Power spectral density (PSD) maps providing the distribution of RF power across space and frequency are constructed using power measurements collected by a network of low-cost sensors. By introducing linear compression and quantization to a small number of bits, sensor measurements can be communicated to the fusion center with minimal bandwidth requirements. Strengths of data- and model-driven approaches are combined to develop estimators capable of incorporating multiple forms of spectral and propagation prior information while fitting the rapid variations of shadow fading across space. To this end, novel nonparametric and semiparametric formulations are investigated. It is shown that PSD maps can be obtained using support vector machine-type solvers. In addition to batch approaches, an online algorithm attuned to real-time operation is developed. Numerical tests assess the performance of the novel algorithms.