Robust Processing and Learning: Principles, Methods, and Wireless Applications

TL;DR

This paper develops a unified perspective on robustness for wireless sensing and communications by formalizing $(\epsilon, l, t)$-robustness and distributional robustness, and by connecting robust statistics, robust optimization, and robust machine learning. It presents min–max and DRO formulations to protect performance under model mismatch, data scarcity, and adversarial perturbations, and demonstrates these ideas through concrete WSC applications such as robust ranging localization, GAN-based channel estimation, distributionally robust receive combining, robust ISAC waveform design, and robust federated learning. The analysis highlights the costs of robustness, including nominal performance trade-offs and added computational burden, while emphasizing the potential for more reliable and trustworthy WSC systems in uncertain environments. The work also discusses practical aspects of uncertainty set design, computational tractability, and the interplay between robustness and adaptivity, outlining open challenges and avenues for future research in robust sensing and communication systems.

Abstract

This tutorial-style overview article examines the fundamental principles and methods of robustness, using wireless sensing and communication (WSC) as the narrative and exemplifying framework. First, we formalize the conceptual and mathematical foundations of robustness, highlighting the interpretations and relations across robust statistics, optimization, and machine learning. Key techniques, such as robust estimation and testing, distributionally robust optimization, and regularized and adversary training, are investigated. Together, the costs of robustness in system design, for example, the compromised nominal performances and the extra computational burdens, are discussed. Second, we review recent robust signal processing solutions for WSC that address model mismatch, data scarcity, adversarial perturbation, and distributional shift. Specific applications include robust ranging-based localization, modality sensing, channel estimation, receive combining, waveform design, and federated learning. Through this effort, we aim to introduce the classical developments and recent advances in robustness theory to the general signal processing community, exemplifying how robust statistical, optimization, and machine learning approaches can address the uncertainties inherent in WSC systems.

Figures (7)

• Figure 1: Among the various aspects of trustworthy wireless sensing and communications, robustness holds a critical position.
• Figure 2: Driving in uncertain environments is dangerous, so is developing WSC systems without uncertainty awareness. When driving in foggy conditions, an adaptive strategy reduces the uncertainty by sensing the visibility (i.e., fog density) and adjusting the brightness of the fog lights accordingly. In contrast, a robust strategy tolerates the uncertainty by maintaining a consistently low moving speed, regardless of the fog density. (Figure credit: ChatGPT 5.)
• Figure 3: Visual illustrations of notions of robustness within the uncertainty set $\Xi \coloneqq [\underline \xi, \bar{\xi}]$; note that $\xi_0$ is unknown and can vary on $\Xi$. In (a), the object (e.g., system, method, etc.) is robust against the uncertain factor $\xi$ because the cost does not exceed the tolerance threshold $t$ for all $\xi \in \Xi$. Moreover, a small factor perturbation $|\hat{\xi} - \xi_0|$ can only cause a small performance deviation. In (b), the object is non-robust because a tiny factor perturbation can lead to a severe performance deviation. In (c), the object is non-robust because the cost can exceed the tolerance threshold for some realizations of $\xi$, e.g., at $\xi_0$. In (d), the decision $x^*$ is robust, while $\hat{x}$ is non-robust.
• Figure 4: Comparison between the TV and Wasserstein distances: the former cannot capture the underlying geometry between points, while the latter can. In (a), two distributions are supported on $\xi_0$ and $\xi_1$, with probability mass vectors $p_0 = [1, 0]$ and $p_1 = [1-\epsilon, \epsilon]$, respectively. In this case, the TV distance $d_{\text{TV}}(p_0, p_1) = \epsilon$, no matter how large $|\xi_1 - \xi_0|$ is. In contrast, the Wasserstein distance $d_{\text{W}}(p_0, p_1) = \epsilon \cdot |\xi_1 - \xi_0|$ can be infinite if $|\xi_1 - \xi_0| \to \infty$. In (b), we let $\xi_1 = \xi_0 + \epsilon$ for $\epsilon > 0$, $p_0 = [1, 0]$, and $p_1 = [0, 1]$. In this case, $d_{\text{TV}}(p_0, p_1) = 1$, no matter how small $\epsilon$ is. However, $d_{\text{W}}(p_0, p_1) = \epsilon$.
• Figure 5: Illustration of the worst-case pair across two distributional uncertainty sets, consisting of the two distributions that are most difficult to distinguish.

Theorems & Definitions (8)

• Definition 1: Robust Object
• Definition 2: Purely Robust Decision
• Definition 3: Robust Decision
• Definition 4: Local Robustness of Decision
• Definition 5: Global Robustness of Decision
• Definition 6: Distributionally Robust Object
• Definition 7: Local Distributional Robustness of Decision
• Definition 8: Global Distributional Robustness of Decision