Using millimeter wave (mmWave) signals for imaging has an important advantage in that they can penetrate through poor environmental conditions such as fog, dust, and smoke that severely degrade optical-based imaging systems. However, mmWave radars, contrary to cameras and LiDARs, suffer from low angular resolution because of small physical apertures and conventional signal processing techniques. Sparse radar imaging, on the other hand, can increase the aperture size while minimizing power consumption and read-out bandwidth. This article presents CoIR, an analysis by synthesis method that leverages the implicit neural network bias in convolutional decoders and compressed sensing to perform high-accuracy sparse radar imaging. The proposed system is data set-agnostic and does not require any auxiliary sensors for training or testing. We introduce a sparse array design that allows for a $5.5\times$5.5× reduction in the number of antenna elements needed compared to conventional MIMO array designs. We demonstrate our system's improved imaging performance over standard mmWave radars and other competitive untrained methods on both simulated and experimental mmWave radar data.