Sometimes you CAN see through the wall — if you know the picture is simple. A signal with only a couple of nonzero pieces (a ‘sparse’ one) can be recovered EXACTLY from far fewer measurements than it has slots, by insisting on the sparsest explanation. That is compressed sensing, the math behind fast MRI and single-pixel cameras. Slide the number of measurements and watch the two hidden spikes snap back — once you have enough.
An n=8 signal with only k=2 nonzero entries (a 2-sparse vector) is sampled by m linear measurements y = Ax. When m < n the system is underdetermined — infinitely many x fit — but among them only one is 2-sparse, and it can be found by seeking the sparsest solution (here: least-squares over every 2-column support, pick the best fit). A fail-loud self-check throws unless, at m=4 ≥ 2k, the recovery returns the exact support {2,5} with coefficients (3, −2). Below ~2k the answer is ambiguous; at and above it the spikes come back perfectly. The wall is passable — but only because you supplied the prior ‘it is sparse.’ ◆ real compressed sensing, node-verified.
A brute-force best-support solve on a tiny deterministic system (the honest core of L1 / orthogonal matching pursuit); real reconstructions use convex L1 minimisation and need incoherent/random measurements. The exactness holds because the prior (2-sparse) is TRUE — give it a wrong prior and it fails, honestly.