Advances in Robust Signal Processing and Applications

Robust signal processing and machine learning methodologies are critical for the reliable operation of modern technological systems, particularly in dynamic and uncertain environments such as the Internet of Things (IoT). However, system performance is often compromised by pervasive challenges, including structural perturbations in graph-based models, complex non-Gaussian noise in communication systems, and the structural heterogeneity of high-dimensional tensor data. This thesis addresses these critical challenges by developing a suite of robust methodologies grounded in distinct yet complementary perspectives on robustness.

First, this research establishes a comprehensive analytical framework to quantify the sensitivity of
Graph Convolutional Neural Networks (GCNNs) to probabilistic graph perturbations. Tight, expected bounds for Graph Shift Operator (GSO) errors are derived without requiring eigendecomposition, and a linear relationship between GSO perturbations and GCNN output differences is revealed, providing theoretical stability guarantees for multilayer architectures.

Second, novel robust device activity detection (AD) algorithms are developed for massive random
access systems operating under challenging non-Gaussian noise. By formulating AD objectives
based on robust loss functions (e.g., Huber’s loss) and proving the geodesic convexity of the
conditional objective, efficient fixed-point, coordinate-wise, and matching pursuit algorithms with
proven convergence are proposed. These methods significantly outperform traditional Gaussian-
based approaches in heavy-tailed noise environments.

Third, a generalized Nonnegative Structured Kruskal Tensor Regression (NS-KTR) framework is
introduced for the effective and interpretable modeling of high-dimensional tensor data. This
framework integrates non-negativity constraints with mode-specific hybrid regularizations (e.g.,
LASSO, total variation, ridge), accommodates both linear and logistic regression, and is solved via
an efficient ADMM-based algorithm. Collectively, this thesis advances the theory and practice of robust signal processing by providing novel tools for ensuring stability, resilience to distributional deviations, and robust modeling through structural priors. The developed frameworks and algorithms contribute to the design of more reliable and efficient signal processing systems for real-world applications.