Local Strong Convexity of Source Localization and Error Bound for Target Tracking under Time-of-Arrival Measurements

TL;DR

This paper analyzes the local strong convexity of the source localization problem using TOA measurements and establishes error bounds for target tracking with online gradient descent, supported by theoretical analysis and numerical validation.

Contribution

It proves local strong convexity of the non-convex TOA localization loss function at global minima and derives error bounds for online gradient descent in target tracking.

Findings

OGD effectively tracks moving targets under noisy TOA data.

Loss function is locally strongly convex at global minima.

Theoretical bounds are validated by numerical experiments.

Abstract

In this paper, we consider a time-varying optimization approach to the problem of tracking a moving target using noisy time-of-arrival (TOA) measurements. Specifically, we formulate the problem as that of sequential TOA-based source localization and apply online gradient descent (OGD) to it to generate the position estimates of the target. To analyze the tracking performance of OGD, we first revisit the classic least-squares formulation of the (static) TOA-based source localization problem and elucidate its estimation and geometric properties. In particular, under standard assumptions on the TOA measurement model, we establish a bound on the distance between an optimal solution to the least-squares formulation and the true target position. Using this bound, we show that the loss function in the formulation, albeit non-convex in general, is locally strongly convex at its global minima. To the best of our knowledge, these results are new and can be of independent interest. By combining them with existing techniques from online strongly convex optimization, we then establish the first non-trivial bound on the cumulative target tracking error of OGD. Our numerical results corroborate the theoretical findings and show that OGD can effectively track the target at different noise levels.