Compressive sensing is a technology that allows the original signal to be reconstructed with far fewer multiplexed measurements than the Shannon-Nyquist sampling rate. Compared to conventional methods, it can provide a comparable quality result with a much lower sampling time, while offering the advantage of higher signal-to-noise ratio. The first part of the work focuses on research using manifold secant patterns for classification in the field of Cryogenic electron microscopy (cryo-EM). In certain scenarios, the main focus may not be on reconstructing the original signal, but rather on solving an inference problem in isolation, which classification is an example and offers certain advantage over current methods. Next, the building of a high-resolution compressive Raman spectrometer for gas analysis will be described. A novel cavity-enhanced gas Raman signal generation system was built and developed in a Sagnac spectrometer configuration. Incorporating the idea of compressive sensing into the system further enhances the signal-to-noise ratio and reduces the acquisition time as well as allowing for compactness and portability for use in remote environments. Lastly, improvements and the future potential of this work will be discussed.