Restricted Isometry Property (RIP) in Compressive Sensing
The restricted isometry property states that:
$$\begin{equation}
  (1-\delta_S)||x||_2^2 \le ||A x||_2^2 \le (1+\delta_S)||x||_2^2
\end{equation}$$
for any $S$-sparse vector $x$. The restricted isometry constant is $\delta_S$, $0 &lt; \delta_S &lt; 1$.

This means that the matrix $A$ is guaranteed to only change the length of any vector $x$ "very little" as long as the vector $x$ is at least $S$-sparse (has at most $S$ non-zero coefficients).

Suppose we have arbitrary $\frac{S}{2}$-sparse vectors $x$. In order to be able to reconstruct such vectors in general, from measurements taken as $y = A x$, we need to be sure that it is possible to distinguish between measurements $y_1 = A x_1$ and $y_2 = A x_2$ of any two such vectors. If $y_1 = y_2$ for any two such vectors $x_1$ and $x_2$, we would not be able to distinguish them and reconstruct them unambigously. Therefore we need to make sure that measurements of any two $\frac{S}{2}$-sparse vectors are "sufficiently different".

If we calculate the difference between any two $\frac{S}{2}$-sparse vectors, their difference can be at most $S$-sparse. So for reconstructing any $\frac{S}{2}$-sparse vector correctly from measurements taken with $A$, the restricted isometry property quantifies how well $A$ lets us do that (the smaller $\delta_S$, the better).

For an early introduction to compressed sensing and the restricted isometry property (and other concepts), see Candès &amp; Wakin, 2008.