimates of the $p \times q$ matrix of direct effects and the $q \times q$ residual precision matrix. Our algorithm iteratively solves a sequence of penalized maximum likelihood problems with self-adaptive penalties that gradually filter out negligible regression coefficients and partial covariances. Because it adaptively penalizes model parameters, our method is seen to outperform fixed-penalty competitors on simulated data. We establish the posterior concentration rate for our model, buttressing our method's excellent empirical performance with strong theoretical guarantees. We use our method to reanalyze a dataset from a study of the effects of diet and residence type on the composition of the gut microbiome of elderly adults