Since compressive sampling (CS) is all about signal acquisition, its faithful recoverability of sensed signals is of paramount importance. Thus, "exact recovery" is a good thing to measure when it comes time for experimental work. However, we find in the CS literature at least three definitions of "exact recovery":

- When the signal support is recovered with no false alarms, and no missed detections;
- When the normalized squared model error is less than \(\epsilon^2\).
- When the largest magnitude difference in the model error is less than \(\epsilon\).

In my paper "When 'exact recovery' is exact recovery in compressive sampling", I analyze the first and second "exact recovery" criteria, and show when they are equivalent, how to interpret \(\epsilon^2\), and an appropriate range over which to define it.
In short,

- \(\epsilon^2\) sets the maximum acceptable false detection rate;
- In the noiseless case, \(\epsilon^2 < 1/s\) for the two conditions to be equivalent for \(s\)-sparse signals a majority of the time, independent of how the sparse signal is distributed;
- In the noisy case, with measurement noise of variance \(\sigma_v^2 > 0\), the parameter \(\epsilon^2 \ge \sigma_v^2/s\) so that the second condition can even be met
- In the noisy case, with measurement noise of variance \(\sigma_v^2\), \(\epsilon^2 \le (k/s) + \sigma_v^2(1 - k/s)\) for the two conditions to be equivalent for \(s\)-sparse signals a majority of the time, independent of how the sparse signal is distributed.

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