There has been an explosion of non-invasive biomedical sensing modalities that have revolutionized our ability to probe the biomedical world. Often decisions have to be made on the basis of these increasingly high-dimensional observations. An example would be the determination of cancer or stroke from indirect tomographic projection measurements. The problem is frequently exacerbated by the lack of labeled training samples from which to learn class models. In many cases, however, there exists a latent low-dimensional sensing structure that can potentially be exploited for inferencing aims. This work investigates the impact of latent sensing structure on supervised classification performance when the data dimension scales to infinity faster than the number of samples. In contrast to some existing studies, here the classification difficulty is held fixed and finite as the data dimension scales. For a binary supervised classification problem with Gaussian likelihood functions, it is shown that the asymptotic error probability converges to that of pure guessing if the sensing structure is totally ignored, whereas it converges to the Bayes risk if the sensing structure is sufficiently regular and the classification method is "sensing aware". It is also shown, however, that without suitable regularity in the latent low-dimensional sensing structure, it is impossible to attain nontrivial asymptotic error probability. These findings are validated through various simulations. Additional numerical results for support vector machines and sensitivity to mismatch between true and assumed structure are also provided.