What Is Adaptive Compressive Sampling?
In "ordinary" compressive sensing you take measurements $y = Ax$ and try to reconstruct $x$ from $y$ where $A$ - of dimension $(M \times N)$ - has been generated with no knowledge of x, except that it is sparse. $A$ is typically generated randomly.

In adaptive compressed sensing, you in principle take each individual measurement $y_m,\ m = 0, 1,\ldots M$, trying to adapt you next measurement $y_m$ cleverly so you get the best possible measurement for reconstruction of $x$, given what you have already observed in $y_0, y_1,\ldots, y_{m-1}$. Effectively, this means determining the rows $A_m$ of $A$ one at a time, taking into account the measurements that previous rows $A_0, A_1,\ldots, A_{m-1}$ gave you.

One example of adaptive compressive sensing could be Malloy &amp; Nowak, 2013.