Spectrum sensing is the basis of dynamic spectrum access and sharing for space information networks consisting of various satellite and terrestrial networks. The traditional spectrum sensing method, guided by the Nyquist-Shannon sampling theorem, might not be suitable for the emerging communication systems such as the fifth-generation mobile communications (5G) and space information networks utilizing spectrum from sub-6 GHz up to 100 GHz to offer ubiquitous broadband applications. In contrast, compressed spectrum sensing can not only relax the requirements on hardware and software, but also reduce the energy consumption and processing latency. As for the compressed measurement (low-speed sampling) process of the existing compressed spectrum sensing algorithms, the compression ratio is usually set to a fixed value, which limits their adaptability to the dynamically changing radio environment with different sparseness. In this paper, an adaptive compressed spectrum sensing algorithm based on radio environment map (REM) dedicated for space information networks is proposed to address this problem. Simulations show that the proposed algorithm has better adaptability to the varying environment than the existing compressed spectrum sensing algorithms.