Theoretical Analysis of Diffusion Models for Radio Map Estimation with Ultra-low Sampling Rates

Abstract: Radio maps, which characterize the spatial distribution of radio frequency metrics such as received signal strength, are essential for a wide range of wireless applications. The problem of radio map estimation involves constructing a radio map from sparse sensor measurements at multiple locations. This problem is particularly challenging due to ultra-low sampling rates, where available sensor measurements are far fewer than the high resolution requirement of radio maps to be estimated. Recently, diffusion models have been increasingly adopted for this problem, yet its theoretical performance remains unexamined. This paper bridges this gap by formulating radio map estimation as a non-linear matrix completion problem. Based on this formulation, we first derive a theoretical lower bound on the minimum estimation error achievable by diffusion models, which is fundamentally governed by the discrepancy between the deployment distribution and the true underlying radio propagation law. We then extend this bound to incorporate the effect of sampling sparsity, capturing the additional error introduced by ultra-low sampling rates. Furthermore, we establish a critical sampling rate threshold necessary for diffusion models to achieve performance convergence. Finally, considering that the derived error bounds depend on certain information that is difficult to obtain in practice, we propose empirical approximations that are readily computable from observable data. Extensive simulations based on real-world traces demonstrate that these empirical formulas tightly approximate the theoretical error bounds, validating their effectiveness for practical deployment.