Raman microscopy is a powerful method combining non-invasiveness with no special sample preparation. Because of this remarkable simplicity, it has been widely exploited in many fields, ranging from life and materials sciences to engineering. Notoriously, due to the required imaging speeds for bio-imaging, it has remained a challenge how to use this technique for dynamic and large-scale imaging. Recently, a supervised compressive Raman framework has been put forward, allowing for fast imaging, therefore alleviating the issue of speed. Yet, due to the need for strong a priori information of the species forming the hyperspectrum, it has remained elusive how to apply this supervised method for microspectroscopy of (dynamic) biological tissues. Combining an original spectral under-sampling measurement technique with a matrix completion framework for reconstruction, we demonstrate fast and inexpensive label-free molecular imaging of biological specimens (brain tissues and single cells). Using the matrix completion outcome with the supervised method allows for large compressions (64 x) and bio-imaging speeds surpassing current technology in spontaneous Raman microspectroscopy. Therefore, our results open interesting perspectives for clinical and cell biology applications using the much faster compressive Raman framework. (C) 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement