An Event-Based Approach for the Conservative Compression of Covariance Matrices

TL;DR

The paper tackles data-efficient, conservative transmission of covariance matrices in sensor fusion by introducing an elementwise event-triggered framework with per-element thresholds, subset and combined triggers, and a bounder based on diagonal dominance. It rigorously rederives and extends prior work to allow per-element triggering, introduces a scale-invariant relative-change trigger, and provides a learning mechanism to optimize thresholds from application data, evaluated on real EKF covariance sequences derived from the InD-EKF dataset. Key findings show substantial data reductions (up to ~80%) with only small increases in bound conservativeness (median ~0.1–2.2%), and demonstrate the feasibility of learning trigger thresholds from data. The approach enables flexible, data-driven compression tailored to per-element accuracy requirements, with practical impact for bandwidth-limited, safety-critical fusion systems such as automated driving.

Abstract

This work introduces a flexible and versatile method for the data-efficient yet conservative transmission of covariance matrices, where a matrix element is only transmitted if a so-called triggering condition is satisfied for the element. Here, triggering conditions can be parametrized on a per-element basis, applied simultaneously to yield combined triggering conditions or applied only to certain subsets of elements. This allows, e.g., to specify transmission accuracies for individual elements or to constrain the bandwidth available for the transmission of subsets of elements. Additionally, a methodology for learning triggering condition parameters from an application-specific dataset is presented. The performance of the proposed approach is quantitatively assessed in terms of data reduction and conservativeness using estimate data derived from real-world vehicle trajectories from the InD-dataset, demonstrating substantial data reduction ratios with minimal over-conservativeness. The feasibility of learning triggering condition parameters is demonstrated.

Figures (9)

• Figure 1: Confidence / credible regions associated with covariance matrix $\mathbf{P}$ (blue), covariance matrix $\hat{\mathbf{P}}$ conservative w.r.t. $\mathbf{P}$ (orange), and covariance matrix $\tilde{\mathbf{P}}$ not conservative w.r.t. $\mathbf{P}$ (green). As can be seen the region of $\hat{\mathbf{P}}$ includes that of $\mathbf{P}$ while that of $\tilde{\mathbf{P}}$ does not. The latter corresponds to an underestimation of the uncertainty encoded by $\mathbf{P}$.
• Figure 2: Overview of event-triggered covariance compression approach. Covariance matrix to be transmitted $\mathbf{P}_k$, transmitter buffer matrix $\tilde{\mathbf{P}}_{k-1}$, reduced event-set $E_k'$, receiver buffer matrix $\tilde{\mathbf{P}}_k$ and upper bound $\hat{\mathbf{P}}_k$ on original covariance matrix $\mathbf{P}_k$.
• Figure 3: Exemplary event-triggered covariance transmission. Sent covariance matrix elements ($[\mathbf{P}_k]_{ij}$), changed buffer matrix elements ($[\tilde{\mathbf{P}}_k]_{ij}$), and bound elements ($[\hat{\mathbf{P}}_k]_{ij}$) modified by the bounder compared to the buffer value are shown in red.
• Figure 4: Matrix $|\mathbf{P}_k-\tilde{\mathbf{P}}_{k-1}|$ ($|\mathbf{P}_k-\tilde{\mathbf{P}}_{k-1}|/|\tilde{\mathbf{P}}_{k-1}|$ for relative-change trigger) based on which the trigger decision is made and the elements in $E_k$ (red) for different triggers. Here, the $N$-most-changed trigger uses the absolute buffer deviation. Symmetric elements need not be transmitted. Hence, although $|E_k|=5$ for the $N$-most-changed trigger, only $N=4$ elements are sent.
• Figure 5: Per-element application of triggers. Blue: absolute-change trigger. Orange and green: $N$-most-changed triggers for disjoint subsets of elements. Blue/orange overlap corresponds to a combined trigger.

Theorems & Definitions (32)

• Definition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3: Absolute-Change Bounds