You are talking about compressed sensing which is not lossy compression (compressed sensing can be lossless unless you're dealing with noisy measurements).

But say you're doing noisy measurements, and you are under-measuring like you say, and you have to fabricate non-random non-homogenous reconstruction noise. In that case it would be a very good idea to produce, as they do for lossy compression, both the standard overall bit rate vs. PSNR characterization against alternate direct (non-sparse) measurement ground truths (that have to exist, or else the reconstruction method should be called into question), and the bit rate for each particular sparse measurement. So this way people can see how reliable the reconstruction is. Ideally the image should be labeled at the pixel level with reconstruction probabilities, or presented in other ways to demonstrate the ratio of measured vs. fabricated information, like 95% confidence-interval extremal reconstructions or something.

It's not clear that community is doing this level of due diligence, so then the voices here are right: it's not a good idea to use.