Diffusion Models for Radio Map Estimation: Theoretical Performance Analysis and Sampling Rate Guideline

Radio maps, which characterize the spatial distribution of radio frequency metrics, such as the received signal strength, are essential for a wide range of wireless applications. The problem of radio map estimation involves constructing a radio map from a limited set of radio samples measured by sparsely distributed sensors.Recently, diffusion models have been increasingly adopted for this problem, yet their theoretical performance remains largely unexamined. Consequently, it is difficult to evaluate their performance when ground-truth radio maps are unavailable, as is often the case in practice. To bridge this gap, we first formulate radio map estimation as a non-linear matrix completion problem. We then derive a theoretical expression for the estimation error of a diffusion model, capturing the effects of the mismatch between the training and deployment environments, the model design, and the sampling strategy on the quality of the estimated radio map. Moreover, considering that the derived error expression depends on certain information that is difficult to obtain in practice, we propose an empirical approximation that is readily computable from observable data. Finally, extensive simulations demonstrate that the empirical formula closely approximates the theoretical error expression, validating its effectiveness for practical deployment. Our results provide guidelines not only for evaluating the performance of a diffusion model when ground truth is unavailable, but also for determining how many sensors are required to spatially sample the region to achieve a given estimation accuracy. They further reveal a critical sensor count beyond which the estimation performance of diffusion models converges.