Compressive imagers, e.g. the single-pixel camera (SPC), acquire measurements in the form of random projections of the scene instead of pixel intensities. Compressive Sensing (CS) theory allows accurate reconstruction of the image even from a small number of such projections. However, in practice, most reconstruction algorithms perform poorly at low measurement rates and are computationally very expensive. But perfect reconstruction is not the goal of high-level computer vision applications. Instead, we are interested in only determining certain properties of the image. Recent work has shown that effective inference is possible directly from the compressive measurements, without reconstruction, using correlational features. In this paper, we show that convolutional neural networks (CNNs) can be employed to extract discriminative non-linear features directly from CS measurements. Using these features, we demonstrate that effective high-level inference can be performed. Experimentally, using hand written digit recognition (MNIST dataset) and image recognition (ImageNet) as examples, we show that recognition is possible even at low measurement rates of about 0.1.